A variational principle for problems in Partial differential equations

and Analysis ∗

Abbas Moameni †

Abstract

The aim of this paper is to provide a comprehensive variational principle that allows one toapply critical point theory on closed proper subsets of a given Banach space and yet, to obtaincritical points with respect to the whole space. This variational principle has many applicationsin partial differential equations while unifies and generalizes several results in nonlinear Analysissuch as the fixed point theory, critical point theory on convex sets and the principle of symmetriccriticality. As a consequence, several substantial new results are established. We shall alsoprovide concrete applications in non-linear elliptic and parabolic partial differential equations,including De Giorgi’s conjecture on bounded domains, for which the standard methodologieshave major limitations to be applied.

2010 Mathematics Subject Classification: 35J87, 58E30, 49J40.Key words: Variational principles, Calculus of Variations.

Contents

1 Introduction 2

2 Preliminaries and notations 5

3 Variational Principles 12

4 Applications in differential equations 214.1 A non-local problem with a concave-convex nonlinearity . . . . . . . . . . . . . . . . 214.2 De Georgi’s conjecture on bounded doamins . . . . . . . . . . . . . . . . . . . . . . . 224.3 Multiplicity and sub-super solutions on unbounded domains . . . . . . . . . . . . . . 264.4 Super critical Neumann problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 A nonsmooth principle of symmetric criticality 32

6 Critical points of locally Lipschitz functions on convex sets 36

7 Applications to the fixed point theory and hemi-variational inequalities 39

8 Evolution equations and non-variational operators 428.1 Semi-linear heat equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

∗The author is pleased to acknowledge the support of the National Sciences and Engineering Research Council ofCanada.†School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada, momeni@math.carleton.ca

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1 Introduction

In a wide range of mathematical problems the existence of a solution is equivalent to the existence ofa fixed point for a suitable map or a critical point for an appropriate variational or hemi-variationalproblem. In particular, in real life applications we are interested in finding such solutions whichpossess certain properties. In this paper we provide a comprehensive variational principle thatallows one to apply critical point theory on closed proper subsets of a given Banach space and yet,to obtain critical points with respect to the whole space. This principle together with the standardresults in the calculus of variations provide an efficient and applicable theory to deal with problemsthat do not usually fit within the standard criterion.

Let V be a real Banach space and V ∗ its topological dual and let 〈., .〉 be the pairing betweenV and V ∗. Let Ψ : V → R∪ {+∞} be a proper convex and lower semi-continuous function and letK be a convex and weakly closed subset of V. Assume that Ψ is Gâteaux differentiable on K anddenote by DΨ the Gâteaux derivative of Ψ. Let Φ ∈ C1(V,R) and consider the following problem:

Find u0 ∈ K such that DΨ(u0) = DΦ(u0). (1.1)

The restriction of Ψ to K is denoted by ΨK and defined by

ΨK(u) =

{Ψ(u), u ∈ K,+∞, u 6∈ K.

To find a solution for (1.1), we shall consider the critical points of the functional I : V → R∪{+∞}defined by

I(u) := ΨK(u)− Φ(u).

There is a rich theory in non-linear analysis that deals with the existence and multiplicity resultsof critical points for functionals of the type I = ΨK −Φ defined on real Banach spaces (see Section2). However, a critical point of I is a solution of the inclusion DΦ(u) ∈ ∂ΨK(u) and not necessarilya solution of (1.1) unless K = V. Here, ∂ΨK stands for the subdifferential of the convex functionΨK by means of the theory of convex analysis. One of the main objectives in this paper is toprovide a practical sufficient condition so that the critical points of I are in fact solutions of theoriginal problem (1.1). As a consequence of our results, we have the following two theorems thataddresses this issue.

Theorem 1.1. Let Ψ : V → R ∪ {+∞} be a proper convex and lower semi-continuous functionand let K be a closed and convex subset of V. Assume that Ψ is Gâteaux differentiable at each pointof K and Φ ∈ C1(V ;R). If the following two assertions hold:

i) The functional I : V → R ∪ {+∞} defined as

I(u) = ΨK(u)− Φ(u),

has a critical point at u0, and;

ii) there exists v0 ∈ K such that DΨ(v0) = DΦ(u0),

then u0 is a solution of the problem

DΨ(u0) = DΦ(u0).

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Note that critical points of the functional I = ΨK − Φ are not necessarily minimas. In fact, inpractice, these critical points are obtained by making use of the classical mountain pass or linkingtype minimax theorems for lower semi-continuous functionals. One can replace the convexityassumption on the set K with some assumptions on the functionals Ψ and Φ.

Theorem 1.2. Let Ψ : V → R ∪ {+∞} and Φ : V → R ∪ {+∞} be a proper convex and lowersemi continuous functions. Let K be a closed subset of V and assume that Ψ and Φ are Gâteauxdifferentiable on K. If the following two assertions hold:

i) The functional I : V → R ∪ {+∞} defined by

I(u) = Ψ(u)− Φ(u),

is bounded below on K, and there exists u0 ∈ K with I(u0) = infu∈K I(u);

ii) there exists v0 ∈ K such that DΨ(v0) = DΦ(u0).

then u0 is a solution ofDΨ(u0) = DΦ(u0).

The following remark reveals the flexibility of the sufficient condition given in the above theo-rems.

Remark 1.3. Condition ii) in both Theorems 1.1 and 1.2 can be relaxed and replaced with eitherof the following conditions:

ii-a) there exist v0 ∈ K, and a convex lower semi-continuous Gâteaux differentiable function G :V → R such that

DΦ(u0) +DG(u0) = DΨ(v0) +DG(v0).

ii-b) there exist v0 ∈ K, and a convex lower semi-continuous Gâteaux differentiable function G :V → R with DG(0) = 0 such that

DΦ(u0) = DΨ(v0) +DG(v0 − u0).

Observe that condition ii) in both Theorems 1.1 and 1.2 follows by assuming G = 0 in either ii-a)or ii-b).In general, we say that the triple (Φ,K,Ψ) satisfies the point-wise invariance condition at u0 ∈ K,if either ii − a) or ii − b) holds at u0 ∈ K. The notion of the the point-wise invariance conditionin full generality is stated in Definition 3.5.

In summary, to solve a problem of type (1.1), we first find a critical point say u0 of the functionI = ΨK −Φ using standard methods in the calculus of variations. Then, to show that this criticalpoint u0 ∈ K satisfies the equation DΨ(u0) = DΦ(u0), we shall need to verify that the triple(Φ,K,Ψ) satisfies the point-wise invariance condition at u0. Note that the point-wise invariancecondition is extremely practical as shown by several applications throughout the paper.

The above results are paving the way to do critical point theory on subsets that are notnecessary a linear space. This allows one to deal with a much wider range of problems in partialdifferential equations and non-linear analysis. Indeed, there are frequent situations that for agiven differential equation the corresponding Euler-Lagrange function is not even well-defined onan appropriate space and therefore one can not make use of standard theories in the calculus of

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variations to find a possible critical point. This happens, for instance, if you consider a semi-linearelliptic differential equation so that the non-linear term is super-critical by means of Sobolev spaces.

We shall now address the case when a given problem does not have a full variational structure.To be precise, assume that F : Dom(F) ⊂ V → V ∗ is a general map and consider the followingproblem,

Find u ∈ K such that DΨ(u) = F(u). (1.2)

To find a solution for (1.2), we shall consider it in association with the following hemi-variationalinequality problem:

Find u ∈ K such that: Ψ(v)−Ψ(u) ≥ 〈F(u), v − u〉, ∀v ∈ K. (1.3)

Similar to the case when a problem has a full variational structure, there is a comprehensive theoryon hemi-variational inequalities that provides existence results for problems of the type (1.3). It isremarkable that most equations in partial differential equations can be modeled and set-up in theform (1.3) when the set K is the whole space.

Interesting examples include gradient flows and evolutionary Navier Stokes. However, there is aslim chance for existence if we consider problem (1.3) in the whole space, i.e., K = V. On the otherhand, for smaller subsets K ⊂ V , the solution coming from the inequality (1.3) does not actuallysolve the original problem DΨ(u) = F(u).

As a consequence of our results, we have the following sufficient conditions assuring that undera very manageable and practical hypothesis each solution of (1.3) is indeed a solution of (1.2).

Theorem 1.4. Let Ψ : V → R ∪ {+∞} be a proper convex and lower semi continuous functionand F : Dom(F) ⊂ V → V ∗ be a map. Let K be a convex and weakly closed subset of V such thatΨ is Gâteaux differentiable at each point of K.Assume that the following two assertions hold:

i) there exist u0 ∈ K such that

Ψ(v)−Ψ(u0) ≥ 〈F(u0), v − u0〉, ∀v ∈ K;

ii) and if u0 belongs to the boundary of K then at least one of the following conditions holds,

ii-a) there exist v0 ∈ K, and a convex lower semi-continuous (l.s.c.) Gâteaux differentiablefunction G : V → R such that F(u0) +DG(u0) = DΨ(v0) +DG(v0);

ii-b) there exist v0 ∈ K, and a convex l.s.c. Gâteaux differentiable function G : V → R withDG(0) = 0 such that F(u0) = DΨ(v0) +DG(v0 − u0).

Then u0 is a solution of (1.2), that is,

DΨ(u0) = F(u0).

We shall state and prove our main variational principles in Section 3. There are manypreliminary results and notations from the theory of non-smooth analysis needed throughout thepaper. For the convenience of the reader, we have gathered them all in Section 2.

In Section 4, we study several differential equations of elliptic type. We shall begin by provingthe existence of a positive solution for the super-critical fractional Laplacian with a convex-concave

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non-linearity (−∆)su = u|u|p−2 +µu|u|q−2 with q < 2 < p. We assume no growth assumption on p.In our second example, we consider the classical conjecture of De Giorgi stating that any boundedsolution of the equation ∆u + (1 − u2)u = 0 in Rn with ∂xnu > 0 must be one dimensional. Weshall show that this conjecture may not hold in bounded domains. In our forthcoming paper, weshall extend this result to unbounded domains.Then we turn our attention to the classical sub-super solution method. It is known that for a widerange of elliptic problems, in the presence of a sub-solution and a super-solution, one can generatea solution between the sub and super ones. Using the arguments established in the current paperwe shall show that one can also expect multiplicity of solutions, between sub and super solutions,when the corresponding Euler-Lagrange functional is even.We conclude this section by proving the existence of one radially increasing solution for super-critical equations of the type −∆pu + |u|p−2u = |x||u|s−2u − |u|q−2u defined on a ball with aNeumann boundary condition by choosing an appropriate convex set K.

Section 5 deals with Palais’ Principle of Symmetric Criticality (PSC). This principle assertsthat, under certain conditions for a given group action, for any group-invariant Lagrangianthe equations obtained by restriction of Euler-Lagrange equations to group-invariant fields areequivalent to the Euler-Lagrange equations of a canonically defined, symmetry reduced Lagrangian.As a straight forward application of our results, by first recovering the (PSC), we provide severalgeneralization to non-compact groups and non-smooth Lagrangians.

Section 6 addresses the old problem of “critical points on convex sets vs actual critical points”.This subject was initiated by J. Mawhin by making use of the notion of Schauder invariancecondition on convex sets. This condition is by no means necessary though. We shall establish anecessary and sufficient condition for this problem.

In Section 7, we shall associate a hemi-variational inequality to a given fixed point problem.Then, we establish a one to one correspondence between the two problems, one in the theory of fixedpoints and the other one from hemi-variational inequalities. This strong relation between the twoseemingly different theories provides an extraordinary tool to produce new results in either one byconsidering the corresponding problem in the other theory. As a consequence, we shall provide somenew fixed point theorems. There seem to be a lot of room for extension of our results in this section.

Section 8 is devoted to study problems lacking a variational structure like evolution equations.We shall prove two new abstract existence results in line with our main results in this paper. Then,we provide some concrete applications in semi-linear Heat equations for which we look for solutionswith specific properties based on the nature of our problem.

2 Preliminaries and notations

In this section we recall some important notations and results in non-smooth Analysis required inthe sequel.

Convex Analysis

Let V be a locally convex space (l.c.s.) and V ∗ its topological dual and let 〈., .〉 be the pairingbetween V and V ∗. The weak topology on V induced by 〈., .〉 is denoted by σ(V, V ∗). A function

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Ψ : V → R is said to be weakly lower semi-continuous if

Ψ(u) ≤ lim infn→∞

Ψ(un),

for each u ∈ V and any sequence un approaching u in the weak topology σ(V, V ∗). Let Ψ : V →R ∪ {∞} be a proper convex function. We say that Ψ is proper if

Dom(Ψ) := {v ∈ V ; Ψ(v)

Proof. Since u is not a boundary point of K then for each v ∈ V we have that tv+(1− t)u ∈ Kfor some small t > 0. Thus, by (2.1) we have that

Ψ(tv + (1− t)u

)−Ψ(u) ≥ 〈u∗, tv + (1− t)u− u〉.

It now follows from the convexity of Ψ that

tΨ(v) + (1− t)Ψ(u)−Ψ(u) ≥ t〈u∗, v − u〉.

Simplifying the left hand side of the latter inequality and diving both sides by t > 0 imply that

Ψ(v)−Ψ(u) ≥ 〈u∗, v − u〉.

Since v ∈ V is arbitrary, we obtain that

Ψ(v)−Ψ(u) ≥ 〈u∗, v − u〉, ∀v ∈ V.

This in fact implies that u∗ ∈ ∂Ψ(u) as claimed. �

Proposition 2.2. Let Ψ1 and Ψ2 be convex functions and suppose there is a point in Dom(Ψ1) ∩Dom(Ψ2) at which Ψ1 is continuous. Then

∂(Ψ1 + Ψ2) = ∂Ψ1 + ∂Ψ2.

Finally, here is the definition for the sum of two multi-valued maps used frequently throughoutthe paper.

Definition 2.3. If F1 : V → 2V∗

and F2 : V → 2V∗

are two multi-valued maps then for eachu, v ∈ V such that F1(u) 6= ∅ and F2(v) 6= ∅ we define

F1(u) + F2(v) = {x+ y; x ∈ F1(u) and y ∈ F2(v)}.

If either F1(u) = ∅ or F2(v) = ∅ then we conventionally define

F1(u) + F2(v) = ∅.

Duality mappings and re-norming in Banach spaces

We shall now collect some information about duality mappings on Banach spaces [58].

Definition 2.4. Let V be a real Banach space and V ∗ its topological dual. The normalized dualitymapping J : V → 2V ∗ is defined by

J(u) ={u∗ ∈ V ∗; 〈u∗, u〉 = ‖u‖2V , ‖u‖V = ‖u∗‖V ∗

}.

By a classical result by Asplund, the normalized duality mapping J : V → 2V ∗ can be character-ized via the subdifferential of a convex functions as follows

J(u) = ∂(‖u‖2V

2

).

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A Banach space (V, ‖.‖) is said to be strictly convex if

‖u‖ = ‖v‖ = 1, u 6= v =⇒ ‖u+ v2‖ < 1.

Also, V is said to be uniformly convex if for each � ∈ (0, 2], there exists δ > 0 such that

‖u‖ = ‖v‖ = 1, ‖u− v‖ > � =⇒ ‖u+ v2‖ < 1− δ.

Proposition 2.3. Let J : V → 2V ∗ be the normalized duallity mapping between a real Banachspace V and its dual V ∗. The following assertions hold:

1. J(u) 6= ∅ for all u ∈ V.

2. J is a monotone operator.

3. If V is strictly convex, then J is one to one, that is,

u 6= v =⇒ J(u) ∩ J(v) = ∅.

4. If V is reflexive, then J is a mapping of V onto V ∗.

The following re-norming result in Banach spaces is standard (see [20] for a proof).

Theorem 2.5. The following statements hold:

1. Any separable Banach space admits an equivalent norm that is strictly convex.

2. If V ∗ is separable then V admits an equivalent norm that is strictly convex and whose dual isalso strictly convex.

3. If V is a reflexive Banach space then V has an equivalent strictly convex norm for whichwhose dual is also strictly convex.

Sobolev spaces

Recall that for each p ≥ 1 and an open domain Ω ⊂ Rn, the Sobolev space W 1,p(Ω) consists offunctions in Lp(Ω) whose weak first order partial derivatives are in Lp(Ω), and it is equipped withthe norm

‖u‖W 1,p(Ω) = ‖∇u‖Lp(Ω) + ‖u‖Lp(Ω).

The Sobolev space W 1,p0 (Ω) consists of functions in W1,p(Ω) being zero on the boundary of Ω by

means of the trace ([27], Chapter 7). When p = 2, we set W 1,2(Ω) = H1(Ω) and W 1,20 (Ω) =H10 (Ω). We also have the continuous Sobolev embedding W

1,p(Ω) ⊂ Lq(Ω) for q ∈ [p, p∗] wherep∗ = pn/(n− p) is said to be the critical Sobolev exponent of W 1,p(Ω). The number q is said to besub-critical if p ≤ q < p∗, critical if q = p∗ and supercritical if q > p∗.

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Locally Lipschitz functions

We recall the following definitions from [14].

Definition 2.6. Let U be an open subset of V and Φ : U → R be locally Lipschitz.1. The generalized directional derivative of the function Φ at u ∈ U in the direction v ∈ V is definedby

Φ0(u; v) = lim supw→u,t↘0

Φ(w + tv)− Φ(w)t

.

2. The generalized derivative (also known as the Clarke subdifferential) ∂CΦ(u) of Φ at a pointu ∈ U is the subset of the dual space defined as follows

∂cΦ(u) = {u∗ ∈ V ∗; Φ0(u; v) ≥ 〈u∗, v〉, ∀v ∈ V }.

We would like to remark that if Φ : V → R is convex and continuous then ∂CΦ = ∂Φ. Here welist some important properties of the generalized derivative for locally Lipschitz functions.

Proposition 2.4. Let U be an open subset of V and Φ : U → R be locally Lipschitz. The followingassertions hold:

1. For every u ∈ U, the set ∂cΦ(u)is nonempty, convex and weakly*-compact subset of V ∗ whichis bounded by the Lipschitz constant K > 0 of Φ near u.

2. For every u ∈ U, Φ0(u; .) is the support function of ∂cΦ(u), that is,

Φ0(u; v) = max{〈u∗, v〉; u∗ ∈ ∂cΦ(u)}.

3. If Φ is continuously differentiable at u ∈ U, then ∂cΦ(u) = {Φ′(u)}.

4. For every u ∈ U, the function Φ0(u, .) : V → R is positively homogeneous and sub-additiveand therefore convex.

5. Φ0(.; .) : U × V → R is upper semi-continuous.

6. Φ0(u;−v) = (−Φ)0(u; v), ∀u ∈ U, ∀v ∈ V.

The interested reader is referred to ([14] or Chapter 1 in [48]) for the proof of the above propo-sition and more detailed discussion on the subject.

Non-smooth Minimax theory

In this section we collect some interesting results concerning the critical point theory for lowersemi-continuous functions due to Szulkin [57] and Motreanu-Panagiotopoulos [48]. Indeed, we aredealing with functions of the form I : V → (−∞,+∞] on a real Banach space V satisfying thefollowing property,

(H) I = Ψ − Φ, with Φ : V → R locally Lipschitz and Ψ : V → (−∞,+∞] proper, convex andlower semi-continuous.

Here is what we mean by a critical point for functions of the type (H). This is a generalization forthe notion of critical points for smooth functions.

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Definition 2.7. Let V be a real Banach space, Φ : V → R locally Lipschitz and Ψ : V → (−∞,+∞]be proper (i.e. Dom(Ψ) 6= ∅), convex and lower semi-continuous. A point u ∈ V is said to be acritical point of

I := Ψ− Φ (2.2)

if u ∈ Dom(Ψ) and if it satisfies the inequality

Ψ(v)−Ψ(u) ≥ Φ0(u; v − u), ∀v ∈ V. (2.3)

The Palais-Smale compactness condition for functions of the type (H) reads as follows.

Definition 2.8. We say that the functional I = Ψ − Φ, given in (2.2), satisfies the Palais-Smalecompactness condition (PS) at level c ∈ R if every sequence {un} ⊂ V such that

• I(un)→ c ∈ R,

• Ψ(v)−Ψ(un) ≥ Φ0(un; v − un)− �n‖v − un‖, ∀v ∈ V,

where �n → 0, possesses a convergent subsequence.

As one expects, minimums are always critical points.

Proposition 2.5. Let I : V → (−∞,+∞] be of the form (2.2). Then each local minimum of I isnecessarily a critical point of I.

Proof. Let u be a local minimum of I. Using the convexity of Ψ, it follows that for all smallt > 0,

0 ≤ I((1− t)u+ tv

)− I(u) = Φ

(u+ t(v − u)

)− Φ(u) + Ψ

((1− t)u+ tv

)−Ψ(u)

≤ Φ(u+ t(v − u)

)− Φ(u) + t (Ψ(v)−Ψ(u)) .

Dividing by t and letting t→ 0+ we obtain (2.3). �The following three theorems about the critical points of functions of the form (2.2) are provedin [48]. The first one deals with possible minimums and the second one addresses the case whenthe functional has a mountain pass geometry while the third one covers problems with a linkingstructure.

Theorem 2.9. Suppose that I : V → (−∞,+∞] is of the form (2.2) and satisfies the Palais-Smalecompactness condition (PS). If I is bounded from below then c = infu∈V I(u) is a critical value.

Theorem 2.10. (Mountain Pass Theorem). Suppose that I : V → (−∞,+∞] is of the form (2.2)and satisfies (PS) and the mountain pass geometry:

1. I(0) = 0.

2. there exist α > 0 and ρ > 0 such that for every u ∈ V with ‖u‖ = ρ one has I(u) ≥ α.

3. there exists e ∈ V with ‖e‖ > ρ such that I(e) ≤ 0.

Then I has a critical value c ≥ α which is characterized by

c = inff∈Γ

supt∈[0,1]

I[f(t)],

where Γ = {f ∈ C([0, 1], V ) : f(0) = 0, f(1) = e}.

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Theorem 2.11. (Generalized Mountain Pass Theorem). Assume that V = V1⊕V2 with dim(V1) <∞, and that I : V → (−∞,+∞] is of the form (2.2) and satisfies (PS) and the generalized mountainpass geometry:

1. there exist α > 0 and ρ > 0 such that I(u) ≥ α for every u ∈ V2 with ‖u‖ = ρ.

2. there exists a constant r > ρ and a point e ∈ V2 with ‖e‖ = 1 such that I(u) ≤ 0 for allu ∈ ∂Q, where

Q = {u ∈ V1 : ‖u‖ ≤ r} ⊕ {te : t ∈ [0, r]}.

Then I has a critical value c ≥ α which is characterized by

c = inff∈Γ

supx∈Q

I[f(x)],

where Γ = {f ∈ C(Q,V ) : f|∂Q = Id|∂Q}.

The next two theorems provides multiplicity information for even functionals of the form (2.2),(see [48, 57] for proofs).

Theorem 2.12. Let I : V → (−∞,+∞] be of the form (2.2) and I(0) = 0. Assume that thefunctions Ψ and Φ given in (2.2) are even and I satisfies (PS). Assume also that

cj := infA∈Γj

supu∈A

I(u) ∈ (−∞, 0), ∀j = 1, ..., k, (2.4)

where Γj denotes the closure with respect to the Hausdorff metric of the collection of all nonemptycompact symmetric subsets A of X for which 0 6∈ A and the (Krasnoselski) genus γ(A) is bigger orequal to j. Then I possesses at least k distinct pairs of symmetric nontrivial critical points (whosecorresponding critical values are the numbers cj).

Remark 2.13. As in the proof of the above theorem in [57] for Φ ∈ C1(V ), we would like toemphasize that for i, j ∈ {1, ..., k} with i 6= j, if the critical levels ci and cj given in (2.4) are equalthen the functional I has infinite number of critical points at the level ci = cj .

Theorem 2.14. Let I : V → (−∞,+∞] be of the form (2.2) with Ψ and Φ even. Assume that Isatisfies (PS), I(0) = 0 and the following conditions,

(i) there exists a subspace X1 of X of finite codimension and numbers α > 0, ρ > 0 such that

I(u) ≥ α whenever u ∈ X1, ‖u‖ = ρ; (2.5)

(ii) there exists a finite dimensional subspace X2 of X, dimX2 > codimX1, such that I(u)→ −∞as ‖u‖ → ∞, u ∈ X2.

Then I has at least dimX2 − codimX1 distinct pairs of symmetric nontrivial critical points. Inparticular, if (ii) is replaced by

(iii) there exists a k-dimensional subspace X2 as in (ii) for each positive integer k,

then I admits infinitely many distinct pairs (u,−u) of nontrivial critical points.

Besides references [48, 57], we refer the reader to the work of M. Struwe [56] on Ljusternik-Schnirelman theory on convex sets.

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3 Variational Principles

Let V be a real Banach space and V ∗ its topological dual and let 〈., .〉 be the duality pairing betweenV and V ∗. Let Ψ : V → R ∪ {∞} be a proper convex and lower semi-continuous function and letK be a convex and weakly closed subset of V. Assume that F : Dom(F) ⊆ V → 2V ∗ is a possiblemulti-valued operator and consider the following problem,

Find u ∈ K such that ∂Ψ(u) ∩ F(u) 6= ∅. (3.1)

To find a solution for (3.1), we shall consider it in association with the following hemi-variationalinequality problem,

Find u ∈ K and u∗ ∈ F(u) such that:Ψ(v)−Ψ(u) ≥ 〈u∗, v − u〉, ∀v ∈ K. (3.2)

Note that a solution of the problem (3.2) is not necessarily a solution of (3.1) unless K = V. IfF is a potential (roughly speaking, F is the derivative of a functional) then there is a rich theoryin non-linear Analysis that deals with the existence of critical points for functionals defined onBanach spaces that indeed generates solutions for problems of the type (3.2). Even, if F is not apotential there is well-developed theory of hemi-variational inequalities addressing existence theoryfor problems of the type (3.2). Our main objective in this section is to provide sufficient conditionsso that solutions of the problem (3.2) are indeed solutions of the original problem (3.1). Here isour first result that deals with the case where the operator F may not be a potential.

Theorem 3.1. Let Ψ : V → R ∪ {+∞} be a proper convex and lower semi-continuous functionand let K be a convex and weakly closed subset of V. Assume that F : Dom(F) ⊂ V → 2V ∗ is apossible multivalued operator and the following two assertions hold:

i) There exist u0 ∈ K and u∗0 ∈ F(u0) such that

Ψ(v)−Ψ(u0) ≥ 〈u∗0, v − u0〉, ∀v ∈ K;

ii) and if u0 belongs to the boundary of K then at least one of the following conditions holds,

ii− a) there exist v0 ∈ K and a convex function G : V → R which is Gâteaux differentiable atu0 such that u

∗0 +DG(u0) ∈ ∂Ψ(v0) +DG(v0).

ii− b) there exist v0 ∈ K and a convex function G : V → R with ∂G(0) = {0} such thatu∗0 ∈ ∂Ψ(v0) + ∂G(v0 − u0).

Then u0 is a solution of (3.1), that is,

u∗0 ∈ ∂Ψ(u0) ∩ F(u0) 6= ∅.

We shall need some preliminary results before proving Theorem 3.1.

Lemma 3.2. Let Ψ : V → R ∪ {+∞} be a proper convex function. Let u0, v0 ∈ Dom(Ψ) andassume that for some w∗ ∈ ∂Ψ(v0) the following holds:

Ψ(v0)−Ψ(u0) = 〈w∗, v0 − u0〉. (3.3)

Then w∗ ∈ ∂Ψ(u0). In particular, if Ψ is Gâteaux differentiable at u0 then w∗ = DΨ(u0).

12

Proof. It follows from w∗ ∈ ∂Ψ(v0) and Proposition 2.1 that

Ψ(v0) + Ψ∗(w∗) = 〈w∗, v0〉. (3.4)

It now follows from (3.3) and (3.4) that

〈u0, w∗〉 −Ψ(u0) = 〈v0, w∗〉 −Ψ(v0) = Ψ∗(w∗),

from which one obtainsΨ(u0) + Ψ

∗(w∗) = 〈w∗, u0〉.

This indeed implies that w∗ ∈ ∂Ψ(u0) by virtue of Proposition 2.1. If now Ψ is Gâteauxdifferentiable at u0 we have that ∂Ψ(u0) = {DΨ(u0)} and therefore w∗ = DΨ(u0). �

The following is a crucial property of convex functions. A proof is given in [44] and for theconvenience of the reader we shall also sketch the proof here.

Proposition 3.1. Let Ψ : V → R ∪ {+∞} be a proper convex function. Suppose Ψ is sub-differentiable at u, v ∈ V. If there exist u∗ ∈ ∂Ψ(u) and v∗ ∈ ∂Ψ(v) with

〈u∗ − v∗, u− v〉 = 0 (3.5)

then u∗, v∗ ∈ ∂Ψ(u) ∩ ∂Ψ(v). In particular, if Ψ is Gâteaux differentiable at u, v ∈ X and

〈DΨ(u)−DΨ(v), u− v〉 = 0

then DΨ(u) = DΨ(v).

Proof. It follows from u∗ ∈ ∂Ψ(u) and v∗ ∈ ∂Ψ(v) that

Ψ(u) + Ψ∗(u∗) = 〈u∗, u〉,Ψ(v) + Ψ∗(v∗) = 〈v∗, v〉.

Adding up this equalities, we obtain

〈u∗, u〉+ 〈v∗, v〉 = Ψ(u) + Ψ∗(u∗) + Ψ(v) + Ψ∗(v∗)

It also follows from (3.5) that 〈u∗, u〉 + 〈v∗, v〉 = 〈u∗, v〉 + 〈v∗, u〉, which together with the aboveequation imply that

〈u∗, v〉+ 〈v∗, u〉 = Ψ(u) + Ψ∗(u∗) + Ψ(v) + Ψ∗(v∗)= Ψ(v) + Ψ∗(u∗) + Ψ(u) + Ψ∗(v∗)

and therefore

Ψ(v) + Ψ∗(u∗)− 〈u∗, v〉+ Ψ(u) + Ψ∗(v∗)− 〈v∗, u〉 = 0.

This together with the fact that

Ψ(v) + Ψ∗(u∗)− 〈u∗, v〉 ≥ 0, Ψ(u) + Ψ∗(v∗)− 〈v∗, u〉 ≥ 0,

imply that both terms are indeed zero,

Ψ(v) + Ψ∗(u∗)− 〈u∗, v〉 = 0,Ψ(u) + Ψ∗(v∗)− 〈v∗, u〉 = 0.

13

Thus, it follows from Proposition 2.1 that u∗ ∈ ∂Ψ(v) and v∗ ∈ ∂Ψ(u).

If now Ψ is Gâteaux differentiable at u, v ∈ X then ∂Ψ(u) = {DΨ(u)} and ∂Ψ(v) = {DΨ(v)}from which we obtain that DΨ(u) = DΨ(v). �

Proof of Theorem 3.1. By i), there exist u0 ∈ K and u∗0 ∈ F(u0) such that

Ψ(v)−Ψ(u0) ≥ 〈u∗0, v − u0〉, ∀v ∈ K. (3.6)

If u0 is not a boundary point of K then it follows from (3.6) and Lemma 2.2 that u∗0 ∈ ∂Ψ(u0) and

we are done. Let us now assume that u0 belongs to the boundary of the set K. We first prove thisunder the condition ii− a).

Let M = ∂G and note that M : V → 2V ∗ is a monotone operator with M(u0) = DG(u0). Itnow follows from u∗0 +DG(u0) ∈ ∂Ψ(v0) + ∂G(v0) that there exists v∗0 ∈M(v0) satysfying

u∗0 +M(u0)− v∗0 ∈ ∂Ψ(v0), (3.7)

from which together with the convexity of Ψ we obtain,

Ψ(u0)−Ψ(v0) ≥〈u∗0 +M(u0)− v∗0, u0 − v0

〉= 〈u∗0, u0 − v0〉+

〈M(u0)− v∗0, u0 − v0

〉, (3.8)

Since M is a monotone operator we have that

〈M(u0)− v∗0, u0 − v0〉 ≥ 0,

and therefore,Ψ(u0)−Ψ(v0) ≥ 〈u∗0, u0 − v0〉. (3.9)

It also follows from condition i) in the statement by assuming v = v0 that

Ψ(v0)−Ψ(u0) ≥ 〈u∗0, v0 − u0〉. (3.10)

Combining (3.9) and (3.10) implies that

Ψ(v0)−Ψ(u0) = 〈u∗0, v0 − u0〉. (3.11)

It follows from inequality (3.8) and equality and (3.17) that〈M(u0)− v∗0, u0 − v0

〉≤ 0.

This together with the monotonicity of M yields that〈M(u0)− v∗0, u0 − v0

〉= 0.

Thus, by Proposition 3.1 we obtain that

M(u0) = v∗0. (3.12)

This together with (3.7) imply that u∗0 ∈ ∂Ψ(v0), from which together with (3.11) and Lemma 3.2we obtain that u∗0 ∈ ∂Ψ(u0). Therefore

u∗0 ∈ ∂Ψ(u0) ∩ F(u0),

14

as desired.

We shall now prove the theorem under the condition ii − b). Similar to the previous case, letM = ∂G and note that M : V → 2V ∗ is possibly a multi-valued monotone operator. It now followsfrom u∗0 ∈ ∂Ψ(v0) + ∂G(v0 − u0) that there exists w∗ ∈ M(v0 − u0) such that u∗0 − w∗ ∈ ∂Ψ(v0).By the convexity of Ψ we have that

Ψ(u0)−Ψ(v0) ≥ 〈u∗0 − w∗, u0 − v0〉. (3.13)

Since M(0) = 0, it follows from (3.13) and the monotonicity of M that

Ψ(u0)−Ψ(v0) ≥ 〈u∗0, u0 − v0〉+ 〈w∗, v0 − u0〉= 〈u∗0, u0 − v0〉+

〈w∗ −M(0), v0 − u0 − 0

〉(3.14)

≥ 〈u∗0, u0 − v0〉.

Thus,

Ψ(u0)−Ψ(v0) ≥ 〈u∗0, u0 − v0〉. (3.15)

It also follows from condition i) in the statement by assuming v = v0 that

Ψ(v0)−Ψ(u0) ≥ 〈u∗0, v0 − u0〉. (3.16)

Taking into account inequalities (3.15) and (3.16), one has that

Ψ(v0)−Ψ(u0) = 〈u∗0, v0 − u0〉. (3.17)

This together with (3.14) imply that〈w∗ −M(0), v0 − u0 − 0

〉≤ 0,

from which and the monotonicity of M we obtain〈w∗ −M(0), v0 − u0 − 0

〉= 0.

Since M = ∂G, it follows from Proposition 3.1 that

w∗ ∈ ∂G(0) = M(0) = 0. (3.18)

Thus, we must have u∗0 ∈ ∂Ψ(v0) as u∗0 − w∗ ∈ ∂Ψ(v0). Therefore, by combining u∗0 ∈ ∂Ψ(v0) and(3.17) we obtain that u∗0 ∈ ∂Ψ(u0) by virtue of Proposition 3.2. Therefore,

u∗0 ∈ ∂Ψ(u0) ∩ F(u0).

This completes the proof. �

We would like to remark that both conditions ii− a) and ii− b) can be relaxed in Theorem 3.1.Indeed, as seen in the proof of this Theorem, we have not completely made use of the convexityof the function G except possible use of some properties of the monotone operator M = ∂G. Thisobservation brings us to the following definition.

15

Definition 3.3. An operator M : V → 2V ∗ is said to be paramonotone if M is monotone and foru∗ ∈M(u), v∗ ∈M(v), the identity

〈v∗ − u∗, v − u〉 = 0,

implies that u∗, v∗ ∈M(u) ∩M(v).

Note that by Proposition 2, ifG is a convex function then the operatorM = ∂G is paramonotone.Thus, we have the following result.

Corollary 3.4. Conditions ii − a) and ii − b) in Theorem 3.1 can be replaced by the followingconditions:

ii− a′) there exist v0 ∈ K and a paramonotone operator M : V → 2V∗

with M(u0) being asingleton such that u∗0 +M(u0) ∈ ∂Ψ(v0) +M(v0).

ii− b′) there exist v0 ∈ K, and a paramonotone operator M : V → 2V∗

with M(0) = {0} suchthat u∗0 ∈ ∂Ψ(v0) +M(v0 − u0).

Proof. The proof goes in the same lines as in the proof of Theorem 3.1. The only differenceare the relations (3.12) and (3.18) that follow from the paramonocity of the operator M and notnecessarily from Proposition 3.1. �

In many applications the multi-valued operator F : Dom(F) ⊂ V → 2V ∗ in Theorem 3.1 isthe generalized derivative of a locally Lipschitz function. Recall that as in Definition 2.6, thegeneralized derivative of a locally Lipschitz function Φ : V → R is denoted by ∂cΦ. This togetherwith Corollary 3.4 are the motivations for the following definition.

Definition 3.5. Let V be a Banach space and K be a closed subset of V . Let Ψ : V → R ∪ {+∞}be a proper convex and lower semi continuous function and assume that Φ : V → R is locallyLipschitz.

1. Say that the triple (Φ,K,Ψ) satisfies the point-wise invariance condition of type (I) at u0 ∈ K,if there exist v0 ∈ K and a paramonotone operator M : V → 2V

∗with M(u0) being a singleton

such that {∂cΦ(u0) +M(u0)

}∩{∂Ψ(v0) +M(v0)

}6= ∅.

2. Say that the triple (Φ,K,Ψ) satisfies the point-wise invariance condition of type (II) at u0 ∈ K,if there exist v0 ∈ K and a paramonotone operator M : V → 2V

∗with M(0) = {0} such that

{∂cΦ(u0)

}∩{∂Ψ(v0) +M(v0 − u0)

}6= ∅.

3. In general, say that the triple (Φ,K,Ψ) satisfies the point-wise invariance condition at u0 ∈ K,if it satisfies the point-wise invariance condition of either type (I) or type (II) at u0 ∈ K.

Definition 3.6. Let Ψ : V → R ∪ {+∞} be a proper convex function and K be a subset of V. Therestriction of Ψ to K is denoted by ΨK and defined by

ΨK(u) =

{Ψ(u), u ∈ K,+∞, u 6∈ K.

Here is a direct application of Theorem 3.1 and Corollary 3.4 when F : Dom(F) ⊂ V → 2V ∗ isreplaced by the generalized derivative of a locally Lipschitz function.

16

Theorem 3.7. Let Ψ : V → R∪{+∞} be a proper convex and lower semi continuous function andlet K be a closed convex subset of V. Assume that Φ : V → R is locally Lipschitz. Let the followingtwo assertions hold:

i) The functional I : V → R ∪ {+∞} defined as

I(u) = ΨK(u)− Φ(u),

has a critical point at u0 ∈ K as in Definition 2.7 and;

ii) if u0 belongs to the boundary of the set K then the triple (Φ,K,Ψ) satisfies the point-wiseinvariance condition at u0.

Then u0 is a solution of the problem

∂Ψ(u0) ∩ ∂cΦ(u0) 6= ∅.

Proof. By i), the function I has a critical point at u0. Thus, it follows from Definition 2.8 thatu0 ∈ Dom(ΨK) and

ΨK(v)−ΨK(u0) ≥ Φ0(u0; v − u0), ∀v ∈ V.

It then follows thatΨ(v)−Ψ(u0) ≥ Φ0(u0; v − u0), ∀v ∈ K. (3.19)

If u0 is not a boundary point of K the we are done. Indeed, if u0 is not a boundary point then foreach v ∈ V we have that tv + (1− t)u0 ∈ K for t > 0 small. Thus, by (3.19) we have that

Ψ(tv + (1− t)u0)−Ψ(u0) ≥ Φ0(u0; tv + (1− t)u0 − u0).

It now follows from the convexity of Ψ and the fact that Φ0(u0; .) is homogeneous (Proposition2.4),

tΨ(v)− tΨ(u0) ≥ tΦ0(u0; v − u0).

Dividing both sides by t > 0 imply that

Ψ(v)−Ψ(u0) ≥ Φ0(u0; v − u0).

Since v ∈ V is arbitrary, we obtain that u0 is a critical point of I(u) = Ψ(u) − Φ(u) whichcompletes the proof for the case where u0 is not a boundary point of K.

We now assume that u0 is a boundary point of K. Define the operator F : Dom(F) ⊂ V → 2V∗

by F(u) = ∂cΦ(u) and note that F may be a multivalued operator. By assumption, the triple(Φ,K,Ψ) satisfies the point-wise invariance condition at u0.

Proof under the point-wise invariance condition at u0 of type (I): In this case we have theexistence of a v0 ∈ K, and a paramonotone operator M : V → 2V

∗such that{

∂cΦ(u0) +M(u0)}∩{∂Ψ(v0) +M(v0)

}6= ∅.

Let u∗0 ∈ ∂cΦ(u0) be such that u∗0 +M(u0) ∈ ∂Ψ(v0) +M(v0). It follows from Proposition 2.4 thatfor each v ∈ V,

Φ0(u0; v − u0) = max{〈u∗, v − u0〉 : u∗ ∈ ∂Φ(u0)} ≥ 〈u∗0, v − u0〉. (3.20)

17

Combining (3.19) and (3.20) yield that

Ψ(v)−Ψ(u0) ≥ 〈u∗0, v − u0〉, ∀v ∈ K. (3.21)

The result now follows from Corollary 3.4.

Proof under the point-wise invariance condition at u0 of type (II): In this case there exist v0 ∈ Kand a paramonotone operator M : V → 2V ∗ with M(0) = {0} such that{

∂cΦ(u0)}∩{∂Ψ(v0) +M(v0 − u0)

}6= ∅.

Let u∗0 ∈ ∂cΦ(u0) ∩{∂Ψ(v0) +M(v0 − u0)

}. Similar to the previous case we have that

Ψ(v)−Ψ(u0) ≥ 〈u∗0, v − u0〉, ∀v ∈ K. (3.22)

from which the result will be a direct consequence of Theorem 3.1. �

A very particular version of Theorem 3.7 is already announced by the author in [43]. Theinterested reader is also referred to [44, 45, 46] for further studies on the subject of variationalprinciple.

One can replace the convexity assumption on the set K with some assumption on the locallyLipschtiz functional Φ. Recall that if Φ : V → R is convex and continuous then ∂cΦ = ∂Φ.

Theorem 3.8. Let Ψ : V → R∪{+∞} be a proper convex and lower semi-continuous function andlet K be a closed subset of V. Assume that Φ : V → R is convex and continuous. Let the followingtwo assertions hold:

i) The functional I : V → R ∪ {+∞} defined by

I(u) = Ψ(u)− Φ(u),

is bounded below on K and there exists u0 ∈ K with I(u0) = infu∈K I(u);

ii) if u0 belongs to the boundary of the set K then the triple (Φ,K,Ψ) satisfies the point-wiseinvariance condition at u0.

Then u0 is a solution of∂Ψ(u0) ∩ ∂Φ(u0) 6= ∅.

Proof. First, we assume that u0 is not a boundary point of the set K. Let v ∈ V be arbitrary.It follows that tv + (1− t)u0 ∈ K for small t > 0. Thus,

Ψ(tv + (1− t)u0

)− Φ

(tv + (1− t)u0

)= I(tv + (1− t)u0

)≥ I(u0) = Ψ(u0)− Φ(u0)

Let u∗0 ∈ ∂Φ(u0). It follows that

Ψ(tv + (1− t)u0

)−Ψ(u0) ≥ Φ

(tv + (1− t)u0

)− Φ(u0) ≥ 〈u∗0, tv + (1− t)u0 − u0〉

It now follows from the convexity of Ψ that

tΨ(v)− tΨ(u∗0) ≥ t〈u∗0, v − u0〉.

Dividing both sides by t > 0, and taking into consideration v ∈ V is arbitrary we obtain thatu∗0 ∈ ∂Ψ(u0). Therefore, u∗0 ∈ ∂Ψ(u0) ∩ ∂Φ(u0) as desired.

18

Let us now assume that u0 is a boundary point of K. By assumption, the triple (Φ,K,Ψ) satisfiesthe point-wise invariance condition at u0.

Proof under the point-wise invariance condition at u0 of type (I): Let u∗0 ∈ ∂Φ(u0) be such that

u∗0 +M(u0) ∈ ∂Ψ(v0)+M(v0) for some paramonotone operator M : V → 2V∗

and for some v0 ∈ K.It follows that u∗0 +M(u0)− v∗0 ∈ ∂Ψ(v0) for some v∗0 ∈M(v0) and therefore,

Φ(u0) + Φ∗(u∗0) = 〈u∗0, u0〉,

Ψ(v0) + Ψ∗(u∗0 +M(u0)− v∗0) = 〈u∗0 +M(u0)− v∗0, v0〉. (3.23)

Since v0 ∈ K, it follows that

−∞ < I(u0) = infu∈K

I(u) ≤ I(v0), (3.24)

from which we obtainΨ(u0)− Φ(u0) ≤ Ψ(v0)− Φ(v0).

Substituting Φ(u0) and Ψ(v0) from (3.23) into the latter inequality yields that

Ψ(u0)− 〈u∗0, u0〉+ Φ∗(u∗0) ≤〈u∗0 +M(u0)− v∗0, v0

〉−Ψ∗

(u∗0 +M(u0)− v∗0

)− Φ(v0).

Rearranging the above inequality implies that

Ψ(u0) + Ψ∗(u∗0 +M(u0)− v∗0)− 〈u∗0, u0〉 − 〈M(u0)− v∗0, v0〉 ≤ 〈u∗0, v0〉 − Φ∗(u∗0)− Φ(v0). (3.25)

On the other hand by Proposition 2.1 we have that

Ψ(u0) + Ψ∗(u∗0 +M(u0)− v∗0) ≥ 〈u∗0 +M(u0)− v∗0, u0〉,

from which together with (3.25) imply that〈u∗0 +M(u0)− v∗0, u0

〉− 〈u∗0, u0〉 −

〈M(u0)− v∗0, v0

〉≤ 〈u∗0, v0〉 − Φ∗(u∗0)− Φ(v0).

Simplifying the left hand side of the latter inequality yields that〈M(u0)− v∗0, u0 − v0

〉≤ 〈u∗0, v0〉 − Φ∗(u∗0)− Φ(v0). (3.26)

Note that the left hand side of (3.26) is non-negative and the right hand side of (3.26) is non-positive.Therefore, 〈

M(u0)−M(v0), u0 − v0〉

= 〈u∗0, v0〉 − Φ∗(u∗0)− Φ(v0) = 0 (3.27)

It now follows from paramonocity of M that M(u0) = v∗0. This together with (3.25) and (3.27)

imply thatΨ∗(u∗0) + Ψ(u0)− 〈u∗0, u0〉 ≤ 0.

Since the latter inequality is always non-negative it must be zero and u∗0 ∈ ∂Ψ(u0). Therefore,

u∗0 ∈ ∂Ψ(u0) ∩ ∂Φ(u0),

and the proof is complete for this case.

19

Proof under the point-wise invariance condition at u0 of type (II): In this case there existu∗0 ∈ ∂Φ(u0) and w∗ ∈M(u0 − v0) such that u∗0 −w∗ ∈ ∂Ψ(u0). Thus, similar to the previous casewe have that

Φ(u0) + Φ∗(u∗0) = 〈u∗0, u0〉,

Ψ(v0) + Ψ∗(u∗0 − w∗) = 〈u∗0 − w∗, v0〉. (3.28)

Since v0 ∈ K, it follows thatI(u0) = inf

u∈KI(u) ≤ I(v0), (3.29)

from which we obtainΨ(u0)− Φ(u0) ≤ Ψ(v0)− Φ(v0).

Substituting Φ(u0) and Ψ(v0) from (3.28) into the latter inequality yields that

Ψ(u0)− 〈u∗0, u0〉+ Φ∗(u∗0) ≤ 〈u∗0 − w∗, v0〉 −Ψ∗(u∗0 − w∗)− Φ(v0).

Rearranging the above inequality implies that

Ψ(u0) + Ψ∗(u∗0 − w∗)− 〈u∗0, u0〉+ 〈w∗, v0〉 ≤ 〈u∗0, v0〉 − Φ∗(u∗0)− Φ(v0). (3.30)

On the other hand we have that

Ψ(u0) + Ψ∗(u∗0 − w∗) ≥ 〈u∗0 − w∗, u0〉,

from which together with (3.30) imply that

〈u∗0 − w∗, u0〉 − 〈u∗0, u0〉+ 〈w∗, v0〉 ≤ 〈u∗0, v0〉 − Φ∗(u∗0)− Φ(v0). (3.31)

Thus,−〈w∗, u0 − v0〉 ≤ 〈u∗0, v0〉 − Φ∗(u∗0)− Φ(v0). (3.32)

Note that the left hand side of (3.32) is non-negative as M(0) = 0 and

−〈w∗, u0 − v0〉 = 〈M(0)− w∗, u0 − v0〉 ≥ 0.

Also the right hand side of (3.32) is non-positive. Therefore,

〈w∗, u0 − v0〉 = 〈u∗0, v0〉 − Φ∗(u∗0)− Φ(v0) = 0 (3.33)

Since M(0) = 0 it follows from (3.33) that〈M(0)− w∗, u0 − v0

〉= 0.

It now follows from the paramonocity of M that w∗ = M(0) = 0. This together with (3.30) yieldthat

Ψ∗(u∗0) + Ψ(u0)− 〈u∗0, u0〉 ≤ 0,

from which we obtain that the latter inequality is indeed zero and u∗0 ∈ ∂Ψ(u0). Therefore,

u∗0 ∈ ∂Ψ(u0) ∩ ∂Φ(u0),

as desired. �

20

4 Applications in differential equations

In this section, by making use of our results in Section 3, we shall study several local and non-localproblems in partial differential equations.

4.1 A non-local problem with a concave-convex nonlinearity

We consider the concave-convex problem involving the fractional Laplacian operator{(−∆)su = u|u|p−2 + µu|u|q−2, in Ω,u = 0, on Rn \ Ω, (4.1)

where Ω is a bounded domain with continuous boundary in Rn, 1 < q < 2 < p , 0 < s < 1and 2s < n. For s = 1, this problem was studied by Ambrosetti and etc. in [1] and Bartsch andWillem in [5]. Here (−∆)s denotes the fractional Laplace operator, which is defined as the followingsingular integral

(−∆)su(x) = C(n, s)P.V.∫Rn

u(x)− u(y)|x− y|n+2s

dy,

for all x ∈ Rn, where P.V. is the principal value of the integral and

C(n, s) =

(∫Rn

1− cos(ξ1)|ξ|n+2s

dξ

)−1with ξ = (ξ1, ..., ξn) ∈ Rn. The space Hs(Ω) is the linear space of Lebesgue measurable functionsfrom Rn to R such that the restriction to Ω of any function u in Hs(Ω) belongs to L2(Ω), and themap

(x, y)→ u(x)− u(y)|x− y|

n2

+s,

is in L2(Q, dxdy) where Q := (Rn ×Rn) \ (Ωc ×Ωc). The norm in Hs(Ω) can be defined as follows

‖u‖Hs(Ω) = ‖u‖L2(Ω) +(∫

Q

|u(x)− u(y)|2

|x− y|n+2sdxdy

) 12. (4.2)

We also define

Hs0(Ω) = {u ∈ Hs(Ω) : u = 0 a.e. in Rn \ Ω}.

Consider the Banach space V = Hs0(Ω) ∩ Lp(Ω). Let I : V → R be the Euler-Lagrange functionalcorresponding to (4.1),

I(u) =1

2

∫Rn×Rn

|u(x)− u(y)|2

|x− y|n+2sdxdy − 1

p

∫Ω|u|pdx− µ

q

∫Ω|u|qdx.

For r > 0, define

K(r) := {u ∈ V : u ≥ 0, ‖u‖L∞(Ω) ≤ r}.

We have the following result.

21

Theorem 4.1. Let Ω ⊂ Rn be a bounded open domain with continuous boundary and assume thats ∈ (0, 1), 2s < n, 1 < q < 2 < p. Then there exists µ∗ > 0 such that for each µ ∈ (0, µ∗) problem(4.1) has at least one non-negative solution u ∈ V ∩ L∞(Ω) with a negative energy.

Proof. We apply Theorem 3.7, where Ψ : V → R and Φ : V → R are defined as follows,

Ψ(u) =1

2

∫Rn×Rn

|u(x)− u(y)|2

|x− y|n+2sdxdy, Φ(u) =

1

p

∫Ω|u|p dx+ µ

q

∫Ω|u|q dx,

and K := K(r) for some r > 0 to be determined. By standard methods, there exists u0 ∈ K(r)such that

I(u0) = minu∈K(r)

I(u).

Since 1 < q < 2 < p and µ > 0, it is easily seen that I(u0) < 0. Therefore, by Proposition 2.5,u0 6≡ 0 is a critical point of I restricted to K(r). To verify condition (ii) in Theorem 3.7, we shallshow that the triple (Φ,K,Ψ) satisfies the point-wise invariance condition of type (I) at u0 ∈ Kfor small values of r. To be more precise, we prove that that there exists v0 ∈ K(r) such that

(−∆)sv0 = |u0|p−2u0 + µ|u0|q−2u0.

The existence of such v0 follows by standard arguments. We show that v0 ∈ K(r) for r small. Itfollows from the regularity theory for the fractional lapacian that (see [[38], Theorem 16] or [[6],Lemma 2.5]) that

‖v0‖L∞(Ω) ≤ C∥∥u0|u0|p−2 + µu0|u0|q−2∥∥L∞(Ω),

where C is a constant depending on Ω. Therefore,

‖v0‖L∞(Ω) ≤ C∥∥u0|u0|p−2∥∥L∞(Ω) + µC∥∥u0|u0|q−2∥∥L∞(Ω) ≤ C(rp−1 + µrq−1)

Choose µ∗ > 0 small enough such that for each µ ∈ (0, µ∗), there exist positive numbers r1, r2 ∈ Rwith r1 < r2 such that C(r

p−1 + µrq−1) ≤ r for all r ∈ [r1, r2]. It then follows that v0 ∈ K(r)provided µ ∈ (0, µ∗) and r ∈ [r1, r2]. �

We refer the interested reader to [35, 36] for more details on this problem and where we havealso proved a multiplicity result for both s = 1 and 0 < s < 1 (see also [9, 55]).

4.2 De Georgi’s conjecture on bounded doamins

A celebrated conjecture due to De Giorgi [17] states that if u is a bounded solution of the equation∆u = u3 − u in Rn with ∂xnu > 0 then its level sets {u = λ} are all hyperplanes, at least fordimension n ≤ 8. Here is a version of the De-Georgi’s conjecture proved by Savin in [52].

Theorem 4.2. Let u ∈ C2(Rn) be a solution of

∆u = u3 − u, (4.3)

such that

|u| ≤ 1, ∂u∂xn

> 0, limxn→+∞

u(x′, xn) = 1, limxn→−∞

u(x′, xn) = −1. (4.4)

Then u is one-dimensional if n ≤ 8.

22

The De Giorgi’s conjecture has been fully established in dimensions n = 2 by Ghoussoub andGui [25], and for n = 3 by Ambrosio and Cabré [2]. Partial results in dimensions n = 4, 5 wereobtained by Ghoussoub and Gui [26]. A counter example for n ≥ 9 is obtained by Del Pino,Kowalczyk and Wei in [18]. We also refer to some older works in this direction by Jerison andMonneau [32] and by Cabré and Terra [12]. Our plan is to use Theorem 3.7 and provide a counterexample for this conjecture on bounded domains. In our forthcoming paper we shall extend thisresult to unbouded domains.Let α and β be two positive numbers and, let Ω = Ωα,β be a cylinder in R2n+1 defined by

Ω = Bn(0, α)×Bn(0, α)× (−β, β),

where Bn(0, α) is the ball in Rn with radius α centred at the origin. We shall consider the followingproblem,

∆u = u3 − u, x ∈ Ω, (4.5)

where u(x) = u(y, z) with y ∈ Bn(0, α)×Bn(0, α) ⊂ R2n and z ∈ (−β, β) ⊂ R satisfies

|u| ≤ 1, u(0) = 0, ∂u∂z

> 0 on Ω, and u(y, β) = −u(y,−β) = l, (4.6)

for some fixed l < 1. We observe that the function

f(z) = tanh( z√

2

),

is the unique solution of the one dimensional problem

f ′′ + (1− f2)f = 0; f(0) = 0; f(β) = −f(−β) = l, (4.7)

provided l = tanh( β√

2

). To simplify our computations, we shall fix this value for l and, we note

that l < 1. Therefore, problem (4.5)-(4.6) has always a unique one-dimensional solution f . Ourplan is to apply Theorem 3.8 and show that this problems has at least one more solution which isnot one-dimensional. Here we state our result.

Theorem 4.3. There exist α0 > 0 and β0 > 0 such that for all α ≥ α0 and β ≥ β0 problem(4.5)-(4.6) has a solution which is not one dimensional.

We shall need some preliminaries before proving our result. Let µ2 be the second eigenvalue ofthe Neumann problem {

−∆ξ + ξ = µ2ξ, y ∈ Bn(0, α),∂ξ∂ν = 0, y ∈ ∂Bn(0, α).

(4.8)

and ξ be the corresponding eigenfunction. Note that the first eigenvalue of the above problem isµ1 = 1, and therefore µ2 > µ1 = 1. We also observe ξ is radial

∫Bn(0,α)

ξ(ỹ) dỹ = 0. Indeed, ξ canbe chosen to be radially increasing.

Let ζ be the eigenfunction corresponding to the first eigenvalue of the problem{−ζ ′′(z) = λ1ζ(z), z ∈ (−β, β),ζ(β) = ζ(−β) = 0, (4.9)

Note that ζ is positive and even. The constants α0 and β0 in Theorem 4.3 will be chosen largeenough such that λ1 + µ2 < 2.

23

Proof of Theorem 4.3. We are interested in solutions u = u(y, z) defined on Ω having thefollowing symmetry conditions:

u(x) = u(y, z) = u(r, s, z) with r =

√√√√ n∑i=1

x2i and s =

√√√√ 2n∑i=n+1

x2i , (4.10)

where y = (x1, ..., xn, xn+1, ..., x2n) ∈ Bn(0, α)×Bn(0, α) and z ∈ (−β, β). Let V = H1(Ω)∩L4(Ω).Define Ψ : V → R and Φ : V → R by

Ψ(u) =1

2

∫Ω|∇u|2 dx+ 1

4

∫Ω|u|4 dx,

and

Φ(u) =1

2

∫Ω|u|2 dx.

We shall define the convex set K ⊂ V as follows,

K ={u ∈ V ; |u| ≤ 1, u(x) = u(y, z) = u(r, s, z), u(y, β) = −u(y,−β) = l,

∂u

∂z≥ 0, and u(r, s, z) = −u(s, r,−z) a.e. on Ω

}. (4.11)

It is easily seen that K is a convex and weakly closed subset of V . Set I = Ψ−Φ and assume that

I(u0) = minu∈K

I(u).

for some u0 ∈ K. We show that u0 satisfies (4.5) and (4.6). By Theorem 3.7, u0 is a solution of(4.5) if the triple (Ψ,K,Φ) satisfies the point-wise invariance condition at u0. To verify this, weshow that there exists v0 ∈ K satisfying the following equation,

−∆v + v3 = u0. (4.12)

Note that ∂Ω = Λ1 ∪ Λ2 ∪ Λ2 where

Λ1 = {x ∈ Ω̄; x2n+1 = β}, Λ2 = {x ∈ Ω̄; x2n+1 = −β},

and

Λ3 ={x ∈ Ω̄; r =

√√√√ n∑i=1

x2i = α or s =

√√√√ 2n∑i=n+1

x2i = α, |x2n+1| < β}.

Set,

W ={u ∈ V ; u = l on Λ1, and u = −l on Λ2

}.

Note that, there exists v0 ∈W such that

E(v0) = infv∈W

E(v)

where

E(v) =1

2

∫Ω|∇v|2 dx+ 1

4

∫Ω|v|4 dx−

∫Ωvu0 dx.

24

We shall now proceed in several steps.

Step 1. |v0| ≤ 1. Note that

−∆v0 + v30 = u ≤ 1 = −∆(1) + (1)3,

from which we have that−∆(v0 − 1) + (v30 − 13) ≤ 0.

Now multiplying both sides by (v0 − 1)+ = max{v0 − 1, 0} yields that∫Ω|∇(v0 − 1)+|2 dx+

∫Ω

(v30 − 13)(v0 − 1)+ dx ≤ 0.

Therefore, (v0 − 1)+ = 0 and v0 ≤ 1. By a similar argument we also have that v0 ≥ −1.

Step 2. Note that the functional E is invariant under the compact group actionG = O(n) × O(n) × {id} on R2n+1. Thus, by the principle of symmetric criticality, we havethat the unique minimizer v0 is invariant under the same group action. It then follows thatv0(x) = v0(y, z) = v0(r, s, z).

Step 3. We now show that v0(r, s, z) = −v0(s, r,−z). Define ṽ(r, s, z) = −v0(s, r,−z) and notethat E(v0) = E(ṽ). Since E is strictly convex, the minimizer of E must be unique. This impliesthat ṽ = v0 as desired.

Step 4. It follows from the Elliptic regularity that v0 ∈ H3(Ω). Thus, the function w = ∂v0∂z is asolution of the problem

−∆w + 3v30w = ∂u0∂z , x ∈ Ω,∂w∂ν = 0, x ∈ Λ3,w = 0, x ∈ Λ1 ∪ Λ2,

(4.13)

Indeed, since v0 minimizes E, we obtain that∂v0∂ν = 0 on Λ3 where ν = ν(x) is the outward normal

to Ω at x ∈ ∂Ω. On the other hand, v0 is constant on Λ1 and Λ2 and hence ∂v0∂ν =∂v0∂z = 0 on

Λ1∪Λ2. It now follows from the maximum principle that w ≥ 0 from which we obtain that ∂v0∂z ≥ 0on Ω.

It follows from Steps 1-4 that v0 ∈ K. Thus, v0 = u0 and the original function u0 ∈ K satisfies(4.5). To show that u0 satisfies (4.6) we just need to verify that

∂u0∂z > 0 on Ω. However, as done in

Step 4, the function w = ∂v0∂z =∂u0∂z is a solution of (4.13). Thus, by the strong maximum principle

either w > 0 on Ω or w = 0 on Ω̄. If w = 0 then it follows that u0 is a constant on the z direction,

u0(y, z1) = u0(y, z2), ∀z1, z2 ∈ [−β, β].

However, it contradicts the fact that u0(y, β) = −u0(y,−β) = l. It now remains to show that u0is not one-dimensional. If u0 is one-dimensional then u0 is just a function of z satisfying problem(4.7). Since the solution of (4.7) is unique we must have that

u0(r, s, z) = f(z) = tanh(z√2

).

For small δ > 0, define g : (−δ, δ)→ R by

g(t) = I(qt)− I(q0),

25

whereqt(x) = qt(y, z) = f(z) + tζ(z)ξ(x1, ..., xn)− tζ(z)ξ(xn+1, ..., x2n),

for y = (x1, ..., x2n) ∈ Bn(0, α)×Bn(0, α). Here ξ and ζ are defined in (4.8) and (4.9) respectively.Observe that

g(0) = I(q0)− I(q0) = 0.Note also that

g′(t) =

∫Ω∇(qt).∇

(ζ(z)η

)dx+

∫Ω

3q2t ζ(z)η dx−∫

Ωqtζ(z)η dx,

where η = ξ(x1, ..., xn)− ξ(xn+1, ..., x2n). Since∫Bn(0,α)

ξ(ỹ) dỹ = 0, we have that g′(0) = 0. On theother hand,

g′′(0) =

∫Ω|∇(ζ(z)η

)|2 + 6

∫Ωf(z)ζ2(z)η2 −

∫Ωζ2(z)η2

=

∫Q

∫ β−β

[|∇η|2ζ2(z) + η2(ζ ′(z))2 − η2ζ2(z)

]dzdy + 6

∫Qη2∫ β−βf(z)ζ2(z) dzdy

=

∫Qη2∫ β−β

(µ2 − 2)ζ2(z) + (ζ ′(z))2 dzdy,

where Q = Bn(0, α)×Bn(0, α). Thus, g′′(0) < 0 if and only if

µ2 − 2 < −∫ β−β(ζ

′(z))2 dz∫ β−β ζ

2(z) dz= −λ1.

Now, choose α0 and β0 large enough such that µ2 + λ1 < 2 for all α ≥ α0 and β ≥ β0. In this casewe obtain that g′′(0) < 0. Thus g′ is decreasing around zero and, for small t > 0, we have thatg′(t) < g′(0) = 0. This indeed implies that g(t) < g(0) = 0 for small t > 0 and therefore,

I(qt) < I(u0).

This contradicts the fact that I admits its minimum at u0 on K. �

4.3 Multiplicity and sub-super solutions on unbounded domains

We consider the problem {−∆u = f(x, u), x ∈ Ωu ∈ H10 (Ω)

(4.14)

where Ω is possibly an unbounded open domain in Rn. Our plan is to make use of Theorem 3.7 andprove the existence of at least one solution for the problem (4.14) in the presence of a weak suband a weak super solution. We shall also prove multiplicity results if the Euler-Lagrange functionalcorresponding to (4.14) is even and satisfies a certain mountain pass geometry.

Definition 4.4. Recall that u ∈ H10 (Ω) is called a weak sub-solution of (4.14) if f(., u) ∈ H−1(Ω)and ∫

Ω∇u.∇η dx ≤

∫Ωf(x, u)η dx, ∀ 0 ≤ η ∈ H10 (Ω),

and ū ∈ H10 (Ω) is called a weak super-solution of (4.14) if f(., ū) ∈ H−1(Ω) and∫Ω∇ū.∇η dx ≥

∫Ωf(x, ū)η dx, ∀ 0 ≤ η ∈ H10 (Ω).

26

Here H−1(Ω) is the topological dual of H10 (Ω). Let F (x, s) =∫ s

0 f(x, t) dt. Here is our result.

Theorem 4.5. Let Ω be an open domain in Rn and f : Ω × R → R be a Carathéodory function.Assume that u ∈ H10 (Ω) is a weak sub-solution and ū ∈ H10 (Ω) is a weak super-solution of (4.14)with u ≤ ū. Suppose that the following assertions hold:

1. There exists a Carathéodory function g : Ω × R → R such that for each x ∈ Ω the mapss→ F (x, s) +G(x, s) and s→ G(x, s) are convex, where G(x, s) =

∫ s0 g(x, t) dt.

2. F (x, u) +G(x, u) ∈ L1(Ω) and F (x, ū) +G(x, ū) ∈ L1(Ω).

3.

lim‖u‖→∞

12

∫Ω |∇u|

2 dx+∫

ΩG(x, u) dx

‖u‖= +∞.

4. f(x, u) + g(x, u) ∈ H−1(Ω) and f(x, ū) + g(x, ū) ∈ H−1(Ω).

Then there exists a solution u of (4.14) such that u ≤ u ≤ ū.

It is remarkable that we are not imposing and growth condition on the function f . Theorem4.5 is a new result even for the case when the domain Ω is bounded.

Proof of Theorem 4.5. Let

K ={u ∈ H10 (Ω); u(x) ≤ u(x) ≤ ū(x), a.e.x ∈ Ω

}.

Define,

E(u) =1

2

∫Ω|∇u|2 dx−

∫ΩF (x, u) dx.

We shall use Theorem 3.7 to prove our result. Set Φ(u) =∫

Ω

[F (x, u) +G(x, u)

]dx, and

Ψ(u) =1

2

∫Ω|∇u|2 dx+

∫ΩG(x, u) dx,

and note that E(u) = Ψ(u)−Φ(u). We shall apply Theorem 3.7 for function Ψ, Φ and the convexset K. First, we show that there exists u0 ∈ K such that E(u0) = infu∈K E(u).

Claim. For each u ∈ K,∫Ω

[F (x, u) +G(x, u)

]dx ≤

∫Ω|F (x, u) +G(x, u)| dx+

∫Ω|F (x, ū) +G(x, ū)| dx.∫

Ω

[F (x, u) +G(x, u)

]dx ≥

∫Ω

[F (x, u) +G(x, u)

]dx+

∫Ω

(f(x, u) + g(x, u)

)(u− u) dx.

Proof of the claim. There exists a function θ : Ω→ [0, 1] such that u(x) = θ(x)u+ (1− θ(x))ū fora.e. x ∈ Ω. It follows from the convexity of the maps s→ F (x, s) +G(x, s) that

F (x, u) +G(x, u) ≤ θ(x)(F (x, u) +G(x, u)) + (1− θ(x))(F (x, ū) +G(x, ū))≤ |F (x, u) +G(x, u)|+ |F (x, ū) +G(x, ū)|.

This proves the first assertion of the claim. The second one simply follows from the convexity ofthe map s→ F (x, s) +G(x, s).

27

It follows from the claim that∫ΩDΦ(u)(u− u) dx ≤ Φ(u) ≤ Φ(u) + Φ(ū).

It now follows from the coercivity of Ψ, Condition 3) in Theorem 4.5, and the above inequalities thatE is bounded below and coercive on K. Let un ⊂ K be a minimizing sequence of E over K. Thus, unmust be bounded in H10 (Ω). Hence, there exists u0 ∈ H10 (Ω) such that, up to a subsequence, un ⇀ u0weakly in H10 (Ω) and un(x) → u0(x) for a.e. x ∈ Ω. It then follows that u0 ∈ K. It also followsfrom the claim together with the dominated convergence theorem that limn→∞Φ(un) = Φ(u). Thistogether with lower semi-continuity of Ψ implies that E(u0) ≤ lim infn→∞E(un). Thus, we musthave E(u0) = infu∈K E(u).We now show that (Φ,K,Ψ) satisfies the point-wise invariance condition of type (I) at u0 ∈ Kfrom which the result follows from Theorem 3.7. We shall need to show that there exists v0 ∈ Ksatisfying {

−∆v + g(x, v) = f(x, u0) + g(x, u0), x ∈ Ωv ∈ H10 (Ω)

(4.15)

Since s→ f(x, s) + g(x, s) is monotone we have that

f(x, u) + g(x, u) ≤ f(x, u0) + g(x, u0) ≤ f(x, ū) + g(x, ū),

from which together with Condition 4) we obtain that f(x, u0) + g(x, u0) ∈ H−1(Ω). This indeedimplies that the problem (4.15) has a unique solution v0 ∈ H10 (Ω). We now show that v0 ∈ K. First,we prove that v ≤ ū. It follows that

−∆v + g(x, v) = f(x, u0) + g(x, u0) ≤ f(x, ū) + g(x, ū) ≤ −∆ū+ g(x, ū),

and therefore−∆v + g(x, v) ≤ −∆ū+ g(x, ū). (4.16)

Since v, ū ∈ H10 (Ω), it follows that (v − ū)+ = max{v − ū, 0} ∈ H10 (Ω). Multiplying both sides of(4.16) with (v − ū)+ and integration over Ω implies that∫

Ω∇(v − ū).∇(v − ū)+ dx ≤

∫Ω

(g(x, ū)− g(x, v)

)(v − ū)+ dx.

Since s → G(x, s) is convex, the right hand side of the above inequality is non-positive. Thus,we have that ∫

Ω|∇(v − ū)+|2 dx ≤ 0,

from which we have that v ≤ ū for a.e. x ∈ Ω as desired. By a similar argument we also have thatv ≥ u for a.e. x ∈ Ω. Thus, v ∈ K. This completes the proof. �

In most applications the function g in Theorem 4.5 can be chosen to be g(x, u) = Mu for someM ≥ 0. For instance, if for each x ∈ Ω the map s → f(x, s) is increasingly monotone then thefunction g(x, u) = u does the job. Another example is the case where the map s→ f(x, s) is locallyLipschitz, uniformally with respect to x, and u, ū ∈ L∞(Ω). In this case g(x, u) = Mu for somelarge M depending on the Lipschitz constants of the map s→ f(x, s) around u and ū.

28

Remark 4.6. It is worth noting that similar results to Theorem 4.5 also hold if we replace theLaplacian operator on the left hand side of the equation (4.14) by a possibly non-linear monotonedifferential operator Γ that satisfies the weak maximum principle, i.e.,

Γu ≤ Γv =⇒ u ≤ v, ∀u, v ∈ Dom(Γ).

As another application, we shall now provide a multiplicity result for the following problem.{−∆u = f(x, u) + λu, x ∈ Ωu ∈ H10 (Ω)

(4.17)

in the presence of a super-solution where Ω is a bounded domain in Rn . It is known that if fis sub-critical (i.e. f does not grow faster than p < 2n/(n − 2)) then we have several multiplicityresults for problem 4.17 (see for instanse [56, 59]). Here, we shall provide a multiplicity result inthe presence of a super-solution without imposing any growth condition on f. To do this, let usdenote by λj the j-th eigenvalue of −∆ on H10 (Ω) (counted according to its multiplicity).

Theorem 4.7. Let Ω be an open bounded domain in Rn, a > 0 and f : Ω × [−a, a] → R be aCarathéodory function such that ∂f(x, s)/∂s is bounded on Ω̄ × [−a, a] and

∫ t−a f(x, s) dx ≥ 0 for

t ∈ [−a, a]. Assume that ū ∈ H10 (Ω)∩L∞(Ω) is a continuous positive weak super-solution of (4.17)with ‖ū‖L∞ < a. If λk < λ < λk+1 for some k ∈ N then problem (4.17) has k distinct pair ofsolutions in the set

K ={u ∈ H10 (Ω); −ū(x) ≤ u(x) ≤ ū(x), a.e.x ∈ Ω

}.

Proof. Let F (x, t) =∫ t

0 χ[−a,a](s)f(x, s) dx where χ[−a,a](s) = 1 for s ∈ [−a, a], and χ[−a,a](s) =0 otherwise. We shall apply Theorem 2.12 to show that the functional E : V = H10 (Ω)→ R

E(u) =1

2

∫Ω|∇u|2 dx−

∫ΩF (x, u) dx− λ

2

∫Ω|u|2 dx

restricted to the set K has at least k distinct pair of critical points. Then we apply Theorem 3.7to deduce that each critical point of E restricted to K is indeed an actual critical point of E andsolves problem (4.17). Set

Φ(u) =

∫ΩF (x, u) dx+

λ

2

∫Ω|u|2 dx, Ψ(u) = 1

2

∫Ω|∇u|2 dx,

and note that E(u) = Ψ(u)−Φ(u). As usual, let ΨK be the restriction of Ψ on the set K. We firstshow that EK := ΨK−Φ satisfies the Palais-Smale compactness condition (PS) (see Definition 2.8).Let {un} ⊂ V be a (PS) sequence. Since {un} ⊂ K we have that {un} is bounded in L∞(Ω). Thistogether with the fact that EK(un) is bounded yield that the sequence {un} is bounded in H10 (Ω).It then implies that there exists u ∈ K such that un ⇀ u weakly in H10 (Ω) and un(x) → u(x) fora.e. x ∈ Ω. It follows from the dominated convergence theorem together with the fact that |un| ≤ ūthat

∫Ω(f(x, un) + λun)(u − un) dx → 0 as n → ∞. Since {un} is a (PS) sequence we also have

that

1

2

∫Ω|∇v|2 dx− 1

2

∫Ω|∇un|2 dx ≥

∫Ω

(f(x, un) + λun)(v − un) dx− �n‖v − un‖V , ∀v ∈ K.

By setting v = u in the latter inequality and letting n→∞ we obtain that

1

2

∫Ω|∇u|2 dx ≥ lim sup

n→∞

1

2

∫Ω|∇un|2 dx.

29

This together with the weak lower semi-continuity of the norm imply that un → u strongly inH10 (Ω). Consequently, the (PS) condition holds.

It is obvious that the function Φ is even and continuously differentiable. Also ΨK is a proper,convex and lower semi-continuous even function. For each j ∈ {1, ..., k}, considering the definitionof Γj in Theorem 2.12, we define

cj = infA∈Γj

supu∈A

EK(u).

We shall now prove that −∞ < cj < 0 for all j ∈ {1, ..., k}. It follows from the continuous domaindependence of the eigenvalues of the Dirichlet Laplacian that there exists a domain Ω0 such thatΩ̄0 ⊂ Ω and λk(Ω0) < λ < λk+1(Ω0) (see [24] a proof). Since ū is continuous and positive on Ω0,we obtain that m = minx∈Ω0 ū(x) > 0. let us denote by λ̄j the j-th eigenvalue of −∆ on H10 (Ω0)(counted according to its multiplicity) and by ej a corresponding eigenfunction satisfying |ej | ≤ mand

∫Ω0∇ei.∇ej dx = δij where δij = 0 for i 6= j and δii = 1. We can extend each ej to Ω, still

denoted by ej , by assuming that ej = 0 on Ω \ Ω0. It now follows that∫

Ω∇ei.∇ej dx = δij . It iseasily seen that EK is bounded below. Thus cj > −∞. Let

A :={u = α1e1 + ...+ αjej : ‖u‖2H10 (Ω) = α

21 + ...+ α

2j = ρ

2},

for small ρ > 0 to be determined. Note that for each u ∈ A we have that |u(x)| = 0 ≤ ū(x) forx ∈ Ω \ Ω0. Also for x ∈ Ω̄0 we have that

|u(x)| ≤j∑i=1

|αi|ei ≤j∑i=1

|αi|m ≤ ū(x)j∑i=1

|αi|.

This in fact implies that A ⊂ K for ρ small.Note also that A ∈ Γj by Proposition 4.1 in [57]. Since A is finite dimensional all norms are

equivalent on A, and we can choose positive constants c1, c2 such that ‖u‖Lp(Ω) > c1‖u‖H10 (Ω) and‖u‖L2(Ω) > c2‖u‖H10 (Ω) for all u ∈ A. Therefore, for u ∈ A ⊂ K

EK(u) =1

2‖u‖2H10 (Ω) −

∫ΩF (x, u) dx− λ

2‖u‖2L2(Ω)

≤ 12‖u‖2H10 (Ω) −

λ

2‖u‖2L2(Ω) =

1

2

j∑i=1

(1− λλi(Ω0)

)α2i ≤ (1−λ

λj(Ω0))‖u‖2H10 (Ω) < 0.

It then follows that cj < 0. Thus, by Theorem 2.12, EK has k distinct pair of critical points. Wenow show that each of these critical points is indeed a solution of (4.17). Let u0 be any of thesecritical points. We shall apply Theorem 3.7 to show that u0 solves (4.17). Let

M := ‖∂f(x, s)/∂s‖L∞(Ω̄×[−a,a])

For x ∈ Ω and t ∈ [−a, a], set h(x, t) = f(x, t) + λt + Mt. Note that for each x ∈ Ω, the mapt→ h(x, t) is increasingly monotone. Therefore,

h(x,−ū) ≤ h(x, u0) ≤ h(x, ū). (4.19)

Since ū is a super-solution of (4.17), it follows from (4.18) and (4.19) that

∆ū−Mū ≤ −∆v +Mv ≤ −∆ū+Mū.

and consequently −ū ≤ v ≤ ū. This completes the proof. �To the best of our knowledge, Theorem 4.17 is the first result that addresses multiplicity resultsfor possible super-critical non-linearties when there exists a positive super-solution.

4.4 Super critical Neumann problems

We shall consider the existence of positive solutions of the p-Laplace Neumann problem−∆pu+ |u|p−2u = a(x)|u|s−2u− β|u|q−2u, x ∈ Ω,u > 0, x ∈ Ω,∂u∂ν = 0, x ∈ ∂Ω,

(4.20)

where Ω is the unit ball centered at the origin in RN with N ≥ 3, β ≥ 0 a non-negative constantand s > q ≥ p ≥ 2. We also assume that a is a radially increasing function, i.e. a(x) = a(r) wherer = |x| and the map r → a(r) is increasing.

Theorem 4.8. Assume that a ∈ H1(0, 1) is radially increasing, not constant and a(r) > 0 a.e. in[0, 1]. Then problem (4.20) admits at least one radially increasing positive solution.

Sketch of the proof. Let V = Ls(Ω) ∩W 1,pr (Ω), where W 1,pr (Ω) is the set of radial functions inW 1,p(Ω). We apply Theorem 3.8, where

Ψ(u) =

∫Ω

|∇u|p + up

pdx+

β

q

∫Ω|u|q dx, Φ(u) = 1

s

∫Ωa(x)|u|s dx,

andK =

{u ∈ V : u(r) ≥ 0, u(r1) ≤ u(r2), ∀r1, r2 ∈ [0, 1], r1 ≤ r2

}.

It can be easily deduced that that V ∩ K is continuously embedded in L∞(Ω) from which onecan apply Theorem 2.10 to show that I = Ψ − Φ restricted to K has a critical point u0 ∈ K ofmountain pass type (see [47] for a detailed argument for p = 2). By a similar argument as in([16], Lemma 3.5) there exists v0 ∈ K satisfying −∆pv0 + |v0|p−2v0 + β|v0|q−2v0 = a(|x|)|u0|p−2u0.This means that the triple (Φ,K,Ψ) satisfies the point-wise invariance condition at u0. Thus, byTheorem 3.8, u0 is a non-negative and nontrivial solution of (4.20). �

We remark that finding radially increasing solutions of problems of type (4.20) has been thesubject of many studies in recent years starting the works of [7, 8, 30, 54] (see also [15, 19, 29] fornon-radial domains). In all these studies, by assuming β = 0, the authors took advantage of thefact that the function on the right hand side of the equation (4.20) is an increasingly monotonefunction in terms of u. However, this property is lost when β > 0 as the right hand side of thisequation is the difference of two monotone functions in terms of u. In Theorem 4.8, we are notrequired to assume that β = 0 and our result holds for all β ≥ 0.

31

5 A nonsmooth principle of symmetric criticality

The aim of this section is to generalize the principle of symmetric criticality, due to R.S. Palais [49],so it can be applied to a wider class of variational problems. Let G be a topological group whichacts linearly on the real Banach space V , i.e., the action (g, u)→ gu is continuous from G × V intoV and the mapping u→ gu is continuous for every g ∈ G. Let V ∗ be the topological dual of V and〈., .〉 be the duality pairing between V and V ∗. We shall define the action G on V ∗ by

〈gu∗, u〉 = 〈u∗, g−1u〉, ∀g ∈ G, ∀u ∈ V,∀u∗ ∈ V ∗.

A function h : V → R ∪ {+∞} is said to be G-invariant if h(gu) = h(u) for every g ∈ G. A setD ⊆ V (or D ⊆ V ∗) is G-invariant if

gD = {gu; u ∈ D} ⊆ D,

for every g ∈ G. LetΣ = {u ∈ V ; gu = u, ∀g ∈ G},

andΣ∗ = {u∗ ∈ V ∗; gu∗ = u∗, ∀g ∈ G}.

Let Ψ : V → R ∪ {+∞} be convex, Φ : V → R be locally Lipschitz and let I = Ψ − Φ. Recallthat, by principle of symmetric criticality, if the functionals Φ and Ψ are G-invariant then undercertain conditions every critical point ũ of IΣ will be also a critical point of I in the whole spaceV . Here, IΣ denotes the restriction of I to Σ. Indeed, to the best of our knowledge, the principleof symmetric criticality holds for the following cases when G is compact,

• R.S. Palais [49] for C1 functionals (i.e., Φ is of class C1, and Ψ = 0);

• W. Krawcewicz and W. Marzantowicz [33] for locally Lipschitz functionals on reflexive Banachspaces (i.e., Φ locally Lipschitz, and Ψ = 0);

• J. Kobayashi and M. Ôtani [34] for Φ ∈ C1(V ;R) and Ψ convex and l.s.c. on reflexive Banachspaces.

• A. Kristály, C. Varga and V. Varga [37] for Φ locally lipschitz and Ψ convex and l.s.c. onreflexive Banach spaces.

If the group G is not compact but the action of G over V is isometric (i.e. ‖gu‖ = ‖u‖ for allu ∈ V and g ∈ G), the principle of symmetric criticality was proved by R.S. Palais [49] for Hilbertspaces and by J. Kobayashi and M. Ôtani [34] for reflexive Banach spaces V such that both V andV ∗ are strictly convex.

Our aim is to unify and improve the aforementioned results by making use of Theorems 3.1 and3.7. We begin with the following definition.

Definition 5.1. Let V be a real Banach space and G be a group which acts linearly on V. LetΨ : V → R ∪ {+∞} be a proper convex and l.s.c function and let Φ : V → R be locally Lipschitzsuch that both Ψ and Φ are G-invariant. We say that the quadruple (V ;G; Ψ; Φ) is compatible withthe principle of symmetric criticality if the following assertions hold;

1. Either Φ ∈ C1(V ;R), or there exists a continuous map T : V ∗ → Σ∗ such that T (D) ⊆ D forevery G-invariant closed convex subset D of V ∗.

32

2. For each u ∈ Σ and u∗ ∈ Σ∗, there exists a convex, l.s.c. and G-invariant function G : V → Rwith ∂G(0) = {0} such that the function

v → Ψ(v) +G(v − u)− 〈u∗, v〉

admits its minimum on V at some point v0 ∈ Σ.

Our plan is to show that if the quadruple (V ;G; Ψ; Φ) is compatible with the principle of sym-metric criticality then every critical point of I = Ψ−Φ on Σ is an actual critical point of I. However,we first would like to illustrate that the assumptions in Definition 5.1 are quite general and coverall the known results in the literature. We begin by addressing the first condition in Definition 5.1.

Remark 5.2. If Φ ∈ C1(V ;R) then the first condition always holds. Let us assume thatΦ : V → R is locally Lipschitz but not necessarily C1. We shall consider two cases. In the first casewe address the compact topological groups and in the second one we look into the non-compact ones.

Case I: If G is a compact topological group then the first condition in Definition 5.1 alwaysholds. Indeed, let A : V → V be the averaging operator over G, defined by

Au =

∫Ggu dµ(g),

where µ is the normalized Haar measure on G. The averaging operator A can also be read as follows:

〈Au, u∗〉 =∫G〈gu, u∗〉 dµ(g), ∀u ∈ V, ∀u∗ ∈ V ∗.

It is easy to verify that A is a continuous linear projection from V to Σ and for every G-invariantclosed convex set D ⊂ V we have A(D) ⊆ D. Also by Lemma 3.13 in [34] the adjoint operatorA∗ of A is a mapping from V ∗ into Σ∗ such that A∗(D) ⊂ D for every G-invariant closed convexset D of V ∗. Thus, if G is a compact topological group then the operator T = A∗ satisfies the firstcondition in Definition 5.1.

Case II: Now, if G is not necessarily a compact topological group but (V ∗; ‖.‖∗) is strictly convexthen the first condition in Definition 5.1 is satisfied provided the following isometry conditions holds,

‖gu‖ = ‖u‖, ∀u ∈ V, ∀g ∈ G.

Indeed, for each G-invariant weak∗-compact subset D of V , let u∗ ∈ D be the one with the minimumnorm, i.e.,

‖u∗‖∗ = infv∗∈D

‖v∗‖∗.

Since (V ∗; ‖.‖∗) is strictly convex and D is G-invariant it follows from ‖gu∗‖∗ = ‖u∗‖∗ that gu∗ = u∗for all g ∈ G as desired.

We have the following result for compact groups. The proof is similar to the proof of Proposition3.15 in [34].

Proposition 5.1. Assume that (V, ‖.‖) is a Banach space and G is compact topological group.Then there is an equivalent norm |||.||| to ‖.‖ which is an isometry, i.e.,

|||gu||| = |||u|||, ∀u ∈ V, ∀g ∈ G.

Moreover, if V is reflexive then this equivalent norm |||.||| can be chosen in such a way that both Vand V ∗ are strictly convex.

33

Proof. Let µ be the normalized Haar measure on G and define

|||u||| =(∫

G‖gu‖2 dµ(g)

) 12, ∀u ∈ V.

Note that the norm |||.||| is equivalent to ‖.‖. It is also easily seen that

|||gu||| = |||u|||, ∀u ∈ V, ∀g ∈ G.

By a result of Asplund [3], if V is a reflexive Banach space then we can choose an equivalentnorm ‖.‖ of V (still denoted by ‖.‖) so that both V and V ∗ are strictly convex. Thus, thecorresponding |||.||| norm equivalent to ‖.‖ is strictly convex. �

Here we remark on the second condition in Definition 5.1.

Remark 5.3. We shall again consider two cases.

The compact case: If G is a compact topological group then the second condition in Definition5.1 always holds. Indeed, by Proposition 5.1, the Banach space V has an equivalent norm |||.|||which is an isometry. If we define the function G : V → R by G(u) = |||u|||2, it is standard thatthe map

v → Ψ(v) +G(v − u)− 〈u∗, v〉, (for fixed u ∈ V, u∗ ∈ V ∗), (5.1)

admits its minimum on V for some ṽ ∈ V. Assuming that µ is the normalized Haar measure on G,we can show that v0 ∈ Σ defined by

v0 =

∫Ggṽ dµ(g)

also minimizes the function defined in (5.1). Indeed, we have that for each g ∈ G

Ψ(gṽ) +G(gṽ − gu)− 〈u∗, gṽ〉 = Ψ(ṽ) +G(ṽ − u)− 〈u∗, ṽ〉.

Thus, it follows from the Jensen inequality that

Ψ(ṽ) +G(ṽ − u)− 〈u∗, ṽ〉 =∫G

[Ψ(gṽ) +G(gṽ − gu)− 〈u∗, gṽ〉

]dµ(g)

≥ Ψ(∫

Ggṽ dµ(g)

)+G

(∫G〈g(ṽ − u) dµ(g)

)−〈u∗,

∫Ggṽ dµ(g)

〉= Ψ(v0) +G(v0 − u)− 〈u∗, v0〉,

which shows that v0 ∈ Σ also minimizes the function defined in (5.1).

The non-compact case: Now, if G is not necessarily a compact topological group but (V ; ‖.‖)is reflexive and both V and V ∗ are strictly convex and satisfy the following isometry condition,

‖gu‖ = ‖u‖, ∀u ∈ V, ∀g ∈ G,

then the second condition in Definition 5.1 also holds.Indeed, in this case the function u → ‖u‖2 is strictly convex and Gâteaux differentiable (see [4]for a proof). Therefore, the second condition in Definition 5.1 is satisfied if we define the functionG : V → R by G(u) = ‖u‖2. To be more precise, let v0 be a minimizer of the function v →Ψ(v) + G(v − u) − 〈u∗, v〉 which exists due to the coercivity and lower semi-continuity of this

34

function. Since this function is strictly convex we have that the minimizer v0 is indeed unique.However, for each g ∈ G it follows that

Ψ(v0) +G(v0 − u)− 〈u∗, v0〉 = Ψ(gv0) +G(gv0 − u)− 〈u∗, gv0〉,

from which we obtain that gv0 = v0. Therefore, v0 ∈ Σ as desired.

We are now ready to state our main result of this section.

Theorem 5.4. Let V be a real Banach space and G be a group which acts linearly on V. LetΨ : V → R ∪ {+∞} be a proper convex and l.s.c function and let Φ : V → R be locally Lipschitzsuch that both Ψ and Φ are G-invariant. Assume that the quadruple (V ;G; Ψ; Φ) is compatible withthe principle of symmetric criticality. Let I = Ψ − Φ. Then every critical point of IΣ is indeed acritical point of I.

Proof. We shall use Theorem 3.7 to prove this result. Define

ΨΣ(u) =

{Ψ(u), u ∈ Σ,+∞, u 6∈ Σ.

Note that if u0 is a critical point of I restricted to Σ then u0 is a critical point of the functionIΣ = ΨΣ−Φ. Thus, we just need to show that the triple (Φ,Σ,Ψ) satisfies the point-wise invariancecondition of type (II) at u0. Define the paramonotone operator M : V → 2V

∗by M(u) = ∂G(u)

where G is given in Definition 5.1. We shall show that there exists v0 ∈ Σ such that{∂cΦ(u0)

}∩{∂Ψ(v0) +M(v0 − u0)

}6= ∅,

from which the result will be a consequence of Theorem 3.7.By Proposition 2.4 the set ∂cΦ(u0) is convex closed and weak

∗-compact. We now show that ∂cΦ(u0)is G-invariant, i.e., g∂cΦ(u0) ⊆ ∂cΦ(u0) for every g ∈ G. To this end, let us fix g ∈ G and u∗ ∈∂cΦ(u0). Then, for every v ∈ V we have that

〈gu∗, v〉 = 〈u∗, g−1v〉 ≤ Φ0(u0; g−1v) = Φ0(gu0; v) = Φ0(u0; v),

from which we obtain that gu∗ ∈ ∂cΦ(u0). If Φ ∈ C1(V ;R), it means that ∂cΦ(u0) = Φ′(u0) andtherefore gΦ′(u0) = Φ

′(u0) for all g ∈ G. Thus, u∗0 := Φ′(u0) ∈ Σ∗. Now, if Φ 6∈ C1(V ;R), it followsfrom the first condition in Definition 5.1 that there exists a continuous map T : V ∗ → Σ∗ such thatT (D) ⊆ D for D = ∂cΦ(u0). Since D is convex closed and weak∗-compact we have that T has afixed point u∗0 ∈ D = ∂cΦ(u0). It then follows that u∗0 = T (u∗0) ∈ Σ∗. By the second condition inDefinition 5.1 the function

v → Ψ(v) +G(v − u0)− 〈u∗0, v〉

admits its minimum on V at some point v0 ∈ Σ. It then implies that

u∗0 ∈ ∂(Ψ(.) +G(.− u0)

)(v0) = ∂Ψ(v0) + ∂G(v0 − u0)

where the second equality follows from Proposition 2.2. Thus,

u∗0 ∈{∂cΦ(u0)

}∩{∂Ψ(v0) +M(v0 − u0)

}6= ∅,

as desired. �

35

Proposition 5.2. One can replace the second condition in Definition 5.1 by the following condition:

2′. There exists a convex, l.s.c. and Gâteaux differentiable G-invariant function G : V → R suchthat the function G is strictly convex and coercive, i.e.,

lim‖u‖→∞

G(u)

‖u‖= +∞.

Proof. Let u0 be a critical point of I restricted to Σ. As in the proof of Theorem 5.4, the setΣ∩ ∂cΦ(u0) is non-empty and there exists at least one u∗0 ∈ Σ∩ ∂cΦ(u0). Define the paramonotoneoperator M : V → V ∗ by M(u) = DG(u). We shall show that there exists v0 ∈ Σ such that{

∂cΦ(u0) +M(u0)}∩{∂Ψ(v0) +M(v0)

}6= ∅,

from which the result will be a consequence of Theorem 3.7.It follows from the coercivity and strict convexity of the function Ψ +G that there exists a uniquev0 ∈ V such that

Ψ(v0) +G(v0)−〈u∗0 +DG(u0), v0

〉= inf

v∈V

{Ψ(v) +G(v)−

〈u∗0 +DG(u0), v

〉}.

This indeed implies that u∗0 + DG(u0) ∈ DG(v0) + ∂Ψ(v0). On the other hand for each g ∈ Gwe have that

Ψ(gv0) +G(gv0)−〈u∗0 +DG(u0), gv0

〉= Ψ(v0) +G(v0)−

〈g−1u∗0 + g

−1DG(u0), v0〉

= Ψ(v0) +G(v0)−〈u∗0 +DG(u0), v0

〉.

from which together with the uniqueness of v0 one obtains that gv0 = v0. This in fact implies thatv0 ∈ Σ and therefore

u∗0 +DG(u0) ∈{∂cΦ(u0) +M(u0)

}∩{∂Ψ(v0) +M(v0)

}6= ∅.

This completes the proof. �

6 Critical points of locally Lipschitz functions on convex sets

In this section we investigate the existence of actual critical points of a locally Lipschitz functionalΦ on a given convex closed subset K of a Banach space V. In fact, we are interested in findingconditions under which critical points for Φ on K are indeed critical points on the whole space V.It is easily seen that this problem is nothing but a particular case of Theorem 3.7 when the convexfunction Ψ is identically zero. As a result, we shall introduce a simplified version of the point-wiseinvariance condition as in Section 3 that turns out to be not only sufficient but also necessary forcritical points of Φ on K to be critical points on the whole space V.

Note that u ∈ V is a critical point of Φ on V if 0 ∈ ∂cΦ(u) where ∂c stands for the generalizedderivative of locally Lipschitz functions as in Definition 2.6. Let δK be the indicator functioncorresponding to K, i.e.,

δK(u) =

{0, u ∈ K,

+∞, u 6∈ K.

36

Recall that u0 is a critical point of Φ on K (see Lemma 4.3 in [41]) if u0 is a critical point of thefunctional I : V → (−∞,+∞] defined by

I(u) := δK(u) + Φ(u).

We now recall the well-known Schauder invariance condition.

Definition 6.1. Let H be a Hilbert space and Φ : H → R be a C1−functional. Let K be a convexand closed subset of H. We say that Φ satisfies the Schauder invariance condition with respect toK if (Id −DΦ)(K) ⊆ K, where Id is the identity map on H.

It [42], Mawhin proved that if Φ satisfies the Schauder invariance condition with respect to Kthen every critical point u of Φ on K is a critical point of Φ on V , i.e., 0 ∈ ∂Φ(u) (see also [31, 40]).This result was extended to reflexive Banach spaces by [39, 41]. In fact, the following interestingresult was established in [41].

Theorem 6.2. Suppose V is a reflexive Banach space, K is a closed convex subset of V andΦ : V → R is a locally Lipschitz function. Assume that

(I − J∗∂cΦ)(∂K) ⊆ K,

where J∗ : V ∗ → 2V is the duality mapp