Eigenvalue problems for nonlinear elliptic equations with ... · Eigenvalue problems ... deﬁned a ﬁxed point index ... The following is a nonsmooth version of the well-known mountain

Nonlinear Analysis 69 (2008) 85–109www.elsevier.com/locate/na

Eigenvalue problems for nonlinear elliptic equations withunilateral constraints

Michael E. Filippakisa, Nikolaos S. Papageorgioua,∗, Vasile Staicub

a Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greeceb Department of Mathematics, Aveiro University, 3810-193 Aveiro, Portugal

Received 6 November 2006; accepted 18 May 2007

Abstract

In this paper we study eigenvalue problems for hemivariational and variational inequalities driven by the p-Laplacian differentialoperator. Using topological methods (based on multivalued versions of the Leray–Schauder alternative principle) and variationalmethods (based on the nonsmooth critical point theory), we prove existence and multiplicity results for the eigenvalue problemsthat we examine.c© 2007 Elsevier Ltd. All rights reserved.

MSC: 35J20; 35J60; 35J85

Keywords: Generalized subdifferential; Fixed point; Nonsmooth PS-condition; Nonlinear regularity theory; m-accretive operator; Monotoneoperator; Spectrum of p-Laplacian; Multiple solutions

1. Introduction

The goal of this paper is to study various eigenvalue problems for nonlinear hemivariational andvariational–hemivariational inequalities. In the past eigenvalue problems for hemivariational inequalities were studiedprimarily in the context of semilinear problems. We refer the reader to the works of Barletta and Marano [2],Motreanu and Panagiotopoulos [28], Goeleven, Motreanu and Panagiotopoulos [18], Gasinski and Papageorgiou [14](problems driven by the Laplacian) and Gasinski and Papageorgiou [13] (problems driven by the p-Laplacian).Concerning eigenvalue problems for variational–hemivariational inequalities (i.e. hemivariational inequalities subjectto variational constraints), we are only aware of the semilinear work of Goeleven and Motreanu [17]. We should alsomention that variational and variational–hemivariational inequalities driven by general nonlinear operators whichinclude the p-Laplacian as a special case were studied recently using the method of upper–lower solutions byLe [22,23] (variational inequalities), by Carl, Le and Motreanu [5] (variational–hemivariational inequalities), byDegiovanni [9], Degiovanni, Marzocchi and Radulescu [10], Motreanu and Radulescu [27] and Ciulcu, Motreanuand Radulescu [7].

∗ Corresponding author.E-mail address: [email protected] (N.S. Papageorgiou).

0362-546X/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2007.05.002

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http://dx.doi.org/10.1016/j.na.2007.05.002

86 M.E. Filippakis et al. / Nonlinear Analysis 69 (2008) 85–109

Here we examine nonlinear problems monitored by the p-Laplacian and we seek to prove existence and multiplicityresults for two types of eigenvalue problems. Our approach is a combination of topological and variational methods.The topological techniques are based on a recent nonlinear multivalued version of the Leray–Schauder alternativeprinciple, due to Bader [1]. This permits us to avoid some rather restrictive hypotheses of the previous works onthe subject, which had to assume that the subdifferential of the locally Lipschitz term (the hemivariational constraint)admits a continuous selector (see [17,18]). The variational techniques are based on the nonsmooth critical point theory(see [15]).

Hemivariational inequalities or more generally variational–hemivariational inequalities turned out to be an effectivetool in modelling important situations in mechanics and engineering, where nonmonotone and multivalued boundaryor interface conditions are taken into account as well as nonmonotone, multivalued relations between stress and strainor between reaction and displacement. Several such applications can be found in [29].

In the next section we briefly review some of the analytical tools that we will use in this work. In Section 3, wetreat variational–hemivariational inequalities. More precisely, for Z ⊆ RN a bounded domain with a C2-boundary∂Z , we investigate the following two nonlinear eigenvalue problems:−div(‖Dx(z)‖p−2 Dx(z))− λ|x(z)|p−2x(z) ∈ ∂ j (z, x(z))

−∂G(x(z)) a.e. on Z ,x |∂Z = 0, 2 ≤ p < ∞, λ ∈ R.

(1.1)

and {−∆x(z) ∈ λ∂ j (z, x(z))− ∂G(x(z)) a.e. on Z ,x |∂Z = 0, λ ∈ R.

}. (1.2)

In both problems j (z, x) is a Caratheodory function which is locally Lipschitz in x ∈ R and by ∂ j (z, x) we denotethe generalized subdifferential of j (z, ·) (see Section 2), while G ∈ Γ0(R) (i.e. G : R → R = R ∪ {+∞}, it isnot identically +∞ and it is convex and lower semicontinuous) and by ∂G(x) we denote the subdifferential in thesense of convex analysis. So in problems (1.1) and (1.2) ∂ j (z, x) expresses the nonmonotone unilateral constraint(hemivariational constraint) and ∂G(x) the monotone one (variational constraint). Therefore in both problemswe have the combined effects of variational and hemivariational constraints, which explains the nomenclature“variational–hemivariational inequalities” for problems (1.1) and (1.2).

In Section 4, we deal with the following nonlinear hemivariational inequality:{−div(‖Dx(z)‖p−2 Dx(z)) ∈ λ∂ j (z, x(z)) a.e. on Z ,x |∂Z = 0, 1 < p < ∞.

}. (1.3)

As above, j (z, x) is a Caratheodory function which is locally Lipschitz in x ∈ R and ∂ j (z, x) stands for thegeneralized subdifferential of x → j (z, x). For problem (1.3) we prove existence and multiplicity results usingdegree theoretic and variational methods.

2. Mathematical background

As we already mentioned, the topological approach of problem (1.1) will be based on a multivalued version ofthe well-known Leray–Schauder alternative principle. In [11] p. 98, we can find such a multivalued extension. Theproblem with the result of Dugundji and Granas is that the multifunction needs to be convex-valued. This precludesthe use of the result outside the class of semilinear problems. Recently Bader [1] defined a fixed point index fornonconvex-valued multifunctions and using it produced the following two multivalued versions of the Leray–Schauderalternative principle. First a definition:

Definition 2.1. Let X, Y be Banach spaces and C ⊆ X a nonempty closed and convex set. By Pwkc(Y ) we denotethe family of nonempty, w-compact and convex subsets of Y and by Yw the Banach space Y furnished with the weaktopology.

• (a) A multifunction S : C → Pwkc(Y ) is said to be upper semicontinuous (usc for short) from C with the relativenorm topology into Yw if for all U ⊆ Y weakly open, the set S+(U ) = {x ∈ C : S(x) ⊆ U } is open in C .

M.E. Filippakis et al. / Nonlinear Analysis 69 (2008) 85–109 87

• (b) A map u : Y → X is said to be completely continuous if it is sequentially continuous from Yw into X with thenorm topology, i.e. yn

w→ y in Y implies that u(yn) → u(y) in X .

• (c) A multifunction G : X → 2X\ {∅} is said to be compact if it maps bounded sets into relatively compact sets.

Theorem 2.2. If X, Y are Banach spaces and E : Br (0) = {x ∈ X : ‖x‖ ≤ r} → 2X\{∅} is a compact multifunction

such that E = u ◦ S, where S : Br (0) → Pwkc(Y ) is usc from Br (0) with the relative norm topology into Yw andu : Y → X is completely continuous, then either:

(a) there exist x0 ∈ ∂Br (0) = {x ∈ X : ‖x‖ = r} and 0 < λ < 1 such that x0 ∈ λE(x0), or(b) there exists x ∈ Br (0) such that x ∈ E(x).

Theorem 2.3. If X, Y are Banach spaces, C ⊆ X is a nonempty, closed, convex set with 0 ∈ C and E : C → 2C\{∅}

is a compact multifunction such that E = u ◦ S where S : C → Pwkc(Y ) is usc from C with the relative norm topologyinto Yw and u : Y → X is completely continuous, then either:

(a) the set D = {x ∈ C : x ∈ t E(x) for some t ∈ (0, 1)} is unbounded, or(b) there exists x ∈ X such that x ∈ E(x).

Remark 2.4. Note that although S has convex values, the composition E = u ◦ S in general does not have convexvalues, unless of course u is linear. So Theorems 2.2 and 2.3 are suitable for use in the study of nonlinear problemssuch as (1.1).

The nonsmooth critical point theory is based on the subdifferential calculus for locally Lipschitz functions. So letX be a Banach space. By X∗ we denote its topological dual and by 〈·, ·〉 the duality brackets for the pair (X, X∗). Afunction ϕ : X → R is said to be locally Lipschitz if for every x ∈ X we can find a neighborhood of U of x and aconstant k > 0 (depending on U ) such that

|ϕ(y)− ϕ(z)| ≤ k‖y − z‖ for all y, z ∈ U.

A function ϕ : X → R which is Lipschitz continuous on bounded sets is locally Lipschitz and the two notionscoincide if X is finite dimensional. It is well known that a continuous convex function ϕ : X → R and a smoothfunction ϕ ∈ C1(X) are locally Lipschitz. For a locally Lipschitz function ϕ : X → R, the generalized directionalderivative at x ∈ X in the direction h ∈ X , ϕ0(x; h), is defined by

ϕ0(x; h) = lim supx ′→xλ↓0

ϕ(x ′+ λh)− ϕ(x ′)

λ.

It is easy to check that h → ϕ0(x; h) is continuous and sublinear, and so it is the support function of a nonempty,w∗-compact and convex set ∂ϕ(x) ⊆ X∗ defined by

∂ϕ(x) = {x∗∈ X∗

: 〈x∗, h〉 ≤ ϕ0(x; h) for all h ∈ X}.

The multifunction x → ∂ϕ(x) is called the generalized subdifferential of ϕ. If ϕ ∈ C1(X), then ∂ϕ(x) = {ϕ′(x)}for all x ∈ X . Also if ϕ,ψ : X → R are locally Lipschitz functions and µ ∈ R, then

∂(ϕ + ψ) ⊆ ∂ϕ + ∂ψ and ∂(µϕ) = µ∂ϕ.

A point x ∈ X is a critical point of the locally Lipschitz function ϕ, if 0 ∈ ∂ϕ(x). It is easy to see that if x ∈ X is alocal extremum of ϕ (i.e. x ∈ X is a local minimum or a local maximum of ϕ), then 0 ∈ ∂ϕ(x), i.e. x is a critical pointof ϕ. It is well known that in the smooth critical point theory, a central role is played by a compactness type conditionfor the smooth functional ϕ. In the present nonsmooth setting, this condition takes the following form:

“A locally Lipschitz function ϕ : X → R, satisfies the nonsmooth Palais–Smale condition (the nonsmooth PS-condition for short) if every sequence {xn}n≥1 such that {ϕ(xn)}n≥1 ⊆ R is bounded and m(xn) = inf{‖x∗

‖ :

x∗∈ ∂ϕ(xn)} → 0 as n → ∞ has a strongly convergent subsequence”.


The following is a nonsmooth version of the well-known mountain pass theorem of Ambrosetti and Rabinowitz,and it is due to Chang [6]; see also the books of Gasinski and Papageorgiou [15], Motreanu and Radulescu [26] andthe paper of Radulescu [31].

Theorem 2.5. If X is a reflexive Banach space, ϕ : X → R is a locally Lipschitz function which satisfies thenonsmooth PS-condition and there exist x0, x1 ∈ X and r > 0 such that:

(i) ‖x1 − x0‖ > r and(ii) inf{ϕ(x) : x ∈ ∂Br (x0)} = β > max{ϕ(x0), ϕ(x1)}

then ϕ has a critical point x ∈ X with critical value c = ϕ(x) such that

c ≥ β and cd f= inf

γ∈Γmax

t∈[0,1]

ϕ(γ (t)),

where

Γd f= {γ ∈ C([0, 1], X) : γ (0) = x0, γ (1) = x1} .

In our analysis of problem (1.3), we will need some facts about the spectrum of the negative p-Laplacian withDirichlet boundary condition. So for m ∈ L∞(Z)+ = {m ∈ L∞(Z) : m(z) ≥ 0 a.e. on Z}, m 6= 0, we consider thefollowing weighted eigenvalue problem with weight m:{

−div(‖Dx(z)‖p−2 Dx(z)) = λm(z)|x(z)|p−2x(z) a.e. on Z ,x |∂Z = 0, 1 < p < ∞, λ ∈ R.

}. (2.1)

The least real number λ for which problem (2.1) has a nontrivial solution is said to be the first eigenvalue of(−∆p,W 1,p

0 (Z),m) and it is denoted by λ1 = λ1(m) (if we want to stress the dependence on the weight m). Weknow that λ1 > 0, and it is isolated and simple (i.e. the corresponding eigenspace is one dimensional). Also there is avariational characterization of λ1, via the Rayleigh quotient, namely

λ1(m) = inf

{‖Du‖

pp∫

Z m|u|pdz: u ∈ W 1,p

0 (Z), u 6= 0

}. (2.2)

This infimum is realized for the normalized eigenfunction u1 ∈ W 1,p0 (Z) \ {0} corresponding to λ1(m). Note

that if u1 minimizes (2.2), then so does |u1| and so we conclude that u1 does not change sign on Z . In fact fromnonlinear regularity theory (see for example [16], p. 728), we have that u1 ∈ C1

0(Z). Moreover, using the strongmaximum principle of Vazquez [34], we have that u1(z) > 0 for all z ∈ Z and ∂u1

∂n (z) < 0 for all z ∈ ∂Z . Forevery eigenvalue λ 6= λ1(m), the corresponding eigenfunctions still belong in C1

0(Z) but necessarily change sign.From the Lusternik–Schnirelmann theory (see for example [16], Section 5.5 and [21]), we know that in additionto λ1 = λ1(m) > 0, problem (2.1) has a whole sequence {λn = λn(m)}n≥1 of eigenvalues, known as variationaleigenvalues of (−∆p,W 1,p

0 (Z),m), such that λn → +∞ as n → ∞. If p = 2 (linear eigenvalue problem), then thevariational eigenvalues are all the eigenvalues of (−∆p, H1

0 (Z),m). If p 6= 2 (nonlinear eigenvalue problem), we donot know whether this is the case. Finally if m1 ≤ m2 and m1 6= m2, then λ1(m2) < λ1(m1). For further details werefer the reader to An Le [21].

3. Variational–hemivariational inequalities

First we deal with problem (1.1). Our hypotheses on j (z, x) and G(x) are the following:

H( j)1: j : Z × R → R is a function such that:(i) for all x ∈ R, z → j (z, x) is measurable;

(ii) for almost all z ∈ Z , x → j (z, x) is locally Lipschitz;(iii) for almost all z ∈ Z , all x ∈ R, and all u ∈ ∂ j (z, x), we have

|u| ≤ α(z)+ c|x |p−1 with α ∈ L∞(Z)+, c > 0.


Remark 3.1. We note that our hypotheses on the nonsmooth potential j (z, x) are minimal, in the sense that noasymptotic conditions at zero and/or infinity are imposed on j (z, ·) and/or ∂ j (z, ·) (see for example [13,14]) or thatthe multifunction ∂ j (z, ·) admits a continuous selector (see [17]).

H(G)1: G ∈ Γ0(R) = the cone of all functions G : R → R = R∪{+∞} which are proper (i.e. not identically +∞),lower semicontinuous and convex, G(x) ≥ 0 for all x ∈ R and G(0) = 0.

Remark 3.2. Some possible choices of G arising in applications are the following:

G1(x) =

12|x |

2 if |x | ≤1k

x if |x | >1k

⇒ ∂G1(x) =

−1 if x < −1kx if |x | ≤ 1+1 if x > 1

, k > 0,

G2(x) = |x | ⇒ ∂G2(x) =

−1 if x < 0[−1, 1] if x = 0+1 if x > 0

,

G3(x) = χR+(x) =

{0 if x ≥ 0+∞ if x < 0

⇒ ∂G3(x) = NR+(x) =

0 if x > 0−R+ if x = 0∅ if x < 0

,

G4(x) = x+= max{x, 0} ⇒ ∂G4 =

0 if x < 0[0, 1] if x = 01 if x > 0

G5(x) =

{0 if x ∈ [a, b]

+∞ if otherwise, a < b,⇒ ∂G5(x) =

∅ if x 6∈ [a, b]

0 if x ∈ (a, b)−R+ if x = aR+ if x = b.

Given h ∈ L p′

(Z)( 1p +

1p′ = 1), we consider the following auxiliary problem:{

−div(‖Dx(z)‖p−2 Dx(z)) ∈ h(z)− ∂G(x(z)) a.e. on Z ,x |∂Z = 0, 2 ≤ p < ∞.

}. (3.1)

By a solution of (3.1), we mean a function x ∈ W 1,p0 (Z) such that there exists v ∈ L p′

(Z) with v(z) ∈ ∂G(x(z))a.e. on Z such that −div(‖Dx(z)‖p−2 Dx(z)) = h(z) − v(z) a.e. on Z (strong solution). For this reason, the nextproposition is not an immediate consequence of standard results for variational inequalities and needs some additionalwork.

Proposition 3.3. If hypotheses H(G)1 hold and h ∈ L p′

(Z), then problem (3.1) has a unique solution ξ(h) ∈

W 1,p0 (Z).

Proof. Let A : W 1,p0 (Z) → W −1,p′

(Z) = W 1,p0 (Z)∗ be the nonlinear operator defined by

〈A(x), y〉 =

∫Z

‖Dx‖p−2(Dx, Dy)RNdz for all x, y ∈ W 1,p

0 (Z).

Hereafter by 〈·, ·〉 we denote the duality brackets for the pair (W 1,p0 (Z),W −1,p′

(Z)). It is easy to check that A isstrongly monotone and demicontinuous; hence it is maximal monotone. Let A : D( A) ⊆ L p′

(Z) → L p′

(Z) be thenonlinear operator defined by

A(x) = A(x) for all x ∈ D( A) = {x ∈ W 1,p0 (Z) : A(x) ∈ L p′

(Z)}.

Let A be the restriction of A on L p′

(Z)× L p′

(Z). From [4] (see also [24]), we know that A is m-accretive.


Also let S : D(S) ⊆ L p′

(Z) → L p′

(Z) be the multifunction defined by

S(x) = S p′

∂G(x(·)) = {v ∈ L p′

(Z) : v(z) ∈ ∂G(x(z)) a.e. on Z}

for all x ∈ D(S) = {x ∈ L p′

(Z) : S p′

∂G(x(·)) 6= ∅}. We know that S is m-accretive (see [32], p. 164).

Therefore g ∈ (I+S)(x)with x ∈ L p′

(Z). Because g ∈ L p′

(Z)was arbitrary, we conclude that R(I+S) = L p′

(Z)which implies the m-accretivity of S.

Next let θ : L p(Z) → L p′

(Z) be defined by

θ(x)(·) = |x(·)|p−2x(·).

If F denotes the duality map of the space L p(Z), then θ(x) = ‖x‖p−2F(x) (see Hu and Papageorgiou [19],

p.317). If by 〈·, ·〉pp′ we denote the duality brackets for the pair (L p(Z), L p′

(Z) = L p(Z)∗) and by Sλ, λ > 0, theYosida approximation of the accretive operator S, for every x ∈ D( A) and every λ > 0, we have

( A(x), θ(Sλ(x)))pp′ = 〈A(x), θ(Sλ(x))〉

=

∫Z

‖Dx‖p−2(Dx, D(θ(∂Gλ(x))))RNdz. (3.2)

Recall that for every λ > 0, (∂G)λ = ∂Gλ, with Gλ being the Moreau–Yosida approximation of G ∈ Γ0(R).Therefore ∂Gλ(·) is Lipschitz continuous and monotone. Applying the chain rule for Sobolev functions (see [25]), wehave

‖Dx(z)‖p−2(Dx(z), D(θ(∂Gλ(x(z)))))RN

= (p − 1)|∂Gλ(x(z))|p−1

(d

dx∂Gλ

)(x(z))‖Dx(z)‖p a.e. on Z . (3.3)

Since by virtue of the monotonicity of ∂Gλ(·), we have ( ddx ∂Gλ)(x) ≥ 0 for all x ∈ R, using (3.3) in (3.2), we

obtain

( A(x), θ(Sλ(x)))pp′ ≥ 0 for all x ∈ D( A) and all λ > 0.

Because 0 ∈ D( A) ∩ D(S), we can apply Theorem 3.3.21, p. 350 in [16] and conclude that x → A(x) + S(x) ism-accretive.

Let IG : W 1,p0 (Z) → R+ = R+ ∪ {+∞} be the integral functional defined by

IG(x) =

∫

ZG(x(z))dz if G(x(·)) ∈ L1(Z)

+∞ if otherwise.

Evidently IG ∈ Γ0(W1,p0 (Z)), i.e. IG is proper, convex, lower semicontinuous. Hence ∂ IG : W 1,p

0 (Z) →

W −1,p′

(Z) is maximal monotone and 0 ∈ ∂ IG(0). Then A+∂ IG is maximal monotone, coercive, and hence surjective.So we can find x ∈ W 1,p

0 (Z) such that

h ∈ (A + ∂ IG)(x),

⇒ (x, h) ∈ Gr(A + ∂ IG) ∩ (W 1,p0 (Z)× L p′

(Z)). (3.4)

Clearly

(A + ∂ IG)|W 1,p0 (Z)×L p′

(Z)= A + S. (3.5)

Combining (3.4) and (3.5), we have

h = A(x)+ v with v ∈ S(x).

From this it follows at once that x ∈ W 1,p0 (Z) is a solution of (3.1) and it is unique since A is strongly monotone.

�


Remark 3.4. It is clear from the above proof that the surjectivity of A+∂ IG gives at once a solution of the variationalinequality

〈A(x)− h, y − x〉 ≤ IG(y)− IG(x) for all y ∈ dom IG .

However, it is not immediately clear that from this inequality we have the pointwise interpretation required inproblem (3.1). For this reason, we had to pass through the realizations A and S and use m-accretive operators. Notethat in general ∂ IG(x) ⊆ W −1,p′

(Z) for all x ∈ W 1,p0 (Z).

So we can define the solution map ξ : L p′

(Z) → W 1,p0 (Z), which to each h ∈ L p′

(Z) assigns the unique solutionξ(h) of the auxiliary problem (3.1).

Proposition 3.5. If hypotheses H(G)1 hold and ξ : L p′

(Z) → W 1,p0 (Z) is the solution map for problem (3.1), then

ξ is completely continuous (i.e. hnw→ h in L p′

(Z) implies ξ(hn) → ξ(h) in W 1,p0 (Z).

Proof. Suppose that hnw→ h in L p′

(Z) and set xn = ξ(hn) for all n ≥ 1. We have

A(xn) = hn − vn with vn ∈ S(xn).

Taking duality brackets in (L p(Z), L p′

(Z)) with xn ∈ W 1,p0 (Z) ⊆ L p(Z), we obtain

( A(xn), xn)pp′ +

∫Zvn xndz =

∫Z

hn xndz,

⇒ ‖Dxn‖pp ≤ ‖hn‖p′‖xn‖p

(via Holder’s inequality and since

∫Zvn xndz ≥ 0

)⇒ {xn}n≥1 ⊆ W 1,p

0 (Z) is bounded (use Poincare’s inequality).

By passing to a suitable subsequence if necessary, we may assume that

xnw→ x in W 1,p

0 (Z) and xn → x in L p′

(Z).

Recall that since 2 ≤ p < ∞, by the Sobolev embedding theorem W 1,p0 (Z) is embedded compactly in L p′

(Z). Ifwe set V = A + S we have

(xn, hn) ∈ GrV for all n ≥ 1.

Because L p(Z) = L p′

(Z)∗ is uniformly convex and V is m-accretive (see the proof of Proposition 3.3), we havethat GrV is sequentially closed in L p′

(Z) × L p′

(Z)w (see [16], p. 344; by L p′

(Z)w we denote the Lebesgue spaceL p′

(Z) furnished by the weak topology). So in the limit as n → ∞, we obtain

(x, h) ∈ GrV,

⇒ A(x)+ v = h for some v ∈ S(x).

Therefore x = ξ(h). We have

〈A(xn), xn − x〉 +

∫Zvn(xn − x)dz =

∫Z

hn(xn − x)dz. (3.6)

By virtue of the monotonicity of S, we have∫Zv(xn − x)dz ≤

∫Zvn(xn − x)dz,

⇒ 0 ≤ lim infn→∞

∫Zvn(xn − x)dz. (3.7)

Also it is clear that

limn→∞

∫Z

hn(xn − x)dz = 0. (3.8)


Hence if we return to (3.6) and pass to the limit as n → ∞, using (3.7) and (3.8), we obtain

lim supn→∞

〈A(xn), xn − x〉 ≤ 0. (3.9)

Recall that A is maximal monotone; hence it is generalized pseudomonotone (see [3]). So from (3.9) we infer that

〈A(xn), xn〉 → 〈A(x), x〉,

⇒ ‖Dxn‖p → ‖Dx‖p.

Because xnw→ x in W 1,p

0 (Z), we have Dxnw→ Dx in L p(Z ,RN ). The space L p(Z ,RN ) is uniformly

convex and so it has the Kadec–Klee property. Therefore it follows that Dxn → Dx in L p(Z ,RN ), and hencexn → x in W 1,p

0 (Z). Since every subsequence of {xn = ξ(hn)}n≥1 has a further subsequence which converges

in W 1,p0 (Z) to x = ξ(h), by Urysohn’s criterion for convergent sequences, we have that the original sequence

{xn = ξ(hn)}n≥1 ⊆ W 1,p0 (Z) converges to x = ξ(h), and hence ξ is completely continuous. �

For every t ∈ [0, 1], we consider the following boundary value problem:−div(‖Dx(z)‖p−2 Dx(z))− tλ|x(z)|p−2x(z) ∈ t∂ j (z, x(z))−t∂G(x(z)) a.e. on Z ,x |∂Z = 0, 2 ≤ p < ∞, λ ∈ R.

(3.10)

Proposition 3.6. If hypotheses H( j)1, H(G)1 hold and λ < λ1 − c with λ1 = λ1(1) (see Section 2) and c > 0 asin hypothesis H( j)1(iii), then there exists r > 0 large and independent of t ∈ [0, 1] such that problem (3.10) has nosolution x ∈ W 1,p

0 (Z) with ‖x‖ = r.

Proof. We argue indirectly. Suppose that we can find {(tn, xn)}n≥1 ⊆ [0, 1]×W 1,p0 (Z) such that (tn, xn) solves (3.10)

and also we have

tn → t in [0, 1], ‖xn‖ → ∞ as n → ∞.

We have

A(xn)− tnλ|xn|p−2xn = tnun − tnvn

with un ∈ K (xn) = S p′

∂ j (·,xn(·))= {u ∈ L p′

(Z) : u(z) ∈ ∂ j (z, xn(z)) a.e. on Z} and vn ∈ S(xn) = S p′

∂G(xn(·)). Then

( A(xn), xn)pp′ + tn

∫Zvn xndz = tn

∫Z

un xndz + tnλ‖xn‖pp

⇒ ‖Dx‖pp ≤ tn

∫Z

un xndz + tnλ‖xn‖pp

(since

∫Zvn xndz ≥ 0, see H(G)1

). (3.11)

By virtue of hypothesis H( j)1(iii), we have

un(z)xn(z) ≤ α(z)|xn(z)| + c|xn(z)|p a.e. on Z ,

⇒

∫Z

un xndz ≤ ‖α‖∞‖xn‖1 + c‖xn‖pp

≤ ‖α‖∞|Z |

1p′

N ‖xn‖p +c

λ1‖Dxn‖

pp

(see (2.2) with m = 1 and note that | · |N denotes the Lebesgue measure on RN )

≤‖α‖∞|Z |

1p′

N

λ

1p′

1

‖Dxn‖p +c

λ1‖Dxn‖

pp. (3.12)


Returning to (3.11) and using (3.12) and (2.2) (with m = 1), we obtain

‖Dxn‖pp ≤

tnλ1(λ+ c)‖Dxn‖

pp +

‖α‖∞|Z |

1p′

N

λ

1p′

1

‖Dxn‖p. (3.13)

Since ‖xn‖ → ∞, we may assume that for all n ≥ 1, ‖xn‖ 6= 0, and so by Poincare’s inequality ‖Dxn‖p 6= 0 forall n ≥ 1. Dividing (3.13) by ‖Dxn‖

pp, we obtain

1 ≤tnλ1(λ+ c)+

‖α‖∞|Z |

1p′

N

λ

1p′

1

1

‖Dxn‖p−1p .

(3.14)

Because ‖xn‖ → ∞, we have ‖Dxn‖p → ∞ and so if we pass to the limit in (3.14), we have

1 ≤t

λ1(λ+ c) ≤

λ+ c

λ1(since 0 ≤ t ≤ 1),

⇒ λ1 − c ≤ λ,

which contradicts the hypothesis that λ < λ1 − c. So the claim of the proposition is true. �

Now we are ready for the existence theorem concerning problem (1.1).

Theorem 3.7. If hypotheses H( j)1, H(G)1 hold and λ < λ1 − c, then problem (1.1) has a solution x ∈ W 1,p0 (Z).

Proof. Let r > 0 be as in Proposition 3.5 and let Br (0) = {x ∈ W 1,p0 (Z) : ‖x‖ ≤ r}. We consider the multifunction

K : Br (0) → Pwkc(L p′

(Z)) defined by

K (x) = S p′

∂ j (·,x(·)) = {u ∈ L p′

(Z) : u(z) ∈ ∂ j (z, x(z)) a.e. on Z}.

From [15], p. 220, we know that K is usc from Br (0) with the relative W 1,p0 (Z)-norm topology into L p′

(Z) with

the weak topology. Also as before let θ : W 1,p0 (Z) → L p′

(Z) be defined by θ(x)(·) = |x(·)|p−2x(·). Evidently θ is

continuous and so x → θ(x) + K (x) is still usc from Br (0) with relative W 1,p0 (Z)-norm topology into L p′

(Z) withthe weak topology. Then we introduce the multifunction

E(x) = ξ ◦ (λθ + K )(x).

Because of Propositions 3.5 and 3.6, we can use Theorem 2.2 and see that alternative (b) holds, which givesx ∈ W 1,p

0 (Z) such that

x ∈ E(x) = ξ ◦ (λθ + K )(x),

⇒ A(x)− λ|x |p−2x ∈ K (x)− S(x).

From this, we conclude that x ∈ W 1,p0 (Z) solves problem (1.1). �

Remark 3.8. Suppose p = 2, j (z, x) = α(z)x for all x ∈ R with α ∈ L∞(Z) and G = iR+. Then hypotheses H( j)1

(with c = 0) and H(G)1 are satisfied and so according to Theorem 3.7, problem (1.1) has a solution x ∈ H10 (Z) with

x(z) ≥ 0 a.e. on Z . Moreover, in this case from the regularity theory for semilinear variational inequalities (see [20],p. 108) we have that x ∈ H2(Z) ∩ C1

0(Z). This is then Theorem 1(i) of Szulkin [33]. Theorem 3.7 also extendswell-known results from the theory of semilinear variational inequalities; see [12] (Theorem 1.2.7) and [20] (TheoremII.2.1).

Next we will examine problem (1.2). The hypotheses on the nonsmooth potential j (z, x) are the following:


(ii) for almost all z ∈ Z , x → j (z, x) is locally Lipschitz;


(iii) for almost all z ∈ Z , all x ∈ R, and all u ∈ ∂ j (z, x), we have

|u| ≤ α(z)+ c|x | with α ∈ L∞(Z)+, c > 0;

(iv) for almost all z ∈ Z , all x ∈ R, and all u ∈ ∂ j (z, x), we have

ux ≤ c0x2 with 0 < c0 < 1.

Theorem 3.9. If hypotheses H( j)2, H(G)1 hold and λ ≤ λ1, then problem (1.2) has a solution x ∈ H10 (Z).

Proof. As before we consider the integral functional IG : H10 (Z) → R = R ∪ {+∞} defined by

IG(x) =

∫

ZG(x(z))dz if G(x(·)) ∈ L1(Z)

+∞ if otherwise.

We know that IG ∈ Γ0(H10 (Z)). For every y ∈ dom IG = {y ∈ H1

0 (Z) : IG(y) < +∞}, we introduce the set

Lλ(y) =

{x ∈ H1

0 (Z) : 〈A(x), x − y〉 − λ

∫Z

u(x − y)dz + IG(x)− IG(y) ≤ 0 for some u ∈ S2∂ j (·,x(·))

}.

Since IG ∈ Γ0(H10 (Z)), we see that Lλ(y) is closed in H1

0 (Z). Suppose that y1, . . . , yn ∈ dom IG andw ∈ conv{yk}

nk=1 such that w 6∈ ∪

nk=1 Lλ(yk). Then for all k ∈ {1, . . . , n}, we have

〈A(w),w − yk〉 − λ

∫Z

u(w − yk)dz + IG(w)− IG(yk) > 0 for all u ∈ S2∂ j (·,w(·)). (3.15)

We consider the set

E =

{v ∈ H1

0 (Z) : 〈A(w),w − v〉 − λ

∫Z

u(w − v)dz + IG(w)− IG(v) > 0 for all u ∈ S2∂ j (·,w(·))

}.

Clearly E is convex. Moreover, because of (3.15), we have

y1, . . . , yn ∈ E,

⇒ w ∈ E, a contradiction.

Therefore it follows that if y1, . . . , yn ∈ dom IG , then conv{yk}nk=1 ⊆ ∪

nk=1 Lλ(yk). Hence the multifunction

Lλ : dom IG → 2H10 (Z) \ {∅} is a KKM-multifunction (see [11], p. 73). Because of hypotheses H(G)1, we see that

0 ∈ dom IG and since IG(0) = 0, we have

Lλ(0) =

{x ∈ H1

0 (Z) : ‖Dx‖22 − λ

∫Z

uxdz + IG(x) ≤ 0 for some u ∈ S2∂ j (·,x(·))

}.

Hypothesis H( j)2(iv) implies that

u(z)x(z) ≤ c0x(z)2.

So we obtain

‖Dx‖22 − λc0‖x‖

22 + IG(x) ≤ 0,

⇒ ‖Dx‖22 ≤ λc0‖x‖

22 (since IG ≥ 0). (3.16)

If λ ≤ 0, then ‖Dx‖2 = 0 and by Poincare’s inequality we have that x = 0.If 0 < λ ≤ λ1, then since c0 < 1, from (3.16) we have

‖Dx‖22 < λ1‖x‖

22 (3.17)

unless x = 0. But (3.17) contradicts the variational characterization of λ1 > 0 (see (2.2) with m ≡ 1). Thereforeagain x = 0 and so we conclude that Lλ(0) = {0}. This then allows us to apply Ky Fan’s Theorem on KKM-maps


(see [11], p. 73) and infer that⋂

y∈dom IGLλ(y) 6= ∅. If x ∈

⋂y∈dom IG

Lλ(y), then we have

〈A(x), x − y〉 − λ

∫Z

u(x − y)dz ≤ IG(y)− IG(x) for all y ∈ dom IG ,

⇒ −A(x)+ λu ∈ ∂ IG(x).

But we know that ∂ IG(x) = S2∂G(x(·)).

So we have

A(x) = λu − v with u ∈ S2∂ j (·,x(·)), v ∈ S2

∂G(x(·)),

⇒ x ∈ H10 (Z) is a solution of problem (1.2) (see the proof of Proposition 3.3). �

4. Hemivariational inequalities

In this section we focus our attention on problem (1.3).The first existence theorem is obtained using a topological approach based on Theorem 2.3.The hypotheses on the nonsmooth potential j (z, x) are the following:



|u| ≤ α(z)+ c|x |p−1 with α ∈ L∞(Z)+, c > 0;

(iv) there exists θ ∈ L∞(Z)+ such that θ(z) ≤ 1 a.e. on Z with strict inequality on a set of positive measureand

lim sup|x |→∞

u

|x |p−2x≤ θ(z)

uniformly for a.a. z ∈ Z and all u ∈ ∂ j (z, x).

Theorem 4.1. If hypotheses H( j)2 hold and λ ≤ λ1, then problem (1.3) has a solution x ∈ C10(Z).

Proof. Given h ∈ L p′

(Z), we consider the following nonlinear boundary value problem:{−div(‖Dx(z)‖p−2 Dx(z)) = h(z) a.e. on Z ,x |∂Z = 0, 1 < p < ∞.

}(4.1)

Let A : W 1,p0 (Z) → W −1,p′

(Z) be the strictly monotone (since now 1 < p < ∞), maximal monotone operator,introduced in Proposition 3.3. Evidently (4.1) can be equivalently rewritten as the following operator equation:

A(x) = h. (4.2)

Note that 〈A(x), x〉 = ‖Dx‖pp; hence by the Poincare’s inequality it follows that A is coercive. But a maximal

monotone coercive operator is surjective (see [3]). So (4.2) has a solution x ∈ W 1,p0 (Z) and this solution is unique

since A is strictly monotone. Equivalently then (4.1) has a unique solution x ∈ W 1,p0 (Z). So we can define the solution

map ξ : L p′

(Z) → W 1,p0 (Z) as before. Arguing as in the proof of Proposition 3.5, we can check that ξ is completely

continuous.Let Sλ : W 1,p

0 (Z) → Pwkc(L p′

(Z)) be the multifunction defined by

Sλ(x) = S p′

λ∂ j (·,x(·)).

We know that Sλ is usc from W 1,p0 (Z) with the norm topology into L p′

(Z) furnished with the weak topology(see [15], p. 220). We consider the following fixed point problem:

x ∈ (ξ ◦ Sλ)(x). (4.3)


We will solve problem (4.3), using Theorem 2.3. To this end for λ < λ1, we need to show that the set

Cλ = {x ∈ W 1,p0 (Z) : x ∈ t (ξ ◦ Sλ)(x), 0 < t < 1}

is bounded. We proceed by contradiction. So suppose that Cλ is not bounded. We can find a sequence {xn}n≥1 ⊆ Cλsuch that ‖xn‖ → +∞ as n → ∞. We have

xn ∈ tn(ξ ◦ Sλ)(xn) with 0 < tn < 1, n ≥ 1.

We may assume that tn → t ∈ [0, 1]. For every n ≥ 1, we have

1tn

xn = ξ(λun) with un ∈ S p′

∂ j (·,x(·))

⇒ A

(1tn

xn

)= λun,

⇒ A(xn) = t pn λun . (4.4)

Set yn =xn

‖xn‖, n ≥ 1. By passing to a suitable subsequence if necessary, we may assume that

ynw→ y in W 1,p

0 (Z), yn → y in L p(Z), yn(z) → y(z) a.e. on Z

and |yn(z)| ≤ k(z) a.e. on Z for all n ≥ 1, with k ∈ L p(Z).

Because of hypothesis H( j)3(iii), we have

|un(z)|

‖xn‖p−1 ≤α(z)

‖xn‖p−1 + c|yn(z)|p−1 a.e. on Z ,

⇒

{un(·)

‖xn‖p−1

}n≥1

⊆ L p′

(Z) is bounded. (4.5)

Therefore we may assume that

un

‖xn‖p−1w→ h in L p′

(Z).

Given ε > 0 and n ≥ 1, we introduce the sets

C+ε,n =

{z ∈ Z : xn(z) > 0,

un(z)

xn(z)p−1 ≤ θ(z)+ ε

}and C−

ε,n =

{z ∈ Z : xn(z) < 0,

un(z)

|xn(z)|p−2xn(z)≤ θ(z)+ ε

}.

Note that for almost all z ∈ {y > 0}, we have xn(z) → +∞ as n → ∞. So because of hypothesis H( j)3(iv), wehave that

χ+ε,n(z) = χC+

ε,n(z) → 1 a.e. on {y > 0} as n → ∞.

Similarly we obtain

χ−ε,n(z) = χC−

ε,n(z) → 1 a.e. on {y < 0} as n → ∞.

Also from the dominated convergence theorem, we have∥∥∥∥(1 − χ+ε,n)

un

‖xn‖p−1

∥∥∥∥L1({y>0})

→ 0

and

∥∥∥∥(1 − χ−ε,n)

un

‖xn‖p−1

∥∥∥∥L1({y<0})

→ 0 as n → ∞.


So it follows that

χ+ε,n

un

‖xn‖p−1w→ h in L1({y > 0})

and χ−ε,n

un

‖xn‖p−1w→ h in L1({y < 0}) as n → ∞.

We have

χ+ε,n(z)

un(z)

‖xn‖p−1 = χ+ε,n(z)

un

xn(z)p−1 yn(z)p−1

≤ χ+ε,n(z)(θ(z)+ ε)yn(z)

p−1 a.e. on Z .

Taking weak limits in L1({y > 0}) and using Mazur’s lemma, we obtain

h(z) ≤ (θ(z)+ ε)y(z)p−1 a.e. on {y > 0}.

Let ε ↓ 0. We have

h(z) ≤ θ(z)y(z)p−1 a.e. on {y > 0}. (4.6)

In a similar fashion we show that

h(z) ≥ θ(z)|y(z)|p−2 y(z) a.e. on {y < 0}. (4.7)

Finally from (4.5), it is clear that

h(z) = 0 a.e. on {y = 0}. (4.8)

From (4.6)–(4.8) it follows that

h(z) = g(z)|y(z)|p−2 y(z) a.e. on Z

with g ∈ L∞(Z)+ satisfying g(z) ≤ θ(z) a.e. on Z .We return to (4.4), divide by ‖xn‖

p−1 and then take duality brackets with yn . We obtain

‖Dyn‖pp = t p

n λ

∫Z

un

‖xn‖p−1 yndz,

⇒ ‖Dy‖pp ≤ t pλ

∫Z

g|y|pdz ≤ t pλ

∫Zθ |y|

pdz. (4.9)

We may always assume without any loss of generality that θ(z) > 0 a.e. on Z . If λ ≤ 0 or t = 0 or y = 0, thenfrom the above argument we see that Dyn → 0 in L p(Z ,RN ); hence yn → 0 in W 1,p

0 (Z), a contradiction to the factthat ‖yn‖ = 1. So λ ∈ (0, λ1), t ∈ (0, 1] and y 6= 0. Suppose that 0 < t < 1. Then we have

t pλ

∫Zθ |y|

pdz < λ

∫Zθ |y|

pdz ≤ λ1

∫Zθ |y|

pdz ≤ λ1‖y‖pp,

⇒ ‖Dy‖pp < λ1‖y‖

pp (see (4.9)),

a contradiction to the variational characterization of λ1 > 0 (see (2.2) with m = 1). Hence we have t = 1 and so from(4.9) it follows that

‖Dy‖pp ≤ λ

∫Zθ |y|

pdz,

⇒ λ1(θ) ≤ λ (see (2.2) with m = θ). (4.10)

On the other hand from the strict monotonicity of the principal eigenvalue on the weight function, we haveλ1(1) = λ1 < λ1(θ) ≤ λ (see (4.10)), a contradiction to the hypothesis that λ ≤ λ1. Therefore the set Cλ ⊆ W 1,p

0 (Z)

is bounded. We apply Theorem 2.3 and obtain x ∈ W 1,p0 (Z) which satisfies (4.3). As before we can check that

x ∈ W 1,p0 (Z) solves problem (2.2). From nonlinear regularity theory we have that x ∈ C1(Z). �


In the previous theorem at infinity we allowed only partial interaction of the “slope” u|x |p−2x

with λ1 > 0 (seehypothesis H( j)3(iv).) This condition can be viewed as a nonuniform nonresonance condition. If we strengthen ourhypotheses on j (z, x), by imposing a uniform nonresonance condition at zero, we can permit full interaction at infinitywith λ1 > 0 (resonant problems). The new hypotheses on the nonsmooth potential are the following:

H( j)4: j : Z × R → R is a function such that j (z, 0) = 0 a.e. on Z and:(i) for all x ∈ R, z → j (z, x) is measurable;


|u| ≤ α(z)+ c|x |p−1 with α ∈ L∞(Z)+, c > 0;

(iv) lim sup|x |→∞u

|x |p−2x≤ 1 uniformly for a.a. z ∈ Z and all u ∈ ∂ j (z, x);

(v) lim supx→0u

|x |p−2x< 1 uniformly for a.a. z ∈ Z and all u ∈ ∂ j (z, x).

Theorem 4.2. If hypotheses H( j)4 hold and λ ≤ λ1, then problem (1.3) has a solution x ∈ C1(Z).

Proof. A careful reading of the proof of Theorem 4.1 reveals that it applies in the present situation when λ < λ1 andit gives us a solution x ∈ C1(Z).

So we need to consider the case λ = λ1. We will treat this case using Theorem 2.2 (as we did in Theorem 3.7), butthis time for small balls. Again we argue indirectly. So suppose we can find {tn}n≥1 ⊆ (0, 1) and {xn}n≥1 ⊆ W 1,p

0 (Z)such that

tn → t in [0, 1], xn → 0 in W 1,p0 (Z)

and A

(1tn

xn

)= λ1un with un ∈ S p′

∂ j (·,xn(·)).

Then for all n ≥ 1, we have

A(xn) = λ1t pn un,

⇒ ‖Dx‖pp = λ1t p

n

∫Z

un xndz. (4.11)

Let yn =xn

‖xn‖n ≥ 1. We may assume that

ynw→ y in W 1,p

0 (Z) and yn → y in L p(Z).

We divide (4.11) with ‖xn‖p and obtain

‖Dyn‖pp =

λ1t pn

‖xn‖p

∫Z

un xndz. (4.12)

By virtue of hypothesis H( j)4(v), we can find µ < 1 and δ = δ(µ) > 0 such that

ux ≤ µ|x |p (4.13)

for almost all z ∈ Z , all |x | < δ and all u ∈ ∂ j (z, x). Combining this with hypothesis H( j)4(iii), we see that

ux ≤ c1|x |p (4.14)

for some c1 = c1(δ, ‖α‖∞, c) > 0 and for almost all z ∈ Z , all |x | ≥ δ and all u ∈ ∂ j (z, x). Then we have

λ1t pn

‖xn‖p

∫Z

un xndz =λ1t p

n

‖xn‖p

∫{|xn |<δ}

un xndz +λ1t p

n

‖xn‖p

∫{|xn |≥δ}

un xndz

≤λ1t p

n

‖xn‖p

∫{|xn |<δ}

µ|xn|pdz +

λ1t pn

‖xn‖p

∫{|xn |≥δ}

c1|xn|pdz (4.15)

(see (4.13) and (4.14)).


Since xn → 0 in W 1,p0 (Z), by passing to a suitable subsequence if necessary, we may assume that xn(z) → 0 a.e.

on Z as n → ∞. Then

|{|xn| < δ}|N → |Z |N and |{|xn| ≥ δ}|N → 0 as n → ∞.

Recall that by | · |N , we denote the Lebesgue measure on RN . Hence if we pass to the limit as n → ∞ in (4.15),we obtain

lim supn→∞

λ1t pn

‖xn‖p

∫Z

unxndz ≤ λ1t p∫

Zµ|y|

pdz. (4.16)

Returning to (4.12), passing to the limit as n → ∞ and using (4.15), we have

‖Dy‖pp ≤ λ1µt p

‖y‖pp. (4.17)

If t = 0 or y = 0, then yn → 0 in W 1,p0 (Z), a contradiction to the fact that ‖yn‖ = 1 for all n ≥ 1. So t ∈ (0, 1)

and y 6= 0. Because 0 < µ < 1, we have

‖Dy‖pp < λ1‖y‖

pp ≤ ‖Dy‖

pp (see (4.17) and (2.2) with m = 1),

a contradiction. So for r > 0 small, we can apply Theorem 2.2 and obtain x ∈ W 1,p0 (Z) such that

x ∈ (ξ ◦ Sλ1)(x),

⇒ A(x) = λ1u with u ∈ S p′

∂ j (·,x(·)).

As before, from the above equation we deduce that x ∈ W 1,p0 (Z) is a solution of problem (1.3) and in addition

nonlinear regularity theory implies that x ∈ C10(Z). �

Remark 4.3. In Theorems 4.1 and 4.2 we cannot guarantee in general that the solution x ∈ C10(Z) is nontrivial.

Indeed, if in Theorem 4.1 j (z, x) = θ(z)|x |p−2x with θ ∈ L∞(Z)+ as in hypothesis H( j)3(iv), then for λ < λ1(θ)

problem (1.3) has only the trivial solution. Similarly, if in Theorem 4.2 j (z, x) = j (x) = |x |p−2x , then for λ < λ1

problem (1.3) has only the trivial solution.

Thus far the existence theorems that we have proved guarantee solutions for problem (1.3) as λ ranges on the half-line below λ1 > 0. Next we will produce existence results for when λ belongs in half-lines above λ1 > 0. To do thiswe will switch from the topological to variational methods in our approach.

The hypotheses on the nonsmooth potential are now the following:

H( j)5: j : Z × R → R is a function such that j (·, 0) ∈ L∞(Z),∫

Z j (z, 0)dz ≥ 0 and:(i) for all x ∈ R, z → j (z, x) is measurable;

(ii) for almost all z ∈ Z , x → j (z, x) is locally Lipschitz;(iii) for almost all z ∈ Z , all x ∈ R and all u ∈ ∂ j (z, x), we have

|u| ≤ α(z)+ c|x |p−1 with α ∈ L∞(Z)+, c > 0;

(iv) there exist functions θ1, θ2 ∈ L∞(Z)+ such that 1 ≤ θ1(z) ≤ θ2(z) a.e. on Z , with the first inequalitybeing strict on a set of positive measure and

θ1(z) ≤ lim infx→+∞

u

x p−1 ≤ lim supx→+∞

u

x p−1 ≤ θ2(z) and limx→−∞

u

|x |p−2x= 0

uniformly for a.a. z ∈ Z and all u ∈ ∂ j (z, x);(v) lim supx→0

pj (z,x)|x |p ≤ 0 uniformly for a.a. z ∈ Z .

Remark 4.4. The following nonsmooth locally Lipschitz function satisfies hypotheses H( j)5. For simplicity we dropthe z-dependence:

j (x) =

sin

π

2x if x < −1

−|x |p if |x | ≤ 1

c|x |p

− c − 1 if x > 1,

with c > 1.


As we already mentioned, the approach will be variational. For this purpose we introduce the Euler functionalϕλ : W 1,p

0 (Z) → R for problem (1.3), which is defined by

ϕλ(x) =1p‖Dx‖

pp − λ

∫Z

j (z, x(z))dz for all x ∈ W 1,p0 (Z).

We know that ϕλ is Lipschitz continuous on bounded sets, and hence locally Lipschitz (see [8], p. 83).

Proposition 4.5. If hypotheses H( j)5 hold and λ ≥ λ1, then ϕλ satisfies the nonsmooth PS-condition.

Proof. Let {xn}n≥1 ⊆ W 1,p0 (Z) be a sequence such that

|ϕλ(xn)| ≤ M1 for some M1 > 0, all n ≥ 1 and mλ(xn) → 0 as n → ∞.

Since ∂ϕ(xn) ⊆ W −1,p′

(Z) is weakly compact and the norm functional in a Banach space is weakly lowersemicontinuous, by the Weierstrass theorem we can find x∗

n ∈ ∂ϕλ(xn) such that mλ(xn) = ‖x∗n‖ for all n ≥ 1.

We have

x∗n = A(xn)− λun

with A : W 1,p0 (Z) → W −1,p′

(Z) as in the proof of Proposition 3.3 and un ∈ L p′

(Z), un(z) ∈ ∂ j (z, xn(z)) a.e. on Z .

We claim that {xn}n≥1 ⊆ W 1,p0 (Z) is bounded. Suppose that this is not true. Then by passing to a subsequence if

necessary, we may assume that ‖xn‖ → +∞ as n → ∞. Set yn =xn

‖xn‖, n ≥ 1. We may assume that

ynw→ y in W 1,p

0 (Z), yn → y in L p(Z), yn(z) → y(z) a.e. on Z

and |yn(z)| ≤ k(z) a.e. on Z , for all n ≥ 1 and with k ∈ L p(Z).

As before using hypothesis H( j)5(iii), we have

|un(z)|

‖xn‖p−1 ≤α(z)

‖xn‖p−1 + c|yn(z)|p−1 a.e. on Z , (4.18)

⇒

{un

‖xn‖p−1

}n≥1

⊆ L p′

(Z) is bounded.

So we may assume thatun

‖xn‖p−1w→ h in L p′

(Z) as n → ∞.

Given ε > 0 and n ≥ 1, we introduce the sets

C+ε,n =

{z ∈ Z : xn(z) > 0, θ1(z)− ε ≤

un(z)

xn(z)p−1 ≤ θ2(z)+ ε

}and C−

ε,n =

{z ∈ Z : xn(z) ≤ 0,−ε ≤

un(z)

|xn(z)|p−2xn(z)≤ ε

}.

Note that for a.a. z ∈ {y > 0} xn(z) → +∞, while for a.a. z ∈ {y < 0} xn(z) → −∞ as n → ∞. So hypothesisH( j)5(iv) implies that

χ+ε,n(z) = χC+

ε,n(z) → 1 a.e. on {y > 0}

and χ−ε,n(z) = χC−

ε,n(z) → 1 a.e. on {y < 0}.

Arguing as in the proof of Theorem 4.1, we obtain

θ1(z)y(z)p−1

≤ h(z) ≤ θ2(z)y(z)p−1 a.e. on {y > 0} (4.19)

and h(z) = 0 a.e. on {y < 0}. (4.20)

Moreover, from (4.18), it is clear that

h(z) = 0 a.e. on {y = 0}. (4.21)


From (4.19)–(4.21) it follows that

h(z) = g(z)y+(z)p−1 a.e. on Z , (4.22)

with g ∈ L∞(Z)+ satisfying θ1(z) ≤ g(z) ≤ θ2(z) a.e. on Z (recall y+= max{y, 0}).

From the choice of the sequence {xn}n≥1 ⊆ W 1,p0 (Z), we have

|〈x∗n , yn − y〉| =

∣∣∣∣〈A(xn), yn − y〉 − λ

∫Z

un(yn − y)dz

∣∣∣∣ ≤ εn‖yn − y‖ with εn ↓ 0.

Dividing by ‖xn‖p−1, we obtain∣∣∣∣〈A(yn), yn − y〉 − λ

∫Z

un

‖xn‖p−1 (yn − y)dz

∣∣∣∣ ≤εn

‖xn‖p−1 ‖yn − y‖.

Since∫

Zun

‖xn‖p−1 (yn − y)dz → 0, it follows that

limn→∞

〈A(yn), yn − y〉 = 0.

From this as in the proof of Proposition 3.5 exploiting the generalized pseudomonotonicity of A and theKadec–Klee property of L p(Z ,RN), we infer that yn → y in W 1,p

0 (Z). For every n ≥ 1 and every v ∈ W 1,p0 (Z), we

have ∣∣∣∣〈A(yn), v〉 − λ

∫Z

un

‖xn‖p−1 vdz

∣∣∣∣ ≤ εn‖v‖,

⇒ 〈A(y), v〉 = λ

∫Z

g(y+)p−1vdz for all v ∈ W 1,p0 (Z) (see (4.21)). (4.23)

It follows that{−div(‖Dy+(z)‖p−2 Dy+(z)) = λg(z)y+(z)p−1 a.e. on Z ,y+

|∂Z = 0

}(4.24)

(recall that Dy+(z) = 0 a.e. on {y ≤ 0}). Exploiting the strict monotonicity of the weight function on the weight, wehave

λ1(g) ≤ λ1(θ1) < λ1(1) = λ1 ≤ λ.

So λ cannot be the principal eigenvalue of (−∆p,W 1,p0 (Z)) with weight g ∈ L∞(Z)+ and so any eigenfunction

corresponding to λ must change sign. This combined with (4.24) implies that y+= 0. So from (4.23) we have

A(y) = 0, and hence y = 0, a contradiction to the fact that ‖yn‖ = 1 for all n ≥ 1. This proves that{xn}n≥1 ⊆ W 1,p

0 (Z) is bounded and so we may assume that

xnw→ x in W 1,p

0 (Z) and xn → x in L p(Z).

As above from the choice of {xn}n≥1 ⊆ W 1,p0 (Z) and since

∫Z un(xn − x)dz → 0, we obtain

〈A(xn), xn − x〉 → 0,

⇒ xn → x in W 1,p0 (Z).

So ϕλ satisfies the nonsmooth PS-condition for λ ≥ λ1. �

Proposition 4.6. If hypotheses H( j)5 hold and λ ∈ R, then there exists ρ = ρ(λ) > 0 small such that inf{ϕλ(x) :

‖x‖ = ρ} ≥ β0(λ) > 0.

Proof. By virtue of hypothesis h( j)5(v), given ε > 0, we can find δ = δ(ε) > 0 such that

j (z, x) ≤ε

p|x |

p for a.a. z ∈ Z and all x ∈ (−δ, δ).


On the other hand because of hypothesis H( j)5(iii) and the mean value theorem for locally Lipschitz functions

(see [8], p. 41), we have j (z, x) ≤ c0|x |τ for a.a. z ∈ Z , all |x | ≥ δ with c0 > 0 and p < τ < p∗

=

{N p

N − pif p < N

+∞ if p ≥ N.

Therefore it follows that

j (z, x) ≤ε

p|x |

p+ c0|x |

τ for a.a. z ∈ Z and all x ∈ R. (4.25)

For x ∈ W 1,p0 (Z), we have

ϕλ(x) =1p‖Dx‖

pp − λ

∫Z

j (z, x(z))dz

≥1p‖Dx‖

pp −

λε

p‖x‖

pp − c1‖Dx‖

τp for some c1 > 0.

Here in obtaining the inequality we used the continuous (in fact compact) embedding of W 1,p0 (Z) into Lτ (Z)

(recall that p < τ < p∗) and Poincare’s inequality. Then

ϕλ(x) ≥1p

(1 −

λε

λ1

)‖Dx‖

pp − c1‖Dx‖

τp (see (2.2) with m = 1). (4.26)

For given λ > 0, we choose ε = ε(λ) > 0 such that ε < λ1λ

. Then from (4.26) and Poincare’s inequality we have

ϕλ(x) ≥ c(λ)‖x‖p

− c3‖x‖τ for some c(λ), c3 > 0, all x ∈ W 1,p

0 (Z). (4.27)

Since p < τ , if we choose ρ = ρ(λ) > 0 small from (4.27), we see that

inf{ϕλ(x) : ‖x‖ = ρ} ≥ β0(λ) > 0. �

Proposition 4.7. If hypotheses H( j)5 hold and λ ≥ λ1, then ϕλ|R+u1 is weakly anticoercive, i.e. ϕλ(tu1) → −∞ ast → +∞.

Proof. From hypothesis H( j)5(iv), we see that given ε > 0, we can find M = M(ε) > 0 such that

(θ1(z)− ε)x p−1≤ u for a.a. z ∈ Z , all x ≥ M and all u ∈ ∂ j (z, x). (4.28)

On the other hand from hypothesis H( j)5(iii), we have

|u| ≤ c4 for a.a. z ∈ Z , all 0 ≤ x ≤ M and all u ∈ ∂ j (z, x), with c4 > 0. (4.29)

Putting together (4.28) and (4.29), we obtain

(θ1(z)− ε)x p−1− c4 ≤ u for a.a. z ∈ Z , all x ≥ 0 and all u ∈ ∂ j (z, x). (4.30)

Because of hypothesis H( j)5(ii), for all z ∈ Z \ D, |D|N = 0, the function j (z, ·) is differentiable a.e. on R andat every point of differentiability r ∈ R, we have d

dr j (z, r) ∈ ∂ j (z, r). So for almost all z ∈ Z and all x ≥ 0, we have

j (z, x) = j (z, 0)+

∫ x

0

ddr

j (z, r)dr

≥ j (z, 0)+

∫ x

0(θ1(z)− ε)r p−1dr − c4x

= j (z, 0)+1p(θ1(z)− ε)x p

− c4x . (4.31)

Let u1 ∈ C10(Z) be the principal eigenfunction of (−∆p,W 1,p

0 (Z)) and let t > 0. We have


ϕλ(tu1) =t p

p‖Du1‖

pp − λ

∫Z

j (z, tu1(z))dz

≤t p

p‖Du1‖

pp −

λt p

p

∫Z(θ1(z)− ε)u1(z)

pdz − λtc5‖u1‖p − λc6 for some c5, c6 > 0 (see (4.31))

=t p

p

[∫Z(λ1 − λθ1(z))u1(z)

pdz

]+λt pε

p‖u1‖

pp − λtc5‖u1‖p − λc6 (since ‖Du1‖

pp = λ1‖u1‖

pp)

≤t p

p

[∫Zλ1(1 − θ(z))u1(z)

pdz + λε

]− λtc5 − λc6

(since ‖u1‖p = 1, λ ≥ λ1 and θ ∈ L∞(Z)+). (4.32)

Note that∫Zλ1(1 − θ(z))u1(z)

pdz = γ < 0.

(see hypothesis H( j)5(iv) and recall that u1(z) > 0 for all z ∈ Z ). So from (4.32) we have

ϕλ(tu1) ≤t p

p(γ + λε)− λtc5 − λc6. (4.33)

Choose ε ∈ (0,− γλ). Then γ + λε < 0 and so from (4.33) we conclude that

ϕλ(tu1) → −∞ as t → +∞. �

So the geometry of the nonsmooth mountain pass theorem (see Theorem 2.5) is in place and we will use it toproduce a solution for problem (1.3). We have the following existence result.

Theorem 4.8. If hypotheses H( j)5 hold and λ ≥ λ1, then problem (1.3) has a nontrivial solution x ∈ C10(Z).

Proof. By virtue of Proposition 4.7, we can find t > 0 large enough so that ‖tu1‖ > ρ and

ϕλ(tu1) < 0 < β0(λ) ≤ inf[ϕλ(x) : ‖x‖ = ρ] (see Proposition 4.6). (4.34)

Moreover, we have

ϕλ(0) = −λ

∫Z

j (z, 0)dz ≤ 0. (4.35)

Then Proposition 4.5, combined with (4.34) and (4.35), permits the use of Theorem 2.5, which gives x ∈ W 1,p0 (Z)

such that

ϕλ(x) ≥ β0(λ) > 0 ≥ ϕλ(0) (4.36)

and 0 ∈ ∂ϕλ(x). (4.37)

From (4.36) we see that x 6= 0, while from (4.37) we have

A(x) = λu with u ∈ S p′

∂ j (·,x(·)).

As before from the above operator equation, we deduce that x ∈ W 1,p0 (Z) is a solution of problem (1.3) and from

nonlinear regularity theory, we have x ∈ C10(Z). �

Thus far the results involved potential functions exhibiting p-growth. The next theorem concerns problems wherethe potential function is p-superlinear. More precisely the hypotheses on the nonsmooth potential j (z, x) are thefollowing:


(ii) for almost all z ∈ Z , x → j (z, x) is locally Lipschitz;


(iii) for almost all z ∈ Z , all x ∈ R, and all u ∈ ∂ j (z, x), we have

|u| ≤ α(z)+ c|x |r−1 with α ∈ L∞(Z)+, c > 0, p < r < p∗

;

(iv) there exists µ > p such that for a.a. z ∈ Z and all x ∈ R, we have

µj (z, x) ≤ − j0(z, x; −x);

(v) there exist η ∈ L∞(Z)+ and δ > 0 such that η(z) ≤ 1 a.e. on Z with strict inequality on a set of positivemeasure and

lim supx→0

pj (z, x)

|x |p ≤ η(z)

uniformly for a.a. z ∈ Z and also

j (z, x) > 0 for a.a. z ∈ Z all x ∈ (0, δ).

Remark 4.9. The first part of hypothesis H( j)6(v) (the one concerning the limit supremum) is similar to H( j)4(v).Nevertheless there are some noteworthy differences. Namely hypothesis H( j)4(v) was a uniform nonresonancecondition, while the new hypothesis H( j)6(v) is a nonuniform nonresonance condition, i.e. we allow interactionwith the spectrum on a set of positive measure. Another difference is that while condition H( j)4(v) was formulatedin terms of the subdifferential ∂ j (z, x), the present condition H( j)6(v) is in terms of the potential j (z, x). The latterformulation is in general more restrictive as the next example illustrates. Consider a potential function j (x) which ina neighborhood of zero has the form j (x) =

12p e|x |

p. Then ∂ j (x) = j ′(x) =

12 |x |

p−2xe|x |p. We have

limx→0

pj (z, x)

|x |p = limx→0

e|x |p

2|x |p = +∞.

but limx→0

∂ j (x)

|x |p−2x= lim

x→0

12

e|x |p

=12< 1.

Also we would like to point out that hypothesis H( j)6(iv) is a nonsmooth global version of the well-knownAmbrosetti–Rabinowitz condition (see [30], p. 9 and [2]). However, here we do not require that j (z, x) > 0 fora.a. z ∈ Z and all x ∈ R (compare with [2], hypotheses ( j4) and ( j5)) The nonsmooth, locally Lipschitz functionj (x) =

1µ|x |

µ− |x | ln |x | with p < µ < p∗, satisfies hypotheses H( j)6.

We start with a lemma, which highlights the significance of the nonuniform nonresonance condition in hypothesisH( j)6(v).

Lemma 4.10. If η1 ∈ L∞(Z)+ and η1(z) ≤ λ1 a.e. on Z with strict inequality on a set of positive measure, thenthere exists ξ > 0 such that

ψ(x) = ‖Dx‖pp −

∫Zη1(z)|x(z)|

pdz ≥ ξ‖Dx‖pp for all x ∈ W 1,p

0 (Z).

Proof. By virtue of the variational characterization of λ1 > 0 (see (2.2) with m = 1), we have that ψ ≥ 0. Supposethat the lemma is not true. Then because ψ is p-homogeneous, we can find {xn}n≥1 ⊆ W 1,p

0 (Z) such that

‖Dxn‖p = 1 for all n ≥ 1 and ψ(xn) ↓ 0.

From the Poincare inequality, we have that {xn}n≥1 ⊆ W 1,p0 (Z) is bounded and so we may assume that xn

w→ x in

W 1,p0 (Z) and xn → x in L p(Z). Because the norm functional in a Banach space is weakly lower semicontinuous, we

see that ψ is weakly lower semicontinuous and so

ψ(x) = ‖Dx‖pp −

∫Zη1(z)|x(z)|

pdz ≤ 0 = limn→∞

ψ(xn), (4.38)

⇒ ‖Dx‖pp ≤ λ1‖x‖

pp.

So x = 0 or x = ±u1 (see (2.2) with m = 0).If x = 0, then ‖Dxn‖p → 0, a contradiction to the fact that ‖Dxn‖p = 1 for all n ≥ 1.


If x = ±u1, then |x(z)| > 0 for all z ∈ Z and so from the first inequality in (4.38) and the hypothesis onη1 ∈ L∞(Z)+, we have

‖Dx‖pp < λ1‖x‖

pp

which contradicts (2.2) (with m = 1). �

Theorem 4.11. If hypotheses H( j)6 hold and 0 < λ ≤ (ξ + 1)λ1 with ξ > 0 as in Lemma 4.10, then problem (1.3)has a nontrivial solution x ∈ C1

0(Z).

Proof. As before we consider the locally Lipschitz Euler functional ϕλ : W 1,p0 (Z)→ R defined by

ϕλ(x) =1p‖Dx‖

pp − λ

∫Z


First we show that ϕλ satisfies the nonsmooth PS-condition. To this end let {xn}n≥1 ⊆ W 1,p0 (Z) be such that

|ϕλ(xn)| ≤ M1 for some M1 > 0, all n ≥ 1 and mλ(xn) → 0 as n → ∞.

We can find x∗n ∈ ∂ϕλ(xn) such that mλ(xn) = ‖x∗

n‖. We have

x∗n = A(xn)− λun with un ∈ S p′

∂ j (·,xn(·)), n ≥ 1.

We have

µ

p‖Dxn‖

pp − λµ

∫Z

j (z, xn(z))dz ≤ µM1 (4.39)

and

∣∣∣∣〈A(xn), xn〉 − λ

∫Z

un xndz

∣∣∣∣ ≤ εn‖xn‖ with εn ↓ 0,

⇒ −‖Dxn‖pp + λ

∫Z

un xndz ≤ εn‖xn‖,

⇒ −‖Dxn‖pp − λ

∫Z

j0(z, xn(z); −xn(z))dz ≤ εn‖xn‖. (4.40)

Adding (4.39) and (4.40), we obtain(µ

p− 1

)‖Dxn‖

pp −

∫Zλ

[µj (z, xn(z))+ j0(z, xn(z); −xn(z))

]dz ≤ εn‖xn‖ + µM1,

⇒

(µ

p− 1

)‖Dxn‖

pp ≤ εn‖xn‖ + µM1 (see hypothesis H( j)6(iv))

⇒ {xn}n≥1 ⊆ W 1,p0 (Z) is bounded (by Poincare’s inequality and since µ > p).

The arguing as in the proof of Proposition 4.5, through the generalized pseudomonotonicity of A and theKadec–Klee property, we conclude that ϕλ satisfies the nonsmooth PS-condition.

From hypotheses H( j)6(iii) and (v) and the mean value theorem for locally Lipschitz functions, we see that givenε > 0, we can find c2(ε) > 0 such that

j (z, x) ≤1p(η(z)+ ε)|x |

p+ c2(ε)|x |

τ (4.41)

for a.a. z ∈ Z and all x ∈ R with p < τ < p∗ (see also the proof of Proposition 4.6). Then for all x ∈ W 1,p0 (Z), we

have

ϕλ(x) =1p‖Dx‖

pp − λ

∫Z

j (z, x(z))dz

≥1p‖Dx‖

pp −

λ

p

∫Zη|x |

pdz −λε

p‖x‖

pp − λc3(ε)‖Dx‖

τp for some cε > 0 (see (4.41))

≥ξ

p‖Dx‖

pp −

(λ− λ1)

p

∫Zη|x |

pdz −λε

λ1 p‖Dx‖


τp (see Lemma 4.10).


Recall that 0 < λ ≤ (ξ + 1)λ1, use again Lemma 4.10 and choose ε < ξ2 λ1λ

. We obtain

ϕλ(x) ≥ ξ1‖Dx‖pp − ξ2‖Dx‖

τp with ξ1, ξ2 > 0 (depending on λ > 0).

So we can find ρ = ρ(λ) > 0 small such that

inf[ϕλ(x) : ‖x‖ = ρ] ≥ β > 0 = ϕλ(0). (4.42)

On the other hand, on R+ \ {0} the function r →1

rµ is continuous, convex, and hence locally Lipschitz. So

r →j (z,r x)

rµ is locally Lipschitz on R+ \ {0} for almost all z ∈ Z and we have

∂r

(1

rµj (z, r x)

)⊆ −

µ

rµ+1 j (z, r x)+1

rµ∂x j (z, r x)x

(see [8], p. 48). Here by ∂r (resp. ∂x ) we denote the generalized subdifferential of the corresponding function withrespect to r ∈ R+ \ {0} (resp. with respect to x ∈ R). Using the mean value theorem for locally Lipschitz functions,for r > 1 we can find γ ∈ (1, r) such that

1rµ

j (z, r x)− j (z, x) ∈

(−

µ

γ µ+1 j (z, γ x)+1γ µ∂x j (z, γ x)x

)(r − 1),

⇒1

rµj (z, r x)− j (z, x) ≥

r − 1

γ µ+1

(−µj (z, γ x)− j0(z, γ x; −γ x)

)≥ 0 (see hypothesis H( j)6(iv))

⇒ rµ j (z, x) ≤ j (z, r x) for a.a. z ∈ Z , all x ∈ R and all r ≥ 1. (4.43)

Choose θ > 0 small so that u1(z) = θu1(z) ≤ δ for all z ∈ Z (recall that u1 ∈ intC10(Z)+). Then for t > 1, we

have

ϕλ(t u1) =t p

p‖Du1‖

pp − λ

∫Z

j (z, t u1(z))dz

≤t p

pλ1‖u1‖

pp − λtµ

∫Z

j (z, u1(z))dz (see (4.43))

= t p[λ1

p‖u1‖

pp − λtµ−p

∫Z

j (z, u1(z))dz

]. (4.44)

By virtue of the second part of the hypothesis H( j)6(v), we have∫

Z j (z, u1(z))dz > 0. So from (4.44) and sinceµ > p, we infer that

ϕλ(t u1) → −∞ as t → +∞ (since 0 < λ)

Therefore we can find t > 0 large such that

‖t u1‖ > ρ and ϕλ(t u1) ≤ 0.

Combining this with (4.42) and the fact that ϕλ satisfies the nonsmooth PS-condition, we see that we can applyTheorem 2.5 (the nonsmooth mountain pass theorem) and obtain x ∈ W 1,p

0 (Z) such that

ϕλ(x) ≥ β > 0 = ϕλ(0) and 0 ∈ ∂ϕλ(x). (4.45)

From the inequality in (4.45), we see that x 6= 0, while from the inclusion we see that x ∈ W 1,p0 (Z) is a solution

of problem (1.3) and from nonlinear regularity theory, we have x ∈ C10(Z). �

We conclude with a multiplicity result for problem (1.3). The hypotheses on the nonsmooth potential are thefollowing:

H( j)7: j : Z × R → R is a function such that j (z, 0) = 0 a.e. on Z and(i) for all x ∈ R, z → j (z, x) is measurable;

(ii) for almost all z ∈ Z , x → j (z, x) is locally Lipschitz;(iii) for almost all z ∈ Z , all x ∈ R and all u ∈ ∂ j (z, x), we have

|u| ≤ α(z)+ c|x |r−1 with α ∈ L∞(Z)+, c > 0, 1 < r < p∗

;


(iv) for almost all z ∈ Z and all x ∈ R, we have j (z, x) ≤ γ (z) with γ ∈ L1(Z);(v) lim supx→0

pj (z,x)|x |p ≤ 0 uniformly for a.a. z ∈ Z;

(vi) there exists x0 ∈ R \ {0} such that∫

Z j (z, x0)dz > 0.

Remark 4.12. The following nonsmooth locally Lipschitz functions satisfy hypotheses H( j)7. For simplicity wedrop the z-dependence.

j1(x) =

{−|x |

p+ |x |

s if |x | ≤ 1

cos(π

2|x |

)if |x | > 1,

with 1 < p < s < p∗

and j2(x) =

− sin

(π2

|x |p)

if x ≤ 11

√x

− 2 if x > 1.

Theorem 4.13. If hypotheses H( j)7 hold, then there exists λ0 > 0 such that for all λ ≥ λ0 problem (1.3) has at leasttwo distinct nontrivial solutions x0, x1 ∈ C1

0(Z).

Proof. As before we consider the locally Lipschitz Euler functional ϕλ : W 1,p0 (Z) → R defined by

ϕλ(x) =1p‖Dx‖

pp − λ

∫Z


Because of hypotheses H( j)7(iv), for all x ∈ W 1,p0 (Z), we have

ϕλ(x) ≥1p‖Dx‖

pp − λ‖γ ‖1,

which by Poincare’s inequality implies that ϕλ is coercive. Also because of the compact embedding of W 1,p0 (Z)

into L p(Z), we can easily check that ϕλ is weakly lower semicontinuous on W 1,p0 (Z). So invoking the Weierstrass

theorem, we can find x0 ∈ W 1,p0 (Z) such that

ϕλ(x0) = µλ = infW 1,p

0 (Z)ϕλ. (4.46)

The integral function I j : L p(Z) → R defined by I j (v) =∫

Z j (z, v(z))dz is continuous and from hypothesis

H( j)7(vi) we have I j (x0) > 0. Since W 1,p0 (Z) is dense in L p(Z), we can find y ∈ W 1,p

0 (Z) such that

I j (y) =

∫Z

j (z, y(z))dz > 0.

Choose λ0 > 0 large so that

1p‖Dy‖

pp − λ0

∫Z

j (z, y(z))dz < 0. (4.47)

Then for all λ ≥ λ0, we have

ϕλ(y) =1p‖Dy‖

pp − λ

∫Z

j (z, y(z))dz

≤1p‖Dy‖

pp − λ0

∫Z

j (z, y(z))dz

< 0 = ϕλ(0) (see (4.47) and recall that j (z, 0) = 0 a.e. on Z).

So it follows that

ϕλ(x0) = µλ ≤ ϕλ(y) < 0 = ϕλ(0) for λ ≥ λ0,

⇒ x0 6= 0.


Also (4.46) implies that

0 ∈ ∂ϕλ(x0),

⇒ A(x0) = λu0 with u0 ∈ Sr ′

∂ j (·,x0(·))

(1r

+1r ′

= 1).

As before this operator equation implies that x0 ∈ W 1,p0 (Z) is a nontrivial solution of problem (1.3) and the

nonlinear regularity theory implies that x0 ∈ C10(Z).

As in the proof of Proposition 4.6, using hypotheses H( j)7(iii) and (v), we see that given ε > 0, we can findc(ε) > 0 such that

j (z, x) ≤ε

p|x |

p+ c(ε)|x |

τ for a.a. z ∈ Z , all x ∈ R and with p < τ < p∗. (4.48)

Then for every λ ≥ λ0 and every x ∈ W 1,p0 (Z), we have

ϕλ(x) =1p‖Dx‖

pp − λ

∫Z

j (z, x(z))dz

≥1p‖Dx‖

pp −

λε

p‖x‖


τp for some c1(ε) > 0 (see (4.48))

≥1p

(1 −

λε

λ1

)‖Dx‖


τp. (4.49)

Choose ε < λ1λ. Then from (4.49) and Poincare’s inequality, we see that we can find ρ = ρ(λ) > 0 small such that

inf{ϕλ(x) : ‖x‖ = ρ} ≥ β1 > 0 = ϕλ(0) > ϕλ(x0). (4.50)

Since ϕλ is coercive, we can easily check that it satisfies the nonsmooth PS-condition. Combining this fact with(4.50), we see that we can apply Theorem 2.5 (the nonsmooth mountain pass theorem) and obtain x1 ∈ W 1,p

0 (Z) suchthat

ϕλ(x1) ≥ β1 > 0 = ϕλ(0) > ϕλ(x0). (4.51)

As before the inequality in (4.51) implies that x1 6= 0, x1 6= x0, while the inclusion means that x1 solves (1.3) andx1 ∈ C1

0(Z) (nonlinear regularity theory). �

Acknowledgments

The authors wish to thank the referee for providing additional references. The first author’s research was supportedby a grant of the National Scholarship Foundation of Greece (I.K.Y.)

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