Ground states of critical and supercritical problems of

Annali di Matematica (2016) 195:1787–1802DOI 10.1007/s10231-015-0548-1

Ground states of critical and supercritical problemsof Brezis–Nirenberg type

Mónica Clapp1 · Angela Pistoia2 · Andrzej Szulkin3

Received: 16 April 2015 / Accepted: 17 December 2015 / Published online: 6 January 2016© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2016

Abstract We study the existence of symmetric ground states to the supercritical problem

−�v = λv + |v|p−2 v in �, v = 0 on ∂�,

in a domain of the form

� = {(y, z) ∈ Rk+1 × R

N−k−1 : (|y| , z) ∈ �},where � is a bounded smooth domain such that � ⊂ (0,∞) × R

N−k−1, 1 ≤ k ≤ N − 3,λ ∈ R, and p = 2(N−k)

N−k−2 is the (k + 1)-st critical exponent. We show that symmetric groundstates exist for λ in some interval to the left of each symmetric eigenvalue and that nosymmetric ground states exist in some interval (−∞, λ∗) with λ∗ > 0 if k ≥ 2. Related tothis question is the existence of ground states to the anisotropic critical problem

−div(a(x)∇u) = λb(x)u + c(x) |u|2∗−2 u in �, u = 0 on ∂�,

where a, b, c are positive continuous functions on �. We give a minimax characterizationfor the ground states of this problem, study the ground state energy level as a function of λ,and obtain a bifurcation result for ground states.

M. Clapp is partially supported by CONACYT Grant 237661 and UNAM-DGAPA-PAPIIT Grant IN104315(Mexico), and A. Pistoia is partially supported by PRIN 2009-WRJ3W7 Grant (Italy).

B Angela [email protected]

Mónica [email protected]

Andrzej [email protected]

1 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U.,04510 México, D.F., Mexico

2 Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, Via AntonioScarpa 16, 00161 Rome, Italy

3 Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden

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http://crossmark.crossref.org/dialog/?doi=10.1007/s10231-015-0548-1&domain=pdf

1788 M. Clapp et al.

Keywords Supercritical elliptic problem · Anisotropic critical problem · Ground states ·Bifurcation

Mathematics Subject Classification 35J61 · 35J20 · 35J25

1 Introduction

We consider the supercritical Brezis–Nirenberg-type problem{−�v = λv + |v|2∗

N ,k−2v in �,

v = 0 on ∂�,(℘λ)

where � is given by

� := {(y, z) ∈ Rk+1 × R

N−k−1 : (|y| , z) ∈ �} (1.1)

for some bounded smooth domain � in RN−k such that � ⊂ (0,∞) × R

N−k−1, 1 ≤ k ≤N − 3, λ ∈ R, and 2∗

N ,k := 2(N−k)N−k−2 is the so-called (k + 1)-st critical exponent.

If k = 0, then 2∗N ,0 = 2∗ is the critical Sobolev exponent, and problem (℘λ) reduces

basically to {−�v = λv + |v|2∗−2 v in �,

v = 0 on ∂�.(1.2)

A celebrated result by Brezis and Nirenberg [1] states that (1.2) has a ground state v > 0 ifand only if λ ∈ (0, λ1) and N ≥ 4, or if λ ∈ (λ∗, λ1) and N = 3, where λ∗ is some numberin (0, λ1). Moreover, they show that λ∗ = λ1

4 > 0 if � is a ball. As usual, λm denotes them-th Dirichlet eigenvalue of −� in �.

Problem (1.2) has been widely investigated. Capozzi, Fortunato, and Palmieri [2] estab-lished the existence of solutions for all λ > 0 if N ≥ 5 and for all λ = λm if N = 4 (seealso [11,23]). Several multiplicity results are also available; see, e.g., [3,7–9,24] and thereferences therein.

Recently, Szulkin, Weth, andWillem [21] gave a minimax characterization for the groundstates of problem (1.2) when λ ≥ λ1. They established the existence of ground states forλ = λm if N = 4 and for all λ ≥ λ1 if N ≥ 5.

Concerning the supercritical problem (℘λ) with k ≥ 1, Passaseo [16,17] showed that anontrivial solution does not exist if λ = 0 and � is a ball. This statement was extended in[5] to more general domains � and to some unbounded domains in [6]. On the other hand,existence of multiple solutions has been established in [4,14,22].

This work is concerned with the existence of symmetric ground states for the supercriticalproblem (℘λ) with k ≥ 1. Note that the domain � is invariant under the action of thegroup O(k + 1) of linear isometries of R

k+1 on the first k + 1 coordinates. A functionv : � → R is called O(k + 1)-invariant if v(gy, z) = v(y, z) for every g ∈ O(k + 1),(y, z) ∈ R

k+1 × RN−k−1. The subspace

H10 (�)O(k+1) := {v ∈ H1

0 (�) : v is O(k + 1)-invariant}of H1

0 (�) is continuously embedded in L2∗N ,k (�), so the energy functional Jλ :

H10 (�)O(k+1) → R given by

Jλ(v) := 1

2

∫�

|∇v|2 − λ

2

∫�

v2 − 1

2∗

∫�

|v|2∗N ,k

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Ground states of critical and supercritical problems of. . . 1789

is well defined. Its critical points are the O(k + 1)-invariant solutions to problem (℘λ). AnO(k + 1)-invariant (PS)τ -sequence for Jλ is a sequence (v j ) such that

v j ∈ H10 (�)O(k+1), Jλ(v j ) → τ and J ′

λ(v j ) → 0 in H−1(�).

We set

O(k+1)λ := inf{τ > 0 : there exists an O(k + 1)-invariant (PS)τ -sequence for Jλ}.

This is the lowest possible energy level for a nontrivial O(k+1)-invariant solution to problem(℘λ). An O(k+1)-invariant ground state of problem (℘λ) is a critical point v ∈ H1

0 (�)O(k+1)

of Jλ such that Jλ(v) = O(k+1)λ . Since Jλ does not satisfy the Palais–Smale condition, an

O(k + 1)-invariant ground state does not necessarily exist.Let 0 < λ

[k]1 < λ

[k]2 ≤ λ

[k]3 ≤ · · · be the O(k + 1)-invariant eigenvalues of the problem

−�v = λv in �, v ∈ H10 (�)O(k+1),

countedwith theirmultiplicity. Setλ[k]0 := 0.We shall prove the following result for O(k+1)-invariant ground states.

Theorem 1.1 For every 1 ≤ k ≤ N − 3, the following statements hold true:

(a) Problem (℘λ) does not have an O(k + 1)-invariant ground state if λ ≤ 0.(b) For each m ∈ N ∪ {0}, there is a number λ

[k]m,∗ ∈ [λ[k]m , λ

[k]m+1) with the property that

problem (℘λ) has an O(k + 1)-invariant ground state for every λ ∈ (λ[k]m,∗, λ[k]m+1) and

does not have an O(k + 1)-invariant ground state for any λ ∈ [λ[k]m , λ[k]m,∗).

(c) Let β := max{dist(x, {0} × RN−k−1) : x ∈ �}. Then,

λ[k]0,∗ ≥

⎧⎪⎨⎪⎩

(k−1)2

4β2 if 3k ≥ N ,

k(2∗N ,kβ

)2((2∗

N ,k − 1)k − 2∗N ,k

)if 3k ≤ N .

In particular, λ[k]0,∗ > 0 if k ≥ 2.

This last statement stands in contrast with the classical Brezis–Nirenberg theorem whichestablishes the existence of a ground state to problem (1.2) for every λ ∈ (0, λ1) if N ≥ 4.We also show that λ[1]0,∗ > 0 if � is thin enough; see Proposition 4.4.

As we shall see, the O(k + 1)-invariant ground states of problem (℘λ) correspond to theground states of the critical problem

− div(a(x)∇u) = λb(x)u + c(x) |u|2∗−2 u in �, u = 0 on ∂�, (1.3)

with 2∗ = 2nn−2 , n := N − k, a(x1, . . . , xn) = xk1 and a = b = c.

The critical problem (1.3) with general coefficients a ∈ C1(�), b, c ∈ C0(�) has aninterest in its own. We study it in Sect. 2 and give a minimax characterization for its groundstates, similar to that in [21]. We study the properties of its ground state energy level as afunction of λ and obtain a bifurcation result for ground states; see Theorem 2.1.

Anisotropic critical problems of the form (1.3) have been studied, for example, by Egnell[10] and, more recently, by Hadiji et al. [12,13]. They obtained existence and multiplicityresults under some assumptions which involve flatness of the coefficient functions at somelocalmaximumorminimumpoint in the interior of�. Note that the function a(x1, . . . , xn) =xk1 attains its minimum on the boundary of �. This produces a quite different behaviorregarding the existence of ground states, as we shall see in the following sections.

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Section 2 is devoted to the study of the general anisotropic critical problem. In Sect. 3,we prove a nonexistence result for supercritical problems. It will be used in Sect. 4 wherewe prove Theorem 1.1. In the last section, we include some questions and remarks.

2 Ground states of the anisotropic critical problem

In this section, we consider the anisotropic Brezis–Nirenberg-type problem{ − div(a(x)∇u) = λb(x)u + c(x) |u|2∗−2 u in �,

u = 0 on ∂�,(2.1)

where � is a bounded smooth domain in Rn , n ≥ 3, λ ∈ R, a ∈ C1(�), b, c ∈ C0(�) are

strictly positive on �, and 2∗ := 2nn−2 is the critical Sobolev exponent in dimension n.

We take

〈u, v〉a :=∫

�

a(x)∇u · ∇v, ‖u‖a :=(∫

�

a(x) |∇u|2)1/2

, (2.2)

to be the scalar product and the norm in H10 (�), and

|u|b,2 :=(∫

�

b(x)u2)1/2

, |u|c,2∗ :=(∫

�

c(x) |u|2∗)1/2∗

,

to be the norms in L2(�) and L2∗(�), respectively. They are, clearly, equivalent to the

standard ones.Let 0 < λ

a,b1 < λ

a,b2 ≤ λ

a,b3 ≤ · · · be the eigenvalues of the problem

−div(a(x)∇u) = λb(x)u in �, u = 0 on ∂�,

counted with their multiplicity, and e1, e2, e3, . . . be the corresponding normalized eigen-functions, i.e.,

∣∣e j ∣∣b,2 = 1. Set

Z0 := {0}, Zm := span{e1, . . . , em},Ym := {w ∈ H1

0 (�) : 〈w, z〉a = 0 for all z ∈ Zm},T0 := (−∞, λ

a,b1 ), and Tm := [λa,b

m , λa,bm+1) if m ∈ N.

The solutions to problem (2.1) are the critical points of the functional Jλ : H10 (�) → R

given by

Jλ(u) := 1

2‖u‖2a − λ

2|u|2b,2 − 1

2∗ |u|2∗c,2∗ .

If λ ∈ Tm , we define

Nλ ≡ Nλ(�) := {u ∈ H10 (�) � Zm : J ′

λ(u)u = 0 and J ′λ(u)z = 0 for all z ∈ Zm}.

This is a C1-submanifold of codimension m + 1 in H10 (�), cf. [21]. If λ < λ

a,b1 it is the

usual Nehari manifold, and if λ ≥ λa,b1 it is the generalized Nehari manifold, introduced

by Pankov in [15] and studied by Szulkin and Weth in [19,20]. Note that J ′λ(z)z < 0 for

all z ∈ Zm � {0}. Clearly, the nontrivial critical points of Jλ belong to Nλ. Moreover, they

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coincide with the critical points of its restriction Jλ|Nλ : Nλ → R. The proof of these factsis completely analogous to the one given in [21] for the autonomous case. Set

λ ≡ a,b,cλ := inf

Nλ

Jλ.

Following [19], one shows that, for every w ∈ Ym � {0}, there exist unique tλ,w ∈ (0,∞)

and zλ,w ∈ Zm such that

tλ,ww + zλ,w ∈ Nλ,

and that

Jλ(tλ,ww + zλ,w) = maxt>0, z∈Zm

Jλ(tw + z).

Let �m := {w ∈ Ym : ‖w‖a = 1} be the unit sphere in Ym . Then,

λ = infw∈�m

maxt>0,z∈Zm

Jλ(tw + z). (2.3)

As usual, we denote the best Sobolev constant for the embedding H1(Rn) ↪→ L2∗(Rn)

by S. We set

κa,c :=(minx∈�

a(x)n2

c(x)n−22

)1

nS

n2 ,

and define

λa,b,cm,∗ := inf{λ ∈ Tm : λ < κa,c}.

Theorem 2.1 For every m ∈ N ∪ {0}, the following statements hold true:

(a) The function λ �−→ λ is nonincreasing in Tm and

0 < λ ≤ κa,c for all λ ∈ Tm .

(b) λ is attained on Nλ if λ < κa,c.(c) The function λ �−→ λ is continuous in Tm and

limλ↗λ

a,bm+1

λ = 0.

Hence, λa,b,cm,∗ < λ

a,bm+1.

(d) λ is not attained if λ ∈ (−∞, λa,b,c0,∗ ) or λ ∈ [λa,b

m , λa,b,cm,∗ ), m ≥ 1, and is attained if

λ ∈ (λa,b,cm,∗ , λ

a,bm+1).

Remark 2.2 It follows from part (c) above that bifurcation (to the left) occurs at each λa,bm .

This fact is essentially known and can be obtained by other methods. However, we wouldlike to emphasize that here we show that our bifurcating solutions are ground states.

Proof of Theorem 2.1 (a): Let λ,μ ∈ Tm . If λ ≤ μ, then Jλ(u) ≥ Jμ(u) for every u ∈H10 (�). So λ ≥ μ according to (2.3). This proves that λ �−→ λ is nonincreasing in Tm .

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If λ ∈ Tm and w ∈ �m , we have that

maxt>0, z∈Zm

Jλ(tw + z) ≥ maxt>0

Jλ(tw) = 1

n

(‖w‖2a − λ |w|2b,2|w|2c,2∗

)n/2

≥ 1

n

(1 − λ

λm+1

|w|2c,2∗

)n/2

. (2.4)

Using Sobolev’s inequality, we conclude that there is a positive constant C such that

maxt>0, z∈Zm

Jλ(tw + z) ≥ C for all w ∈ �m .

Therefore, λ > 0.Let ϕ j ∈ C∞

c (Rn) be a positive function such that supp(ϕ j ) ⊂ B1/j (0) and∫ ∣∣∇ϕ j

∣∣2 →Sn/2,

∫ ∣∣ϕ j∣∣2∗ → Sn/2, where Br (ξ) = {x ∈ R

n : |x − ξ | < r}. Let ξ ∈ � be such that

a(ξ)n2

c(ξ)n−22

= minx∈�

a(x)n2

c(x)n−22

and choose ν ∈ Rn with |ν| = 1 such that ν is the inward pointing unit normal at ξ if ξ ∈ ∂�.

Set ξ j := ξ + 1j ν and u j (x) := ϕ j (x − ξ j ). Then, u j ∈ H1

0 (�) for j large enough, and wehave that

maxt>0

Jλ(tu j ) = 1

n

⎛⎝∥∥u j

∥∥2a − λ

∣∣u j∣∣2b,2∣∣u j

∣∣2c,2∗

⎞⎠

n2

= 1

n

⎛⎜⎝∫B1/j (ξ j )

a(x)∣∣∇u j

∣∣2 − λ∫B1/j (ξ j )

b(x)u2j(∫B1/j (ξ j )

c(x)∣∣u j

∣∣2∗)2/2∗

⎞⎟⎠

n2

−→ 1

n

(a(ξ)

n2

c(ξ)n−22

)S

n2 = κa,c as j → ∞. (2.5)

Hence, λ ≤ κa,c for λ < λa,b1 .

Next, we assume that λ ∈ Tm with m ∈ N. We fix an open subset θ of � such thatθ ∩ B1/j (ξ j ) = ∅ for j large enough. If z ∈ Zm and z = 0 in θ , then z = 0 in �;see [21, Lemma 3.3]. Hence, (

∫θc(x) |z|2∗

)1/2∗is a norm in Zm and, since Zm is finite-

dimensional, this norm is equivalent to ‖z‖a . In particular, there is a positive constant A suchthat

∫θc(x) |z|2∗ ≥ 2∗A ‖z‖2∗

a for all z ∈ Zm . It follows by convexity that, for every t > 0and every z ∈ Zm , we have

∣∣tu j + z∣∣2∗c,2∗ =

∫��θ

c(x)∣∣tu j + z

∣∣2∗ +∫

θ

c(x) |z|2∗

≥ t2∗∫

�

c(x)u2∗j + 2∗t2∗−1

∫�

c(x)u2∗−1j z + 2∗A ‖z‖2∗

a .

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Therefore,

Jλ(tu j + z) ≤ J0(tu j ) − λ

2

∣∣tu j∣∣2b,2 + t

∫�

(a(x)∇u j∇z − λb(x)u j z

)

+ 1

2

(‖z‖2a − λ |z|2b,2) − t2

∗−1∫

�

c(x)u2∗−1j z − A ‖z‖2∗

a

≤ J0(tu j ) + t∫

�

(a(x)∇u j∇z − λb(x)u j z

)

− t2∗−1

∫�

c(x)u2∗−1j z − A ‖z‖2∗

a . (2.6)

Consequently,

Jλ(tu j + z) ≤ B(t2 + t ‖z‖a + t2∗−1 ‖z‖a) − C(t2

∗ + ‖z‖2∗a )

for some positive constants B and C . This implies that there exists R > 0 such that Jλ(tu j +z) ≤ 0 for all t ≥ R, z ∈ Zm and j large enough. On the other hand, for t ≤ R, z ∈ Zm andj large enough, since ϕ j ⇀ 0 weakly in H1

0 (�), inequalities (2.6) and (2.5) imply that

Jλ(tu j + z) ≤ J0(tu j ) + o(1) = κa,c + o(1).

This proves that λ ≤ κa,c for λ ≥ λa,b1 and concludes the proof of statement (a).

(b): Let Iλ : �m → R be the function given by

Iλ(w) := Jλ(tλ,ww + zλ,w).

Then, λ := infw∈�m Iλ(w). It is shown in [19,20] that Iλ ∈ C1(�m, R). Since�m is a smoothsubmanifold of H1

0 (�), Ekeland’s variational principle yields a Palais–Smale sequence (w j )

for Iλ such that Iλ(w j ) → λ, cf. [23, Theorem 8.5]. Set u j := tλ,w j w j +zλ,w j . By Corollary2.10 in [19] or Corollary 33 in [20], (u j ) is a Palais–Smale sequence for Jλ. Now, Corollary3.2 in [4] asserts that every Palais–Smale sequence (u j ) for Jλ such that Jλ(u j ) → τ < κa,c

contains a convergent subsequence. It follows that λ is attained on Nλ if λ < κa,c.(c): Let w ∈ �m . First, we will show that the function λ �−→ Iλ(w) is continuous in Tm .

Let μ j , μ ∈ Tm be such that μ j → μ. A standard argument shows that Jμ j (tw + z) ≤ 0 forevery j ∈ N if t2 + ‖z‖2a is large enough. Therefore, the sequences (tμ j ,w) and (zμ j ,w) arebounded and, after passing to a subsequence, tμ j ,w → t0 in [0,∞) and zμ j ,w → z0 in Zm .Hence,

Iμ j (w) = Jμ j (tμ j ,ww + zμ j ,w) → Jμ(t0w + z0) ≤ Iμ(w).

If Jμ(t0w + z0) < Iμ(w), then, since

Jμ j (tμ,ww + zμ,w) → Jμ(tμ,ww + zμ,w) = Iμ(w),

we would have that, for j large enough,

maxt>0, z∈Zm

Jμ j (tw + z) = Jμ j (tμ j ,ww + zμ j ,w) < Jμ j (tμ,ww + zμ,w),

which is a contradiction. Consequently, Iμ j (w) → Iμ(w). This proves that λ �−→ Iλ(w) iscontinuous in Tm for each w ∈ �m .

Next,weprove that the functionλ �−→ λ is continuous from the left inTm . Letμ j , μ ∈ Tmbe such that μ j ≤ μ and μ j → μ. Since the infimum of any family of continuous functionsis upper semicontinuous and λ �−→ λ is nonincreasing, we have that

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lim supj→∞

μ j ≤ μ ≤ lim infj→∞ μ j .

This proves that λ �−→ λ is continuous from the left in Tm .To prove that λ �−→ λ is continuous from the right in Tm , we argue by contradiction.

Assume there areμ j , μ ∈ Tm such thatμ j ≥ μ,μ j → μ and sup j∈Nμ j < μ. Then, μ j <

κa,c and, by statement (b), there exists w j ∈ �m such that μ j = Jμ j (tμ j ,ww j + zμ j ,w).Inequality (2.4) asserts that

μ > μ j = Jμ j (tμ j ,w j w j + zμ j ,w j ) ≥ 1

n

⎛⎝1 − μ j

λm+1∣∣w j∣∣2c,2∗

⎞⎠

n/2

.

This implies that∣∣w j

∣∣2∗c,2∗ ≥ ε > 0 for all j ∈ N. Denote the closure of Ym in L2∗

(�) by Ym .

Since dim(Zm) < ∞, the projection Ym ⊕ Zm → Ym is continuous in L2∗(�). Hence, there

is a positive constant A0 such that

μ ≤ Jμ(tμ,w j w j + zμ,w j )

= t2μ,w j

2(1 − μ

∣∣w j∣∣2b,2) + 1

2(∥∥zμ,w j

∥∥2a

− μ∣∣zμ,w j

∣∣2b,2

) − 1

2∗∣∣tμ,w j w j + zμ,w j

∣∣2∗c,2∗

≤ t2μ,w j

2− A0

t2∗

μ,w j

2∗∣∣w j

∣∣2∗c,2∗ ≤ t2μ,w j

2− A0ε

t2∗

μ,w j

2∗ for all j ∈ N.

It follows that (tμ,w j ) is bounded. Hence, (‖zμ,w j ‖a) is bounded too. Consequently,

μ ≤ Jμ(tμ,w j w j + zμ,w j ) = Jμ j (tμ,w j w j + zμ,w j ) + (μ − μ j )∣∣tμ,w j w j + zμ,w j

∣∣2b,2

≤ Jμ j (tμ j ,w j w j + zμ j ,w j ) + o(1) = μ j + o(1) ≤ supj∈N

μ j + o(1) < μ + o(1).

This is a contradiction. It follows that the function λ �−→ λ is continuous in Tm .Finally, let μ j ∈ Tm be such that μ j → λm+1. We have that

0 < μ j ≤ Jμ j (tμ j ,em+1em+1 + zμ j ,em+1)

= t2μ j ,em+1

2(λm+1 − μ j ) + 1

2(∥∥zμ j ,em+1

∥∥2a

− μ j∣∣zμ j ,em+1

∣∣2b,2

)

− 1

2∗∣∣tμ j ,em+1em+1 + zμ j ,em+1

∣∣2∗c,2∗

≤ t2μ j ,em+1

2(λm+1 − μ j ) − A0

t2∗

μ j ,em+1

2∗ |em+1|2∗c,2∗ .

It follows that (tμ j ,em+1) is bounded and, hence, that

0 < μ j ≤ t2μ j ,em+1

2(λm+1 − μ j ) = o(1).

This proves that μ j → 0 as μ j → λm+1 from the left.

(d): If λ ∈ Tm , λ < λa,b,cm,∗ , and w ∈ �m were such that λ = Iλ(w), then for μ ∈

(λ, λa,b,cm,∗ (�)) we would have that

κa,c = μ ≤ Iμ(w) < Iλ(w) = λ,

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contradicting (a). It follows that λ is not attained if λ ∈ [λa,bm , λ

a,b,cm,∗ ). Statement (b) implies

that λ is attained if λ ∈ (λa,b,cm,∗ , λ

a,bm+1). ��

Recall that a (PS)τ -sequence for Jλ is a sequence (u j ) in H10 (�) such that Jλ(u j ) → τ

and J ′λ(u j ) → 0 in H−1(�). The value λ is characterized as follows.

Corollary 2.3 λ = inf{τ > 0 : there exists a (PS)τ -sequence for Jλ}.Proof The argument given in the proof of statement (b) of Theorem 2.1 shows that thereexists a (PS)λ -sequence for Jλ. To prove that λ is the smallest positive number with thisproperty, we argue by contradiction. Assume that τ < λ and that there exists a (PS)τ -sequence (u j ) for Jλ. Then, τ < κa,c and Corollary 3.2 in [4] asserts that (u j ) contains asubsequence which converges to a critical point u of Jλ with Jλ(u) = τ . If τ = 0, thenu ∈ Nλ and, hence, λ ≤ τ . This is a contradiction. ��

For the classical Brezis–Nirenberg problem (1.2) (where a = b = c ≡ 1) with n ≥ 4,it is known that λ

a,b,c0,∗ = 0 and λ

a,b,cm,∗ = λm , the m-th Dirichlet eigenvalue of −� in �, for

all m ∈ N. Moreover, λ = 1n S

n2 = κa,c for every λ ≤ 0, but λ < 1

n Sn2 for every λ > 0 if

n ≥ 5; see [1,11,21].As we shall see below, this is not true in general: For the problem (℘#

λ) in Sect. 4 which

arises from the supercritical one, one has that λa,b,c0,∗ > 0 in most cases; see Propositions 4.3

and 4.4. A special feature of that problem is that the value κa,c is attained on the boundaryof �. A different situation was considered by Egnell [10] and Hadiji and Yazidi [13]. Theyshowed for example that, if a attains its minimum at an interior point x0 of�, b = 1 = c, anda is flat enough around x0, then λ

a,b,c0,∗ = 0 for n ≥ 4, as in the classical Brezis–Nirenberg

case.We do not knowwhether, in general, λa,b,c

0,∗ ≥ 0. But this will be true in the special case weare interested in; see Proposition 4.1. The proof uses a nonexistence result for the supercriticalproblem, which we discuss in the following section.

3 Nonexistence of solutions to a supercritical problem

Let� be a bounded smooth domain inRN−k with� ⊂ (0,∞)×R

N−k−1 and 1 ≤ k ≤ N−3.Set

� := {(y, z) ∈ Rk+1 × R

N−k−1 : (|y| , z) ∈ �}and consider the problem { −�u = λu + |u|p−2 u in �,

u = 0 on ∂�.(3.1)

Passaseo [16,17] showed that, if � is a ball, problem (3.1) does not have a nontrivialsolution for λ = 0 and p ≥ 2∗

N ,k := 2(N−k)N−k−2 . In [5], it is shown that this is also true for

doubly starshaped domains.

Definition 3.1 � is doubly starshaped if there exist two numbers 0 < t0 < t1 such thatt ∈ (t0, t1) for every (t, z) ∈ � and � is strictly starshaped with respect to ξ0 := (t0, 0) andto ξ1 := (t1, 0), i.e.,

〈x − ξi , ν�(x)〉 > 0 ∀x ∈ ∂� � {ξi } , i = 0, 1,

where ν� is the outward pointing unit normal to ∂�.

123


We denote the first Dirichlet eigenvalue of −� in � by λ1(�).

Theorem 3.2 If � is doubly starshaped, p ≥ 2∗N ,k and

λ ≤ 2(p − 2∗N ,k)

2∗N ,k(p − 2)

λ1(�),

then problem (3.1) does not have a nontrivial solution.

Wepoint out that the geometric assumption on� cannot be dropped. Existence ofmultiplesolutions to problem (3.1) for λ = 0 and p = 2∗

N ,k in some domains where � is not doublystarshaped has been established in [4,14,22].

The proof of Theorem 3.2 follows the ideas introduced in [5,16,17]. Fix τ ∈ (0,∞) andlet ϕ be the solution to the problem{

ϕ′(t)t + (k + 1)ϕ(t) = 1, t ∈ (0,∞),

ϕ(τ) = 0.

Explicitly, ϕ(t) = 1k+1

[1 − ( τ

t )k+1

]. Note that ϕ is strictly increasing in (0,∞). For y = 0,

we defineχτ (y, z) := (ϕ(|y|)y, z). (3.2)

Lemma 3.3 The vector field χτ has the following properties:

(a) divχτ = N − k,(b) 〈dχτ (y, z) [ξ ] , ξ 〉 ≤ max{1 − kϕ(|y|), 1} |ξ |2 for every y ∈ R

k+1� {0}, z ∈ R

N−k−1,ξ ∈ R

N .

Proof See [17, Lemma 2.3] or [5, Lemma 4.2]. ��Proposition 3.4 Assume there exists τ ∈ (0,∞) such that |y| ∈ (τ,∞) for every (y, z) ∈ �

and 〈χτ , ν�〉 > 0 a.e. on ∂�. If p ≥ 2∗N ,k and

λ ≤ 2(p − 2∗N ,k)

2∗N ,k(p − 2)

λ1(�),

then problem (3.1) does not have a nontrivial solution.

Proof The variational identity (4) in Pucci and Serrin’s paper [18] implies that, if u ∈C2(�) ∩ C1(�) is a solution of (3.1) and χ ∈ C1(�, R

N ), then

1

2

∫∂�

|∇u|2 〈χ, ν�〉 dσ =∫

�

(divχ)

[1

p|u|p + λ

2u2 − 1

2|∇u|2

]dx

+∫

�

〈dχ [∇u] ,∇u〉 dx, (3.3)

where ν� is the outward pointing unit normal to ∂� (in the notation of [18] we have takenF(x, u,∇u) = 1

2 |∇u|2 − 12λu

2 − 1p |u|p , h = χ and a = 0). Let χ := χτ . Then, by

Lemma 3.3,

divχτ = N − k.

Moreover, since 1 − kϕ(t) < 1 for t ∈ (τ,∞), and |y| ∈ (τ,∞) for every (y, z) ∈ �,Lemma 3.3 yields

〈dχτ (y, z) [ξ ] , ξ 〉 ≤ |ξ |2 ∀(y, z) ∈ �, ξ ∈ RN .

123


By assumption, 〈χτ , ν�〉 > 0 a.e. on ∂�. Therefore, if u is a nontrivial solution of (3.1) wehave, using (3.3), that

0 < (N − k)

(1

p− 1

2

)∫�

[|∇u|2 − λu2]dx +

∫�

|∇u|2 dx

= (N − k)

(1

p− 1

2+ 1

N − k

)∫�

|∇u|2 dx − (N − k)

(1

p− 1

2

)λ

∫�

u2dx,

that is,(1

2− 1

p

)λ

∫�

u2dx >

(1

2∗N ,k

− 1

p

)∫�

|∇u|2 dx .

Therefore, if p ≥ 2∗N ,k and

λ ≤ 2(p − 2∗N ,k)

2∗N ,k(p − 2)

infu∈H1

0 (�)

u =0

∫�

|∇u|2 dx∫�u2dx

= 2(p − 2∗N ,k)

2∗N ,k(p − 2)

λ1(�),

problem (3.1) does not have a nontrivial solution in �, as claimed. ��The following result was proved in [5].

Proposition 3.5 If � is doubly starshaped, then � ⊂ (t0,∞) × RN−k−1 and

⟨χt0 , ν�

⟩> 0

a.e. on ∂�, with t0 as in Definition 3.1.

Proof See the proof of (4.11) in [5]. ��Proof of Theorem 3.2 The conclusion follows immediately from Propositions 3.4 and 3.5. ��

4 Existence and nonexistence of symmetric ground states to supercriticalproblems

Next, we come back to our original supercritical problem{ −�v = λv + |v|2∗N ,k−2

v in �,

v = 0 on ∂�,(℘λ)

where

� := {(y, z) ∈ Rk+1 × R

N−k−1 : (|y| , z) ∈ �}for some bounded smooth domain � in R

N−k with � ⊂ (0,∞)× RN−k−1, 1 ≤ k ≤ N − 3,

and 2∗N ,k := 2(N−k)

N−k−2 .An O(k + 1)-invariant function v : � → R can be written as v(y, z) = u(|y| , z) for

some function u : � → R. A straightforward computation shows that

�v = 1

a(x)div(a(x)∇u), (4.1)

where a(x1, . . . , xN−k) := xk1 . Hence, v is an O(k+1)-invariant solution of (℘λ) if and onlyif u solves { −div(xk1∇u) = λxk1u + xk1 |u|2∗−2 u in �,

u = 0 on ∂�,(℘#

λ)

123


where 2∗ = 2∗N ,k is the critical exponent in dimension n := N − k. So this problem is a

special case of the problem treated in Sect. 2 with a(x1, . . . , xn) := xk1 and a = b = c.

For these functions a, b, c, we simplify notation and write [k]λ , κ [k], λ[k]m , λ[k]m,∗ instead of

a,b,cλ , κa,c, λa,b

m , λa,b,cm,∗ . Note that

κ [k] =(minx∈�

xk1

)1

nSn/2.

Proposition 4.1 If α := minx∈� x1 and λ ≤ 0, then [k]λ = αk

n Sn/2 and it is not attained byJλ on Nλ ≡ Nλ(�).

Proof ByTheorem 2.1, it is enough to show this for λ = 0. Arguing by contradiction, assume

that [k]0 < αk

n Sn/2. Then, there exists ϕ ∈ C∞c (�) ∩ N0(�) such that

J0(ϕ) <αk

nSn/2 = κ [k].

Since supp(ϕ) is a compact subset of (α,∞) × Rn−1, there exists a � ∈ (α,∞) such that

supp(ϕ) ⊂ B := {x ∈ R

n : (x1 − �)2 + x22 + · · · + x2n < (α − �)2}. Hence, ϕ ∈ N0(B).

Theorem 3.2 and the discussion given at the beginning of this section imply that problem

−div(xk1∇u) = xk1 |u|2∗−2 u in B, u = 0 on ∂B

does not have a nontrivial solution. So, by Theorem 2.1, infN0(B) J0 = κ [k] = αk

n Sn/2. But

infN0(B)

J0 ≤ J0(ϕ) <αk

nSn/2.

This is a contradiction. We conclude that [k]0 = αk

n Sn/2.Since this value is the same for every � such that α := minx∈� x1, a standard argument

shows that [k]0 is not attained by J0 on N0 ≡ N0(�). ��Set

α := minx∈�

x1, β := maxx∈�

x1.

Lemma 4.2 For every positive function f ∈ C2[α, β] which satisfies

αk f 2(α) ≤ tk f 2(t) and tk f 2∗(t) ≤ αk f 2

∗(α) ∀t ∈ [α, β], (4.2)

we have that

λ[k]0,∗ ≥ min

t∈[α,β]− (

tk f ′(t))′

tk f (t).

Proof Let u ∈ H10 (�), u = 0, and set u(x) = f (x1)w(x). Then,

∫�

xk1 |∇u|2 =∫

�

(xk1 f

2 |∇w|2 + xk1 f′ f ∂w2

∂x1+ xk1 ( f

′)2w2)

=∫

�

(xk1 f

2 |∇w|2 + xk1 f′ ∂( f w2)

∂x1

)

=∫

�

(xk1 f

2 |∇w|2 −(xk1 f

′)′f w2

).

123


So, if λ ≤ −(tk f ′(t)

)′tk f (t)

for all t ∈ [α, β], we have that∫�xk1 |∇u|2 − λ

∫�xk1u

2

(∫�xk1 |u|2∗)2/2∗ =

∫�xk1 f

2 |∇w|2 − ∫�

[(xk1 f

′)′ + λxk1 f]f w2

(∫�xk1 f

2∗ |w|2∗)2/2∗

≥∫�xk1 f

2 |∇w|2(∫�xk1 f

2∗ |w|2∗)2/2∗ ≥ αk f 2(α)∫�

|∇w|2α2k/2∗ f 2(α)

(∫�

|w|2∗)2/2∗

= α2k/n

∫�

|∇w|2(∫�

|w|2∗)2/2∗ ≥ α2k/n S > 0 for all u ∈ H10 (�), u = 0.

This implies that λ < λ[k]1 and, hence, that

maxt>0

Jλ(tu) = 1

n

(∫�xk1 |∇u|2 − λ

∫�xk1u

2

(∫�xk1 |u|2∗)2/2∗

)n/2

≥ αk

nSn/2

for all u ∈ H10 (�), u = 0. Therefore, [k]λ = αk

n Sn/2 for every λ ≤ mint∈[α,β]−(

tk f ′(t))′

tk f (t), and

the conclusion follows. ��

We obtain the following estimates for λ[k]0,∗.

Proposition 4.3 λ[k]0,∗ ≥ 0 and

λ[k]0,∗ ≥

{(k−1)2

4β2 if 2k ≥ n,k

(2∗β)2((2∗ − 1)k − 2∗) if 2k ≤ n.

Therefore, λ[k]0,∗ > 0 if k ≥ 2.

Proof Proposition 4.1 implies that λ[k]0,∗ ≥ 0.

Set f (t) := t−γ with k2∗ ≤ γ ≤ k

2 . This function satisfies (4.2) and, since

− (tk f ′(t)

)′tk f (t)

= γ (k − γ − 1)

t2,

Lemma 4.2 implies that

λ[k]0,∗ ≥ γ (k − γ − 1)

β2 .

Now observe that the function φ(γ ) := γ (k − γ − 1) attains its maximum on the interval[ k2∗ , k

2

]at the point

γ∗ := max

{k − 1

2,k

2∗

}.

Therefore, λ[k]0,∗ ≥ 1β2 φ(γ∗) = 1

β2 γ∗(k − γ∗ − 1), as claimed.

Finally, note that k > 2∗2∗−1 = 2n

n+2 if k ≥ 2. Hence, λ[k]0,∗ > 0 if k ≥ 2. ��

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Proof of Theorem 1.1 Using Corollary 2.3, it is easily seen that, if v(y, z) = u(|y| , z), thenv is an O(k + 1)-invariant ground state for problem (℘λ) if and only if u is a ground statefor problem (℘#

λ). So Theorem 1.1 follows immediately from Proposition 4.1, Theorem 2.1,and Proposition 4.3. ��

The following result shows that λ[1]0,∗ > 0 if the domain is thin enough in the x1-direction.

Proposition 4.4 If βα

≤ nn−2 , then λ

[k]0,∗ ≥ k2

4β2 > 0 for all k ≥ 1.

Proof Set f (t) := e−γ (t−α) with k2∗α ≤ γ ≤ k

2β , and write g(t) := tk f 2(t) and h(t) :=tk f 2

∗(t). Then, g′(t) = tk−1e−2γ (t−α)(k − 2γ t) ≥ 0 and h′(t) = tk−1e−2∗γ (t−α)(k −

2∗γ t) ≤ 0 for all t ∈ [α, β], so f satisfies (4.2). Since

− (tk f ′(t)

)′tk f (t)

= γ tk−1e−γ (t−α)(k − γ t)

tke−γ (t−α)= γ (k − γ t)

t,

Lemma 4.2 implies that

λ[k]0,∗ ≥ γ (k − γβ)

β.

Now observe that the function φ(γ ) := γ (k−γβ) attains its maximum at the point γ∗ := k2β .

Hence, λ[k]0,∗ ≥ k2

4β2 > 0, as claimed. ��

5 Some open questions and comments

Many questions remain open. Here are some of them.

Problem 1 Concerning problem (℘#λ):

(1) Is it true that λ[1]0,∗ > 0 for any domain �, and not only for thin domains?

(2) For m ≥ 1, is λ[k]m,∗ > λ

[k]m , or is λ

[k]m,∗ = λ

[k]m as in the classical Brezis–Nirenberg case?

(3) What happens in general at λ[k]m,∗? Is there, or not, a ground state of problem (℘#

λ) for

λ = λ[k]m,∗?

Problem 2 Concerning the general anisotropic problem (2.1):

(1) Is λa,b,c0,∗ always nonnegative? Or are there examples where a ground state exists for some

λ < 0? For all λ < 0?(2) Can one give lower estimates for λ

a,b,cm,∗ in some cases?

(3) Suppose that c ∈ C1(�) in addition to our earlier assumptions. If κa,c is attained only

at points which are nonstationary for a(x)n/2

c(x)(n−2)/2 and lie on the boundary of �, is it then

true that λa,b,c0,∗ > 0?

Two particular cases of (3) are: c = 1, and a = b = c. If the answer is positive in the firstcase, this would be in contrast to the results in [10] and [13]. A positive answer in the secondcase would be a generalization of our results for (℘#

λ). A partial answer can be given usingProposition 4.3. Consider, for example, the problem

− div(a(x)∇u) = λb(x)u + |u|2∗−2 u in �, u = 0 on ∂�, (5.1)

123


where � is a bounded smooth domain in Rn , n ≥ 3, λ ∈ R, a ∈ C1(�), b ∈ C0(�) are

strictly positive on �, and 2∗ = 2nn−2 . Then, the following statement holds true.

Proposition 5.1 If a(x) ≥ xk1 ≥ b(x) for all x ∈ � and minx∈� a(x) = (minx∈� x1)k > 0

for some k ≥ 2, then λa,b,10,∗ > 0.

Proof Let α := minx∈� x1 > 0. For every u ∈ H10 (�), u = 0, λ ∈ [0, λ[k]

0,∗], we have that∫�a(x) |∇u|2 − λ

∫�b(x)u2

α2k/2∗ (∫�

|u|2∗)2/2∗ ≥∫�xk1 |∇u|2 − λ

∫�xk1u

2

(∫�xk1 |u|2∗)2/2∗ > 0.

Hence, λ < λa,b1 and

1

n

(∫�a(x) |∇u|2 − λ

∫�b(x)u2(∫

�|u|2∗)2/2∗

)n/2

≥ αk(n−2)

21

n

(∫�xk1 |∇u|2 − λ

∫�xk1u

2

(∫�xk1 |u|2∗)2/2∗

)n/2

≥ αk(n−2)

2αk

nSn/2 =

(αk

)n/2 1

nSn/2 = κa,1.

It follows from Theorem 2.1 that

a,b,1λ = κa,1 for all λ ∈ (−∞, λ

[k]0,∗].

Hence, by Proposition 4.3, λa,b,10,∗ ≥ λ

[k]0,∗ > 0, as claimed. ��

References

1. Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolevexponents. Commun. Pure Appl. Math. 36, 437–477 (1983)

2. Capozzi, A., Fortunato, D., Palmieri, G.: An existence result for nonlinear elliptic problems involvingcritical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 463–470 (1985)

3. Cerami, G., Solimini, S., Struwe, M.: Some existence results for superlinear elliptic boundary valueproblems involving critical exponents. J. Funct. Anal. 69, 289–306 (1986)

4. Clapp, M., Faya, J.: Multiple solutions to anisotropic critical and supercritical problems in symmetricdomains. Prog. Nonlinear Differ. Equ. Appl. 86, 99–120 (2015)

5. Clapp, M., Faya, J., Pistoia, A.: Nonexistence and multiplicity of solutions to elliptic problems withsupercritical exponents. Calc. Var. Partial Differ. Equ. 48, 611–623 (2013)

6. Clapp, M., Szulkin, A.: A supercritical elliptic problem in a cylindrical shell. Prog. Nonlinear Differ. Equ.Appl. 85, 233–242 (2014)

7. Clapp, M., Weth, T.: Multiple solutions for the Brezis–Nirenberg problem. Adv. Differ. Equ. 10, 463–480(2005)

8. Devillanova, G., Solimini, S.: Concentration estimates and multiple solutions to elliptic problems atcritical growth. Adv. Differ. Equ. 7, 1257–1280 (2002)

9. Devillanova, G., Solimini, S.: A multiplicity result for elliptic equations at critical growth in low dimen-sion. Commun. Contemp. Math. 5, 171–177 (2003)

10. Egnell, H.: Semilinear elliptic equations involving critical Sobolev exponents. Arch. Ration. Mech. Anal.104, 27–56 (1988)

11. Gazzola, F., Ruf, B.: Lower order perturbations of critical growth nonlinearities in semilinear ellipticequations. Adv. Differ. Equ. 2, 555–572 (1997)

12. Hadiji, R., Molle, R., Passaseo, D., Yazidi, H.: Localization of solutions for nonlinear elliptic problemswith critical growth. C. R. Math. Acad. Sci. Paris 343, 725–730 (2006)

13. Hadiji, R., Yazidi, H.: Problem with critical Sobolev exponent and with weight. Chin. Ann. Math. Ser. B28, 327–352 (2007)

14. Kim, S., Pistoia, A.: Supercritical problems in domains with thin toroidal holes. Discrete Contin. Dyn.Syst. 34, 4671–4688 (2014)

123


15. Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math.73, 259–287 (2005)

16. Passaseo, D.: Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivialdomains. J. Funct. Anal. 114, 97–105 (1993)

17. Passaseo, D.: New nonexistence results for elliptic equations with supercritical nonlinearity. Differ. Inte-gral Equ. 8, 577–586 (1995)

18. Pucci, P., Serrin, J.: A general variational identity. Indiana Univ. Math. J. 35, 681–703 (1986)19. Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal.

257, 3802–3822 (2009)20. Szulkin, A., Weth, T.: The Method of Nehari Manifold. Handbook of Nonconvex Analysis and Applica-

tions, pp. 597–632. International Press, Somerville (2010)21. Szulkin, A., Weth, T., Willem, M.: Ground state solutions for a semilinear problemwith critical exponent.

Differ. Integral Equ. 22, 913–926 (2009)22. Wei, J., Yan, S.: Infinitely many positive solutions for an elliptic problem with critical or supercritical

growth. J. Math. Pures Appl. 96, 307–333 (2011)23. Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications,

24. Birkhäuser Boston, Inc., Boston (1996)24. Zhang, D.: On multiple solutions of �u + λu + |u| 4

n−2 = 0. Nonlinear Anal. TMA 13, 353–372 (1989)

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Ground states of critical and supercritical problems of