Entire nodal solutions to the critical Lane-Emden system

TL;DR

This paper proves the existence of multiple nonradial sign-changing solutions to the critical Lane-Emden system in , using limit profiles of symmetric minimizing sequences for a related higher-order problem.

Contribution

It establishes the existence of finitely many nonradial solutions to the critical Lane-Emden system on , which was previously unknown.

Findings

Existence of finitely many sign-changing solutions.

Solutions are nonradial and arise as limits of symmetric sequences.

Results apply to the critical hyperbola rac{1}{p}+rac{1}{q}=rac{N-2}{N}.

Abstract

We establish the existence of finitely many sign-changing solutions to the Lane-Emden system $$-\Delta u=|v|^{q-2}v,\quad -\Delta v=|u|^{p-2}u \quad \text{ in }\mathbb{R}^N, \ \ N\geq 4,$$ where the exponents $p$ and $q$ lie on the critical hyperbola $\frac{1}{p}+\frac{1}{q}=\frac{N-2}{N}$. These solutions are nonradial and arise as limit profiles of symmetric sign-changing minimizing sequences for a critical higher-order problem in a bounded domain.

Equations

$$-\Delta u=|v{|}^{q-2}v,-\Delta v=|u{|}^{p-2}u\text{ in }{\mathbb{R}}^N,N\ge 4,$$

$$\{\begin{array}{l}-\Delta u=|v{|}^{q-2}v,\\ -\Delta v=|u{|}^{p-2}u,\\ u\in D^{2,q^{\prime }}({\mathbb{R}}^N),v\in D^{2,p^{\prime }}({\mathbb{R}}^N),\end{array}$$

$$\frac{1}{p}+\frac{1}{q}=\frac{N-2}{N}.$$

$$\Vert w{\Vert }_r:={\left({\int }_{{\mathbb{R}}^N}|\Delta w{|}^r\right)}^{\frac{1}{r}}.$$

$$\{\begin{array}{l}\Delta (|\Delta u{|}^{q^{\prime }-2}\Delta u)=|u{|}^{p-2}u,\\ u\in D^{2,q^{\prime }}({\mathbb{R}}^N)\end{array}$$

$$-\Delta u=|u{|}^{2^{\ast }-2}u,u\in D^{1,2}({\mathbb{R}}^N),$$

$${\Delta }^{2}u=|u{|}^{2_{\ast }-2}u,u\in D^{2,2}({\mathbb{R}}^N),$$

$$\Delta (|\Delta u{|}^{q^{\prime }-2}\Delta u)=|u{|}^{p-2}u,u\in D_0^{2,q^{\prime }}(\Omega ),$$

$$(-\Delta {)}^{m}u=|v{|}^{q-2}v,(-\Delta {)}^{m}v=|u{|}^{p-2}u,u\in D^{2m,q^{\prime }}({\mathbb{R}}^N),v\in D^{2m,p^{\prime }}({\mathbb{R}}^N),$$

$${\Delta }^m(|{\Delta }^{m}u{|}^{q^{\prime }-2}{\Delta }^{m}u)=|u{|}^{p-2}u,u\in D^{2m,q^{\prime }}({\mathbb{R}}^N).$$

$$-\mathrm{div}({\Delta }^m(|\nabla {\Delta }^{m}u{|}^{q^{\prime }-2}\nabla {\Delta }^{m}u))=|u{|}^{p-2}u,u\in D^{2m+1,q^{\prime }}({\mathbb{R}}^N),$$

$$D^{2,q^{\prime }}({\mathbb{R}}^N):=\{u\in L^p({\mathbb{R}}^N):u\text{ is twice weakly differentiable},\Delta u\in L^{q^{\prime }}({\mathbb{R}}^N)\},$$

$$\Vert w{\Vert }_{q^{\prime }}:={\left({\int }_{{\mathbb{R}}^N}|\Delta w{|}^{q^{\prime }}\right)}^{\frac{1}{q^{\prime }}}.$$

$$I(u,v):={\int }_{{\mathbb{R}}^N}\nabla u\cdot \nabla v-\frac{1}{p}{\int }_{{\mathbb{R}}^N}|u{|}^p-\frac{1}{q}{\int }_{{\mathbb{R}}^N}|v{|}^q,$$

$$J(u):=\frac{1}{q^{\prime }}{\int }_{{\mathbb{R}}^N}|\Delta u{|}^{q^{\prime }}-\frac{1}{p}{\int }_{{\mathbb{R}}^N}|u{|}^p,$$

$$J^{\prime }(u)\phi ={\int }_{{\mathbb{R}}^N}|\Delta u{|}^{q^{\prime }-2}\Delta u\Delta \phi -{\int }_{{\mathbb{R}}^N}|u{|}^{p-2}u\phi .$$

$${\partial }_{u}I(u,v)\phi$$

$$=J^{\prime }(u)\phi =0,$$

$${\partial }_{v}I(u,v)\phi$$

$$u(gx)=\varphi (g)u(x)\text{for all}g\in G,x\in \Omega .$$

$$D_0^{2,q^{\prime }}(\Omega {)}^{\varphi }:=\{u\in D_0^{2,q^{\prime }}(\Omega ):u\text{ is }\varphi \text{-equivariant}\}.$$

$$\{\begin{array}{l}\Delta (|\Delta {|}^{q^{\prime }-2}\Delta u)=|u{|}^{p-2}u,\\ u\in D_0^{2,q^{\prime }}(\Omega ),\end{array}$$

$${\phi }_{\varphi }(x):=\frac{1}{\mu (G)}{\int }_G\varphi (g)\phi (gx)\mathrm{d}\mu (g),$$

$$J^{\prime }(u)[{\phi }_{\varphi }]=J^{\prime }(u)\phi \text{for every }\phi \in {\mathcal{C}}_c^{\infty }(\Omega ).$$

$$\Delta ({\phi }_{\varphi })(x)=\frac{1}{\mu (G)}{\int }_G\Delta (\varphi (g)\phi \circ g)(x)\mathrm{d}\mu (g)=\frac{1}{\mu (G)}{\int }_G\varphi (g)\Delta \phi (gx)\mathrm{d}\mu (g).$$

$${\int }_{\Omega }|\Delta u{|}^{q^{\prime }-2}\Delta u\Delta ({\phi }_{\varphi })={\int }_{\Omega }|\Delta u{|}^{q^{\prime }-2}\Delta u\Delta \phi ,{\int }_{\Omega }|u{|}^{p-2}u({\phi }_{\varphi })={\int }_{\Omega }|u{|}^{p-2}u\phi .$$

$${\mathcal{N}}^{\varphi }(\Omega ):=\{u\in D_0^{2,q^{\prime }}(\Omega {)}^{\varphi }:u\ne 0,\Vert u{\Vert }_{q^{\prime }}^{q^{\prime }}=|u{|}_p^p\},$$

$$c^{\varphi }(\Omega ):=\underset{u\in {\mathcal{N}}^{\varphi }(\Omega )}{\inf }J(u).$$

$$c^{\varphi }(\Omega )=\underset{\sigma \in \mathcal{T}}{\inf }\underset{t\in [0,1]}{\max }J(\sigma (t)).$$

$${\Omega }^G:=\{x\in \Omega :Gx=\{x\}\}$$

$$c^{\varphi }(\Omega )=c^{\varphi }({\mathbb{R}}^N)=:c_{\infty }^{\varphi }.$$

Full text

[TABLE]

where the exponents $p$ and $q$ lie on the critical hyperbola $\frac{1}{p}+\frac{1}{q}=\frac{N-2}{N}$. These solutions are nonradial and arise as limit profiles of symmetric sign-changing minimizing sequences for a critical higher-order problem in a bounded domain.

Keywords: Hamiltonian system; critical hyperbola; entire nodal solutions; variational methods; concentration-compactness; symmetries.

MSC2010: 35J47; 35J30; 35B33; 35B08; 35B06.

1 Introduction

Consider the Lane-Emden system

[TABLE]

where $N\geq 3$ and $(p,q)$ lies on the critical hyperbola, that is,

[TABLE]

As usual, $p^{\prime}:=\frac{p}{p-1}$, $q^{\prime}:=\frac{q}{q-1}$, and $D^{2,r}(\mathbb{R}^{N})$ is the completion of $\mathcal{C}^{\infty}_{c}(\mathbb{R}^{N})$ with respect to the norm

[TABLE]

The reduction-by-inversion approach allows to reformulate the system (1.1) as a higher-order quasilinear problem. Indeed, $(u,v)$ is a (strong) solution to (1.1) if and only if $u$ is a (weak) solution of

[TABLE]

and $v:=-|\Delta u|^{q^{\prime}-2}\Delta u$; see Lemma 2.1 below.

Using a concentration-compactness argument, P.-L. Lions showed in [15] that (1.3) has a positive solution when $(p,q)$ satisfies (1.2). Thus, a positive solution $(u,v)$ of (1.1)-(1.2) exists. Moreover, $u$ and $v$ are radially symmetric, and they are unique up to translations and dilations [13, 6]. This solution does not have, in general, an explicit formula like in the case of the scalar problem

[TABLE]

where $2^{*}:=\frac{2N}{N-2}$ is the critical Sobolev exponent; but the precise decay rates of $u$ and $v$ at infinity can be deduced and they depend in a subtle way on the value of the exponents $p$ and $q$; see [13, Theorem 2].

In this paper, we establish the existence of sign-changing solutions to (1.1)-(1.2). Our main result is the following one. We use $\lfloor x\rfloor$ to denote the greatest integer less than or equal to $x$.

Theorem 1.1**.**

If $(p,q)$ satisfies (1.2), then the system (1.1) has at least $\lfloor\frac{N}{4}\rfloor$ nonradial sign-changing solutions, i.e., both components $u$ and $v$ change sign.

The solutions given by Theorem 1.1 have some explicit symmetries which provide some information on their shape; see Lemma 4.2 and Remark 4.3.

Theorem 1.1 seems to be the first result regarding the existence of entire sign-changing solutions to (1.1)-(1.2), except for the particular cases $p=q=2^{*}$ and $q=2$ (or $p=2$).

When $p=q=2^{*}$, the solutions to (1.1) are $(u,u)$, where $u$ is a solution to the Yamabe problem (1.4), which is invariant under Möbius transformations. Taking advantage of this fact, W. Ding established the existence of infinitely many sign-changing solutions to (1.4) in [12]. They are invariant under the action of a group of conformal transformations whose orbits have positive dimension. Bubbling sign-changing solutions were obtained by del Pino, Musso, Pacard, and Pistoia in [11], using the Lyapunov-Schmidt reduction method. Their solutions are different from those in [12].

When $q=2$, (1.3) becomes the Paneitz problem

[TABLE]

with $2_{*}:=\frac{2N}{N-4}$. Inspired by Ding’s approach, Bartsch, Schneider and Weth [2] established the existence of infinitely many solutions to (1.5) and to more general polyharmonic problems which, like (1.5), are invariant under conformal transformations. We stress that the solutions given by Theorem 1.1 for $q=2$ are different from those in [2].

Unfortunately, the approach followed in [12, 2] does not apply to arbitrary $(p,q)$ on the critical hyperbola because, even though the problem (1.3) is invariant under Euclidean transformations and dilations, it is not invariant under Möbius transformations in general; see Proposition 4.4 below. On the other hand, the Lyapunov-Schmidt reduction method used in [11] relies on a good knowledge of the linearized problem and on the explicit form of the positive entire solution to (1.4), but this information is not available for the Lane-Emden system (1.1).

Yet another kind of sign-changing solutions to the Yamabe problem (1.4), different from those in [12, 11], was recently discovered in [7]. They arise as limit profiles of symmetric sign-changing minimizing sequences for the purely critical exponent problem in a bounded domain.

To prove Theorem 1.1 we follow the strategy of [7], that is, we analyze the behavior of minimizing sequences, with a specific kind of symmetries, for the critical problem

[TABLE]

when $\Omega$ is the unit ball. These symmetric functions, which we call $\phi$-equivariant, may be chosen to be sign-changing by construction; see Section 3. Unlike the conformal symmetries considered in [12, 2] which prevent blow-up, our symmetries are given by linear isometries which have fixed points, thus allowing blow-up. We impose some conditions on the symmetries to ensure that the blow-up profile of $\phi$-equivariant minimizing sequences is $\phi$-equivariant; see assumptions (S1) and (S2) below.

There are two main sources of difficulties in performing the blow-up analysis: the nonlinear nature of the operator in equation (1.3) and the fact that it is of higher order. Like for the purely critical $p$-Laplacian problem [16, 9], it is delicate to show that the weak limit of a minimizing sequence for (1.3) solves a limit problem when $q\neq 2$. For the $p$-Laplacian this is usually achieved by using suitable truncations, but due to the higher-order nature of (1.3) this approach cannot be applied (a truncation may cause a jump discontinuity of the gradient, preventing the truncated function from being twice weakly differentiable). We circumvent this difficulty using a more general approach based on mollifications.

The concentration and blow-up behavior of $\phi$-equivariant minimizing sequences for (1.6) in a bounded domain is given by Theorem 3.5 below. This result contains an existence alternative: it asserts that there exists a $\phi$-equivariant minimizer for (1.6), either in the unit ball, or in a half-space, or in the whole space $\mathbb{R}^{N}$. Moreover, due to the presence of fixed points, these minimizers have the same energy; see Lemma 3.3. Therefore, anyone of them is a $\phi$-equivariant least energy solution to the problem (1.3) in the whole space $\mathbb{R}^{N}$. We stress that, unlike for the Laplacian, a general unique continuation property is not available, as far as we know, for the problem (1.6). So one cannot discard the possibility of having solutions to (1.3) which vanish outside a ball or in a half-space.

Finally, we point out two limitations of our method. Firstly, it cannot be applied when $N=3$, because there are no groups in this dimension with the properties that we need; see Remark 4.3. Secondly, in contrast with the cases $q=2^{*}$ and $q=2$ considered in [12, 2], our approach yields only finitely many solutions. The questions whether the system (1.1) has a sign-changing solution in dimension $3$, or whether it has infinitely many solutions in every dimension, remain open.

To close this introduction we mention some possible generalizations of Theorem 1.1. For the sake of clarity, in this paper we have focused on the system (1.1) and the associated higher-order problem (1.3); but an inspection of the proofs shows that the same approach can be used to study the existence of finitely many entire nodal solutions to the Hardy-Littlewood-Sobolev system

[TABLE]

with $m\in{\mathbb{N}}$ and $\frac{1}{p}+\frac{1}{q}=\frac{N-2m}{N}$, or the associated higher-order problem

[TABLE]

The left-hand side of the equation above can be regarded as a quasilinear version of the polyharmonic operator. Similarly, one could also consider the problem

[TABLE]

where $m\in{\mathbb{N}}\cup\{0\}$ and $\frac{1}{p}+\frac{1}{q}=\frac{N-2m-1}{N}$. Note that this problem reduces to the critical $p$-Laplacian problem if $m=0$. The existence of finitely many sign-changing solutions, in this particular case, was shown in [9]. The approach we present here can be used to extend Theorem 1.1 to anyone of these problems.

The paper is organized as follows. In Section 2 we explain the reduction-by-inversion approach and show the equivalence between (1.1) and (1.3). In Section 3 we introduce our symmetric variational framework and we give a precise description of the concentration and blow-up behavior of $\phi$-equivariant minimizing sequences for the higher-order problem (1.6) in a bounded domain. Our main result, Theorem 1.1, is proved in Section 4. Finally, in an appendix, we give conditions which guarantee that the weak limit of a minimizing sequence for the variational functional is a critical point of this functional.

2 Reduction by inversion

From now on, we assume that $(p,q)$ lies on the critical hyperbola (1.2). Then $p,q>\frac{N}{N-2}$ and $p=\frac{Nq^{\prime}}{N-2q^{\prime}}$, where $q^{\prime}:=\frac{q}{q-1}$.

We consider the Banach space

[TABLE]

endowed with the norm

[TABLE]

This space is the completion of $\mathcal{C}^{\infty}_{c}(\mathbb{R}^{N})$ with respect to $\|\cdot\|_{q^{\prime}}$, and $p$ is the critical exponent for the Sobolev embedding $D^{2,q^{\prime}}(\mathbb{R}^{N})\hookrightarrow L^{p}(\mathbb{R}^{N})$.

A solution to the system (1.1) is a critical point $(u,v)$ of the functional

[TABLE]

defined in the space $D^{1,(q^{\prime})^{*}}(\mathbb{R}^{N})\times D^{1,(p^{\prime})^{*}}(\mathbb{R}^{N})$, where $r^{*}:=\frac{Nr}{N-r}$. The critical points belong to $D^{2,q^{\prime}}(\mathbb{R}^{N})\times D^{2,p^{\prime}}(\mathbb{R}^{N})$ and are, therefore, strong solutions.

By a solution $u$ to the problem (1.3) we mean a weak solution, i.e., a critical point of the functional

[TABLE]

defined in the Banach space $D^{2,q^{\prime}}(\mathbb{R}^{N})$. Its derivative is given by

[TABLE]

The following lemma establishes the equivalence between solutions to the higher-order quasilinear problem (1.3) and to the system (1.1). This relation is sometimes called reduction-by-inversion. We refer to the surveys [4, 10, 17] for an overview of the diversity of methods used in the study of (1.1) and more general Hamiltonian systems.

Lemma 2.1**.**

$u$* is a solution of (1.3) and $v=-|\Delta u|^{q^{\prime}-2}\Delta u$ iff $(u,v)$ is a solution of (1.1).*

Proof.

Let $u\in D^{2,q^{\prime}}(\mathbb{R}^{N})$ be a solution of (1.3) and set $v:=-|\Delta u|^{q^{\prime}-2}\Delta u$. It is easy to see that $v$ coincides a.e. in $\mathbb{R}^{N}$ with the (unique) solution to the problem $-\Delta w=|u|^{p-2}u$, $w\in D^{1,p^{\prime}}(\mathbb{R}^{N})$, which belongs to $D^{2,p^{\prime}}(\mathbb{R}^{N})$. Therefore, $v\in D^{2,p^{\prime}}(\mathbb{R}^{N})$. Note that $|v|^{q-2}v=-\Delta u$. Hence, for every $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N})$, we have that

[TABLE]

i.e., $(u,v)$ solves (1.1). The converse is proved in a similar way. ∎

3 Sign-changing minimizing sequences

To produce sign-changing solutions we introduce suitable symmetries, as in [7, 9].

Let $G$ be a closed subgroup of the group $O(N)$ of linear isometries of $\mathbb{R}^{N}$ and let $\phi:G\rightarrow\mathbb{Z}_{2}:=\{1,-1\}$ be a continuous homomorphism of groups. We write $Gx:=\{gx:g\in G\}$ for the $G$-orbit of a point $x\in\mathbb{R}^{N}$. From now on, we assume that $G$ and $\phi$ have the following properties:

• $(\mathbf{S1})$

For each $x\in\mathbb{R}^{N}$, either $\dim(Gx)>0$ or $Gx=\{x\}$.

• $(\mathbf{S2})$

There exists $\xi\in\mathbb{R}^{N}$ such that $\{g\in G:g\xi=\xi\}\subset\ker\phi$.

Now, let $\Omega$ be a $G$-invariant domain in $\mathbb{R}^{N}$, i.e., $Gx\subset\Omega$ for every $x\in\Omega$. A function $u:\Omega\to\mathbb{R}$ is called $\phi$-equivariant if

[TABLE]

If $\phi\equiv 1$, then a $\phi$-equivariant function is simply a $G$-invariant function. On the other hand, if $\phi:G\to\mathbb{Z}_{2}$ is surjective, then every nontrivial $\phi$-equivariant function is nonradial and changes sign.

We denote the closure of $\mathcal{C}_{c}^{\infty}(\Omega)$ in $D^{2,q^{\prime}}(\mathbb{R}^{N})$ by $D_{0}^{2,q^{\prime}}(\Omega)$, and set

[TABLE]

We say that $u$ is a solution of

[TABLE]

if $u$ is a critical point of the $\mathcal{C}^{1}$-functional $J:D_{0}^{2,q^{\prime}}(\Omega)\to\mathbb{R}$ given by (2.1).

Let $\mathcal{C}_{c}^{\infty}(\Omega)^{\phi}:=\{u\in\mathcal{C}_{c}^{\infty}(\Omega):u\text{ is }\phi\text{-equivariant}\}$. Given $\varphi\in\mathcal{C}_{c}^{\infty}(\Omega)$, we define

[TABLE]

where $\mu$ is the Haar measure on $G$. Then $\varphi_{\phi}\in\mathcal{C}_{c}^{\infty}(\Omega)^{\phi}$.

Lemma 3.1**.**

If $u\in D^{2,q^{\prime}}_{0}(\Omega)^{\phi}$, then

[TABLE]

Consequently, if $J^{\prime}(u)\vartheta=0$ for every $\vartheta\in\mathcal{C}_{c}^{\infty}(\Omega)^{\phi}$, then $J^{\prime}(u)\varphi=0$ for every $\varphi\in\mathcal{C}_{c}^{\infty}(\Omega)$, i.e., $u$ is a solution to the problem (3.1).

Proof.

Note that $\Delta(v\circ g)=(\Delta v)\circ g$ for every $v\in D^{2,q^{\prime}}_{0}(\Omega)$, $g\in G$. So, as $u$ is $\phi$-equivariant, we have that $\Delta u$ is $\phi$-equivariant. Also,

[TABLE]

Fubini’s theorem and a suitable change of variables yield

[TABLE]

Hence, $J^{\prime}(u)[\varphi_{\phi}]=J^{\prime}(u)\varphi$, as claimed. ∎

The nontrivial $\phi$-equivariant solutions to (3.1) belong to the set

[TABLE]

where $|\cdot|_{p}$ denotes the norm in $L^{p}(\mathbb{R}^{N})$. Define

[TABLE]

Property $(\mathbf{S2})$ guarantees that the space $D_{0}^{2,q^{\prime}}(\Omega)^{\phi}$ is infinite dimensional; see [5]. Therefore, $\mathcal{N}^{\phi}(\Omega)\neq\emptyset$ and $c^{\phi}(\Omega)\in\mathbb{R}$. Note that $(\mathbf{S2})$ is trivially satisfied if $\phi\equiv 1$.

Next we describe the limit profile of minimizing sequences for the functional $J$ on $\mathcal{N}^{\phi}(\Omega)$. As we shall see, property $(\mathbf{S1})$ guarantees that the limit profile is $\phi$-equivariant; see Theorem 3.5. So it will be sign-changing if $\phi$ is surjective.

We start by listing some properties of $\mathcal{N}^{\phi}(\Omega)$.

Lemma 3.2**.**

1. $(a)$

There exists $a_{0}>0$ such that $\|u\|_{q^{\prime}}\geq a_{0}$ for every $u\in\mathcal{N}^{\phi}(\Omega)$. 2. $(b)$

$\mathcal{N}^{\phi}(\Omega)$* is a $\mathcal{C}^{1}$-Banach submanifold of $D_{0}^{2,q^{\prime}}(\Omega)^{\phi}$, and a natural constraint for $J$.* 3. $(c)$

Let $\mathscr{T}:=\left\{\sigma\in\mathcal{C}^{0}\left([0,1],D_{0}^{2,q^{\prime}}(\Omega)^{\phi}\right):\sigma(0)=0,\,\sigma(1)\neq 0,\,J(\sigma(1))\leq 0\right\}.$ Then,

[TABLE]

Proof.

The proof is similar to that of [9, Lemma 2.1] and we omit it. ∎

As usual, let

[TABLE]

denote the set of $G$-fixed points in $\Omega$. The proof of the next lemma is similar to that of [9, Lemma 2.3]. We include it here for the sake of completeness.

Lemma 3.3**.**

If $\Omega$ is a $G$-invariant smooth domain in $\mathbb{R}^{N}$ and $\Omega^{G}\neq\emptyset$, then

[TABLE]

Proof.

As

[TABLE]

one has that $c_{\infty}^{\phi}\leq c^{\phi}(\Omega)$. For the opposite inequality, we use the translation and dilation invariance of the problem (1.3). Fix $x_{0}\in\Omega^{G}$ and let $(\varphi_{k})$ be a sequence in $\mathcal{N}^{\phi}(\mathbb{R}^{N})\cap\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N})$ such that $J(\varphi_{k})\to c_{\infty}^{\phi}$. Since $\varphi_{k}$ has compact support, we may choose $\varepsilon_{k}>0$ such that the support of $\tilde{\varphi}_{k}(x):=\varepsilon_{k}^{-N/p}\varphi_{k}(\varepsilon_{k}^{-1}(x-x_{0}))$ is contained in $\Omega$. As $x_{0}$ is a $G$-fixed point, $\tilde{\varphi}_{k}$ is $\phi$-equivariant. Thus, $\tilde{\varphi}_{k}\in\mathcal{N}^{\phi}(\Omega)$ and, hence,

[TABLE]

Letting $k\to\infty$ we conclude that $c^{\phi}(\Omega)\leq c_{\infty}^{\phi}$. ∎

Lemma 3.4**.**

If $G$ satisfies $(\mathbf{S1})$ then, for every pair of sequences $(\varepsilon_{k})$ in $(0,\infty)$ and $(x_{k})$ in $\mathbb{R}^{N}$, there exists a positive constant $C_{0}>0$ and a sequence $(\xi_{k})$ in $\mathbb{R}^{N}$ such that, after passing to a subsequence,

[TABLE]

Furthermore, one of the following statements holds true:

1. $(a)$

either $\xi_{k}\in(\mathbb{R}^{N})^{G}$ for all $k\in{\mathbb{N}}$, 2. $(b)$

or, for each $m\in\mathbb{N}$, there exist $g_{1},\ldots,g_{m}\in G$ such that

[TABLE]

Proof.

A detailed proof can be found in [9, Lemma 2.4]. Here we just give a sketch. For each $k\in{\mathbb{N}}$ we write $x_{k}=z_{k}+y_{k}$, where $z_{k}\in({\mathbb{R}}^{N})^{G}$ and $y_{k}\in[({\mathbb{R}}^{N})^{G}]^{\bot}$. If $(\varepsilon_{k}^{-1}y_{k})$ does not contain a bounded subsequence, an easy argument shows that $\xi_{k}:=x_{k}$ satisfies $(b)$ and (3.4). On the other hand, if $(\varepsilon_{k}^{-1}y_{k})$ has a bounded subsequence, then $\xi_{k}:=z_{k}$ satisfies $(a)$ and (3.4). ∎

The next theorem is our main concentration result. The proof follows the general strategy of [18, Theorem 8.13] and [9, Theorem 2.5], but some parts require a different and more careful treatment.

Theorem 3.5**.**

Assume $(\mathbf{S1})$ and $(\mathbf{S2})$. Let $\Omega$ be a $G$-invariant bounded smooth domain in $\mathbb{R}^{N}$ and $(u_{k})$ be a sequence such that

[TABLE]

Then, up to a subsequence, one of the following two possibilities occurs:

1. (I)

either $(u_{k})$ converges strongly in $D_{0}^{2,q^{\prime}}(\Omega)$ to a minimizer of $J$ on $\mathcal{N}^{\phi}(\Omega)$, 2. (II)

or there exist a sequence of $G$-fixed points $(\xi_{k})$ in $\mathbb{R}^{N}$, a sequence $(\varepsilon_{k})\in(0,\infty)$, and a nontrivial solution $w$ to the problem

[TABLE]

with the following properties:

1. (i)

$\varepsilon_{k}\to 0$,  $\xi_{k}\to\xi$,   $\xi\in(\bar{\Omega})^{G}$,  and  $\varepsilon_{k}^{-1}\mathrm{dist}(\xi_{k},\Omega)\to d\in[0,\infty]$. 2. (ii)

If $d=\infty$, then $\mathbb{E}=\mathbb{R}^{N}$ and  $\xi_{k}\in\Omega$. 3. (iii)

If $d\in[0,\infty)$, then $\xi\in\partial\Omega$ and $\mathbb{E}=\{x\in\mathbb{R}^{N}:x\cdot\nu>\bar{d}\}$, where $\nu$ is the inward pointing unit normal to $\partial\Omega$ at $\xi$ and $\bar{d}\in\{d,-d\}$. Moreover, $\mathbb{E}$ is $G$-invariant, $\mathbb{E}^{G}\neq\emptyset$, and $\Omega^{G}\neq\emptyset$. 4. (iv)

$w\in\mathcal{N}^{\phi}(\mathbb{E})$* and $J(w)=c_{\infty}^{\phi}$.* 5. (v)

$\lim\limits_{k\to\infty}\left\|u_{k}-\varepsilon_{k}^{-N/p}w\left(\frac{x-\xi_{k}}{\varepsilon_{k}}\right)\right\|=0$.

Proof.

As $q^{\prime}>1$ and

[TABLE]

the sequence $(u_{k})$ is bounded in $D_{0}^{2,q^{\prime}}(\Omega)^{\phi}$ and, after passing to a subsequence, there is $u\in D_{0}^{2,q^{\prime}}(\Omega)^{\phi}$ such that $u_{k}\rightharpoonup u$ weakly in $D_{0}^{2,q^{\prime}}(\Omega)^{\phi}$. Hence, by Proposition A.1, $J^{\prime}(u)\varphi=0$ for all $\varphi\in C^{\infty}_{c}(\Omega)^{\phi}$. We consider two cases:

(I) If $u\neq 0$, then $u\in\mathcal{N}^{\phi}(\Omega)$. So, from (3.6) and our assumptions, we get

[TABLE]

Hence, $u_{k}\to u$ strongly in $D_{0}^{2,q^{\prime}}(\Omega)^{\phi}$ and $J(u)=c^{\phi}(\Omega)$.

(II) Assume that $u=0$. Fix $\delta\in(0,\frac{N}{4}c^{\phi}_{\infty})$, where $c^{\phi}_{\infty}:=c^{\phi}(\mathbb{R}^{N})\leq c^{\phi}(\Omega)$. Since

[TABLE]

there are bounded sequences $(\varepsilon_{k})$ in $(0,\infty)$ and $(x_{k})$ in $\mathbb{R}^{N}$ such that, after passing to a subsequence,

[TABLE]

where $B_{r}(x):=\{z\in\mathbb{R}^{N}:|z-x|<r\}$. For this choice of $(x_{k})$ and $(\varepsilon_{k})$ let $C_{0}>0$ and $(\xi_{k})$ as in Lemma 3.4. Then, $|g_{k}x_{k}-\xi_{k}|\leq C_{0}\varepsilon_{k}$ for some $g_{k}\in G$ and, as $|u_{k}|$ is $G$-invariant, setting $C_{1}:=C_{0}+1$, we have that

[TABLE]

This implies, in particular, that

[TABLE]

We claim that $\xi_{k}\in(\mathbb{R}^{N})^{G}$. Otherwise, for each $m\in\mathbb{N}$, Lemma 3.4 yields $m$ elements $g_{1},\ldots,g_{m}\in G$ such that $B_{C_{1}\varepsilon_{k}}(g_{i}\xi_{k})\cap B_{C_{1}\varepsilon_{k}}(g_{j}\xi_{k})=\emptyset$ if $i\neq j$, for $k$ large enough, and from (3.7) we would get that

[TABLE]

for every $m\in\mathbb{N}$, which is a contradiction. This proves that $\xi_{k}\in(\mathbb{R}^{N})^{G}$.

Define

[TABLE]

Since $u_{k}$ is $\phi$-equivariant and $\xi_{k}$ is a $G$-fixed point, we have that $w_{k}$ is $\phi$-equivariant. Note that

[TABLE]

Moreover, $(w_{k})$ is a bounded sequence in $D^{2,q^{\prime}}(\mathbb{R}^{N})$. Hence, there is $w\in D^{2,q^{\prime}}(\mathbb{R}^{N})^{\phi}$ such that, up to a subsequence, $w_{k}\rightharpoonup w$ weakly in $D^{2,q^{\prime}}(\mathbb{R}^{N})^{\phi}$, and, by the Rellich-Kondrashov theorem,

[TABLE]

We claim that $w\neq 0$. To prove this, first note that, for any given $\varphi\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N})$, if we set

[TABLE]

then, using the notation in (3.2), we have that $(\varphi w_{k})_{\phi}=\vartheta w_{k}$,  $\vartheta_{k}u_{k}$ is $\phi$-equivariant, and the sequence $(\vartheta_{k}u_{k})$ is bounded in $D_{0}^{2,q^{\prime}}(\Omega)^{\phi}$. So Lemma 3.1 and a suitable rescaling yield

[TABLE]

because $J^{\prime}(u_{k})\to 0$ in $(D^{2,q^{\prime}}_{0}(\Omega)^{\phi})^{\prime}$. Now, arguing by contradiction, assume that $w=0$. Then, for any $\varphi\in\mathcal{C}_{c}^{\infty}(B_{1}(z))$ with $z\in\mathbb{R}^{N}$, we have by (3.9), (3.10), (3.11), Hölder’s inequality, and Sobolev’s inequality, that

[TABLE]

Therefore, $\|\varphi^{2}w_{k}\|_{q^{\prime}}=o(1)$ and, hence, $|\varphi^{2}w_{k}|_{p}=o(1)$ for any $\varphi\in\mathcal{C}_{c}^{\infty}(B_{1}(z))$, $z\in\mathbb{R}^{N}$. It follows that $w_{k}\to 0$ in $L^{p}_{\mathrm{loc}}(\mathbb{R}^{N})$, contradicting (3.9).

Since $u_{k}\rightharpoonup 0$ and $w_{k}\rightharpoonup w\neq 0$ we deduce that $\varepsilon_{k}\to 0$. Moreover, passing to a subsequence, we have that $\xi_{k}\to\xi\in(\mathbb{R}^{N})^{G}$. Let

[TABLE]

We consider two cases:

• (a)

If $d=\infty$ then, by (3.8), we have that $\xi_{k}\in\Omega$. Hence, for every compact subset $X$ of $\mathbb{R}^{N}$, there exists $k_{0}$ such that $X\subset\Omega_{k}$ for all $k\geq k_{0}$. In this case, we set $\mathbb{E}:=\mathbb{R}^{N}$.

• (b)

If $d\in[0,\infty)$ then, as $\varepsilon_{k}\to 0$, we have that $\xi\in\partial\Omega$. If a subsequence of $(\xi_{k})$ is contained in $\bar{\Omega}$ we set $\bar{d}:=-d$, otherwise we set $\bar{d}:=d$. We consider the half-space

[TABLE]

where $\nu$ is the inward pointing unit normal to $\partial\Omega$ at $\xi$. Since $\xi$ is a $G$-fixed point, so is $\nu$. Thus, $\Omega^{G}\neq\emptyset$, $\mathbb{H}$ is $G$-invariant and $\mathbb{H}^{G}\neq\emptyset$. It is easy to see that, if $X$ is compact and $X\subset\mathbb{H}$, there exists $k_{0}$ such that $X\subset\Omega_{k}$ for all $k\geq k_{0}$. Moreover, if $X$ is compact and $X\subset\mathbb{R}^{N}\smallsetminus\bar{\mathbb{H}}$, then $X\subset\mathbb{R}^{N}\smallsetminus\Omega_{k}$ for $k$ large enough. As $w_{k}\to w$ a.e. in $\mathbb{R}^{N}$, this implies that $w=0$ a.e. in $\mathbb{R}^{N}\smallsetminus\mathbb{H}$. So $w\in D_{0}^{2,q^{\prime}}(\mathbb{H})^{\phi}$. In this case, we set $\mathbb{E}:=\mathbb{H}$.

For $\varphi\in C^{\infty}_{c}({\mathbb{E}})^{\phi}$ and $\psi\in C^{\infty}_{c}({\mathbb{E}})^{G}$ define

[TABLE]

Then $\varphi_{k}$ and $\psi_{k}$ are $\phi$-equivariant. Observe that ${\rm supp}(\varphi_{k})$ and ${\rm supp}(\psi_{k})$ are contained in $\Omega$ for $k$ sufficiently large, and $(\varphi_{k})$ and $(\psi_{k})$ are bounded in $D^{2,q^{\prime}}_{0}(\Omega)$. Therefore,

[TABLE]

Then, by Proposition A.1, $w$ is a nontrivial solution of (3.5). Lemma 3.3 asserts that $c^{\phi}(\Omega)=c^{\phi}({\mathbb{E}})=c^{\phi}_{\infty}$. Hence,

[TABLE]

This implies that $J(w)=c^{\phi}_{\infty}$ and $w_{k}\to w$ strongly in $D^{2,q^{\prime}}({\mathbb{R}}^{N})$. Consequently,

[TABLE]

and the proof is complete. ∎

An immediate consequence of the previous theorem is the following existence result.

Corollary 3.6**.**

Assume that $G$ and $\phi$ satisfy $(\mathbf{S1})$ and $(\mathbf{S2})$, and let $\Omega$ be a $G$-invariant bounded smooth domain in $\mathbb{R}^{N}$ such that $\Omega^{G}=\emptyset$. Then the problem

[TABLE]

has a least energy solution. This solution is sign-changing if $\phi:G\to\mathbb{Z}_{2}$ is surjective.

Proof.

By statements $(a)$ and $(c)$ of Lemma 3.2, and [18, Theorem 2.9], there exists a sequence $(u_{k})$ such that

[TABLE]

As $\Omega$ does not contain $G$-fixed points, the statement (II) in Theorem 3.5 cannot hold true. Hence, $J$ attains its minimum on $\mathcal{N}^{\phi}(\Omega)$. ∎

In fact, arguing as in [8, Corollary 3.2], one should be able to prove that, under the assumptions of Corollary 3.6, problem (3.12) has an unbounded sequence of solutions.

Note that the solution $u$ given by Corollary 3.6 does not yield a solution of the Dirichlet system

[TABLE]

due to the incompatibility of the boundary conditions. To obtain a solution to this system we would need to replace $D^{2,q^{\prime}}_{0}(\Omega)$ with the Navier space $Y(\Omega)=D^{2,q^{\prime}}(\Omega)\cap D^{1,q^{\prime}}_{0}(\Omega)$ in problem (3.12); see, e.g., [4, Section 4]. Observe, however, that there is no energy-preserving embedding of $Y(\Omega)$ into $Y({\mathbb{R}}^{N})$, and this is an important property required in our method; see (3.3).

4 Entire nodal solutions

In this section we prove our main theorem. We start with the following existence result.

Theorem 4.1**.**

Let $G$ be a closed subgroup of $O(N)$ and $\phi:G\rightarrow\mathbb{Z}_{2}$ be a continuous homomorphism satisfying $\mathbf{(S1)}$ and $\mathbf{(S2)}$. Then $J$ attains its minimum on $\mathcal{N}^{\phi}(\mathbb{R}^{N})$. Consequently, the problem (1.3) has a nontrivial $\phi$-equivariant solution. This solution is sign-changing if $\phi:G\to\mathbb{Z}_{2}$ is surjective.

Proof.

The unit ball $\mathbb{B}=\{x\in\mathbb{R}^{N}:|x|<1\}$ is $G$-invariant for every subgroup $G$ of $O(N)$. Note that, as $0\in\mathbb{B}^{G}$, we have that $c^{\phi}(\mathbb{B})=c_{\infty}^{\phi}$ by Lemma 3.3. Furthermore, by statements $(a)$ and $(c)$ of Lemma 3.2, and [18, Theorem 2.9], there exists a sequence $(u_{k})$ such that

[TABLE]

Now, applying Theorem 3.5 we have the following existence alternative: there exists $u\in\mathcal{N}^{\phi}(\Theta)$ with $J(u)=c_{\infty}^{\phi}$, either for $\Theta=\mathbb{B}$, or for some half-space $\Theta$, or for $\Theta=\mathbb{R}^{N}$. As $\mathcal{N}^{\phi}(\Theta)\subset\mathcal{N}^{\phi}(\mathbb{R}^{N})$ for any $G$-invariant domain $\Theta$, we conclude that, in any case, $J$ attains its minimum on $\mathcal{N}^{\phi}(\mathbb{R}^{N})$. ∎

Note that, if $\phi\equiv 1$, the solution given by the previous theorem is a least energy $G$-invariant solution. The ground state solution obtained by Lions in [15] is radial, hence, it is $G$-invariant. So Theorem 4.1 says nothing new in this case.

The next lemma exhibits surjective homomorphisms which yield different sign-changing minimizers. It was proved in [9, Lemma 3.2]. We give the proof here again, to make the symmetries explicit.

Lemma 4.2**.**

For each $j=1,\ldots,\lfloor\frac{N}{4}\rfloor$, there exist a closed subgroup $G_{j}$ of $O(N)$ and a continuous homomorphism $\phi_{j}:G_{j}\to\mathbb{Z}_{2}$ with the following properties:

• (a)

$\phi:G\to\mathbb{Z}_{2}$* is surjective,*

• (b)

$G_{j}$* and $\phi_{j}$ satisfy $\mathbf{(S1)}$ and $\mathbf{(S2)}$,*

• (c)

If $u,v:\mathbb{R}^{N}\to\mathbb{R}$ are nontrivial functions, $u$ is $\phi_{i}$-equivariant, and $v$ is $\phi_{j}$-equivariant with $i<j$, then $u\neq v$.

Proof.

Let $\Gamma$ be the group generated by $\{\mathrm{e}^{\mathrm{i}\theta},\varrho:\theta\in[0,2\pi)\}$, acting on $\mathbb{C}^{2}$ by

[TABLE]

and let $\phi:\Gamma\to\mathbb{Z}_{2}$ be the homomorphism given by $\phi(\mathrm{e}^{\mathrm{i}\theta}):=1$ and $\phi(\varrho):=-1$. Note that the $\Gamma$-orbit of a point $z\in\mathbb{C}^{2}$ is the union of two circles that lie in orthogonal planes if $z\neq 0$, and it is $\{0\}$ if $z=0$.

Set $n=\lfloor\frac{N}{4}\rfloor$, $\Lambda_{j}:=O(N-4j)$ if $j=1,...,n-1$, and $\Lambda_{n}:=\{1\}$. Then the $\Lambda_{j}$-orbit of a point $y\in\mathbb{R}^{N-4j}$ is an $(N-4j-1)$-dimensional sphere if $j=1,\ldots,n-1$, and it is a single point if $j=n$.

Define $G_{j}:=\Gamma^{j}\times\Lambda_{j}$, acting on $\mathbb{R}^{N}\equiv(\mathbb{C}^{2})^{j}\times\mathbb{R}^{N-4j}$ by

[TABLE]

where $\gamma_{i}\in\Gamma$, $\eta\in\Lambda_{j}$, $z_{i}\in\mathbb{C}^{2}$, and $y\in\mathbb{R}^{N-4j}$, and let $\phi_{j}:G_{j}\rightarrow\mathbb{Z}_{2}$ be the homomorphism

[TABLE]

The $G_{j}$-orbit of $(z_{1},\ldots,z_{j},y)$ is the product of orbits

[TABLE]

Clearly, $\phi_{j}$ is surjective, and $G_{j}$ and $\phi_{j}$ satisfy $\mathbf{(S1)}$ and $\mathbf{(S2)}$ for each $j=1,\ldots,n$.

Now we prove (c). If $u$ is $\phi_{i}$-equivariant, $v$ is $\phi_{j}$-equivariant with $i<j$, and $u(x)=v(x)\neq 0$ for some $x=(z_{1},\ldots,z_{j},y)\in(\mathbb{C}^{2})^{j}\times\mathbb{R}^{N-4j}$, then, as

[TABLE]

we have that $u(z_{1},\ldots,\varrho z_{j},y)\neq v(z_{1},\ldots,\varrho z_{j},y)$. This proves that $u\neq v$. ∎

Proof of Theorem 1.1.

Apply Theorem 4.1 to each of the $\phi_{j}:G_{j}\to\mathbb{Z}_{2}$ given by Lemma 4.2 to obtain pairwise distinct sign-changing solutions $u_{1},\ldots,u_{n}$ to the problem (1.3). Set $v_{i}:=-|\Delta u_{i}|^{q^{\prime}-2}\Delta u_{i}$. Since $u_{i}$ is $\phi_{i}$-equivariant, $\Delta u_{i}$ is also $\phi_{i}$-equivariant and, by Lemma 2.1, $(u_{i},v_{i})$ is a sign-changing solution to the system (1.1). ∎

Remark 4.3**.**

1) At first glance, the symmetries given by Lemma 4.2 may seem a bit involved. To illustrate the general shape of a $\phi$-equivariant function we give an explicit example. Let $f:[0,\infty)\to{\mathbb{R}}$ be any function and $u:\mathbb{C}^{2}\to{\mathbb{R}}$ be given by

[TABLE]

for $z=(z_{1},z_{2})\in\mathbb{C}^{2}$. Clearly, $u(\mathrm{e}^{\mathrm{i}\theta}z_{1},\mathrm{e}^{\mathrm{i}\theta}z_{2})=u(z_{1},z_{2})$ and $u(-\bar{z_{2}},\bar{z_{1}})=-u(z_{1},z_{2})$, i.e., $u$ is $\phi$-equivariant. Note that $u$ is nonradial and changes sign.

2) Theorem 1.1 is not optimal since, as the proof of Lemma 4.2 suggests, there can be other symmetries yielding different solutions.

3) Our approach cannot be used to obtain sign-changing solutions when $N=3$ because no closed subgroup $G$ of $O(3)$ satisfying $(\mathbf{S1})$ and $(\mathbf{S2})$ admits a surjective homomorphism $\phi:G\to\mathbb{Z}_{2}$, as can be verified by analyzing each subgroup of $O(3)$. The complete list of them is given in [1, Section 8].

To close this section we analyze the lack of Möbius invariance of problem (1.3). A Möbius transformation $\tau:\mathbb{R}^{N}\cup\{\infty\}\to\mathbb{R}^{N}\cup\{\infty\}$ is a finite composition of inversions on spheres and reflections on hyperplanes. Recall that the inversion on the sphere $S_{\varrho}(\xi):=\{x\in\mathbb{R}^{N}:|x-\xi|=\varrho\}$, $\xi\in\mathbb{R}^{N}$, $\varrho>0$, is the map $\iota_{\varrho,\xi}$ defined by

[TABLE]

Since Euclidean isometries are compositions of reflections on hyperplanes, they are Möbius transformations. Dilations $x\mapsto\lambda x$, $\lambda>0$, are also Möbius transformations; see [3].

If $\tau$ is a Möbius transformation and $u:\mathbb{R}^{N}\to\mathbb{R}$, we define $u_{\tau}$ by

[TABLE]

Then, the map $u\mapsto u_{\tau}$ is a linear isometry of $L^{p}(\mathbb{R}^{N})$, i.e., $|u_{\tau}|_{p}=|u|_{p}$ for every $u\in L^{p}(\mathbb{R}^{N})$. Next we investigate, for which values of $q$ this map is also an isometry of $D^{2,q^{\prime}}(\mathbb{R}^{N})$, i.e., $\|u_{\tau}\|_{q^{\prime}}=\|u\|_{q^{\prime}}$, as, for such values, the functional $J$ is Möbius-invariant. The answer is given by the following proposition.

Proposition 4.4**.**

Let $\iota$ be the inversion on the unit sphere $S_{1}(0)$. Then, $\|u_{\iota}\|_{q^{\prime}}=\|u\|_{q^{\prime}}$ for every $u\in D^{2,q^{\prime}}(\mathbb{R}^{N})$ if and only if $q\in\{2,2^{*}\}$.

Proof.

As $\iota(x)=\frac{x}{|x|^{2}}$, the map $u\mapsto u_{\iota}$ is the Kelvin-type transform given by

[TABLE]

Assume that $\|u_{\iota}\|_{q^{\prime}}=\|u\|_{q^{\prime}}$ for every $u\in D^{2,q^{\prime}}(\mathbb{R}^{N})$. Since $u\mapsto u_{\iota}$ is a continuous linear map, differentiating the identity $\|u_{\iota}\|_{q^{\prime}}^{q^{\prime}}=\|u\|_{q^{\prime}}^{q^{\prime}}$ and applying the chain rule we obtain

[TABLE]

Set $A_{a,b}:=\{x\in\mathbb{R}^{N}:a<|x|<b\}$ and $\alpha:=-\frac{2N}{p}$. Let $u\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N})$ be such that $u(x)=1$ if $x\in A_{1/2,2}$, and define $Lu:=\Delta(|\Delta u|^{q^{\prime}-2}\Delta u)$. Then $u_{\iota}(x)=|x|^{\alpha}$ and $L[u_{\iota}](x)=C_{N,p,q}|x|^{(q^{\prime}-1)(\alpha-2)-2}$ for every $x\in A_{1/2,2}$, where

[TABLE]

Therefore,

[TABLE]

for every $\varphi\in\mathcal{C}_{c}^{\infty}(A_{1/2,1})$. This implies that $C_{N,p,q}=0$. Hence, either $\frac{2N}{p}=N-2$, or $(q^{\prime}-1)(\frac{2N}{p}+2)=N-2$. Recall that $\frac{N}{p}+\frac{N}{q}=N-2$ and $\frac{N}{q}+\frac{N}{q^{\prime}}=N$. Thus, $\frac{2N}{p}=N-2$ iff $2^{*}=p=q$, and $(q^{\prime}-1)(\frac{2N}{p}+2)=N-2$ iff $q=2$, as claimed.

The opposite statement is the Kelvin-invariance for the Yamabe equation (1.4) and the Paneitz equation (1.5), which is well known; see [2, 12]. ∎

Appendix A The weak limits are solutions

In [16, 9] a truncation is used to show that the weak limits $u$ and $w$ in the proof of Theorem 3.5 are solutions of a limit problem. Truncations are commonly used in the study of the $p$-Laplacian, but they do not work well in the higher-order setting because gradient discontinuities prevent the truncated function from being twice weakly differentiable. Here we give a different argument, that can also be applied to more general problems, like those described in the introduction.

Let $\Theta$ be a $G$-invariant smooth domain in $\mathbb{R}^{N}$, not necessarily bounded, and let $C^{\infty}_{c}(\Theta)^{\phi}$ and $C^{\infty}_{c}(\Theta)^{G}$ denote the spaces of functions in $C^{\infty}_{c}(\Theta)$ which are $\phi$-equivariant and $G$-invariant respectively. The main result in this appendix is the following one.

Proposition A.1**.**

Let $v_{k},v\in D^{2,q^{\prime}}({\mathbb{R}}^{N})^{\phi}$ be such that $v_{k}\rightharpoonup v$ weakly in $D^{2,q^{\prime}}({\mathbb{R}}^{N})$. Assume that

[TABLE]

for every $\varphi\in C^{\infty}_{c}(\Theta)^{\phi}$ and $\psi\in C^{\infty}_{c}(\Theta)^{G}$. Then $J^{\prime}(v)\varphi=0$ for all $\varphi\in C^{\infty}_{c}(\Theta)^{\phi}$.

We start with the following lemmas. For a set $U\subset{\mathbb{R}}^{N}$, we use $|U|$ to denote its Lebesgue measure.

Lemma A.2**.**

Let $U\subset{\mathbb{R}}^{N}$ be a measurable bounded set with $|U|>0$, let $(f_{k})$ be a sequence of nonnegative functions which is bounded in $L^{1}(U)$, and let $\alpha>0$. Then, there exists $\kappa>0$ such that

[TABLE]

Proof.

Let $C\geq\int_{U}f_{k}$ for all $k\in{\mathbb{N}}$. Fix $n\in{\mathbb{N}}$ such that $(n-1)|U|>C$ and, for each $k\in{\mathbb{N}}$, set

[TABLE]

Since $f_{k}\geq 0$, we have that

[TABLE]

Then $|U_{k}|\leq\frac{C}{n-1}$ and therefore,

[TABLE]

as claimed. ∎

Lemma A.3**.**

Let $v_{k},v\in D^{2,q^{\prime}}({\mathbb{R}}^{N})^{\phi}$ be such that $v_{k}\rightharpoonup v$ weakly in $D^{2,q^{\prime}}({\mathbb{R}}^{N})$. Assume that

[TABLE]

Then, after passing to a subsequence, $\Delta v_{k}\to\Delta v$ a.e. in $\Theta$.

Proof.

As shown in [14], there is a constant $C_{0}>0$, which depends only on $q^{\prime}$, such that, for every $s,t\in\mathbb{R}$,

[TABLE]

Let $v_{k}$ and $v$ as in the statement and set $f_{k}:=|\Delta v_{k}|^{q^{\prime}-2}\Delta v_{k}-|\Delta v|^{q^{\prime}-2}\Delta v$ and

[TABLE]

From (A.2), (A.1), and the fact that $v_{k}\rightharpoonup v$ weakly in $D^{2,q^{\prime}}({\mathbb{R}}^{N})$ we get that

[TABLE]

for every nonnegative $\psi\in C^{\infty}_{c}(\Theta)^{G}.$ If $q^{\prime}\geq 2$, this immediately implies that $\Delta v_{k}\to\Delta v$ a.e. in $\Theta$. If $1<q^{\prime}<2$, we argue by contradiction. Assume that, after passing to a subsequence, there is a compact set $K\subset\Theta$ with positive measure, and a constant $\mu>0$, such that

[TABLE]

Fix $\psi\in C^{\infty}_{c}(\Theta)^{G}$ nonnegative with $\psi(x)=1$ for every $x\in K$. Then, (A.3) implies that

[TABLE]

contradicting Lemma A.2. The proof is complete. ∎

Proof of Proposition A.1.

We show first that, after passing to a subsequence, $\Delta v_{k}\to\Delta v$ a.e. in $\Theta$. Let $\psi\in C^{\infty}_{c}(\Theta)^{G}$. To simplify notation, we write

[TABLE]

Note that $(f_{k})$ is bounded in $L^{q}(\mathbb{R}^{N})$, $h_{k}$ is bounded in $L^{p^{\prime}}(\mathbb{R}^{N})$, and

[TABLE]

Let $\gamma>0$. We choose mollifiers $\eta_{\varrho}\in C^{\infty}_{c}(\mathbb{R}^{N})$ with $\eta_{\varrho}\geq 0$, $\mathrm{supp}(\eta_{\varrho})\subset\overline{B_{\varrho}(0)}$ and $\int_{\mathbb{R}^{N}}\eta_{\varrho}=1$. Then, since $J^{\prime}(v_{k})[\psi(v_{k}-v)]=o(1)$, $\eta_{\varrho}*(v_{k}-v)\to v_{k}-v$ in $L^{p}(\mathbb{R}^{N})$ as $\varrho\to 0$, and $(h_{k})$ is bounded in $L^{p^{\prime}}(\mathbb{R}^{N})$, we may fix $\varrho>0$ such that, for $k$ large enough,

[TABLE]

where $K:=\mathrm{supp}(\psi)$. Now, for any $r\in[1,p]$ and $x\in K$, Hölder’s inequality yields

[TABLE]

As $v_{k}\rightharpoonup v$ weakly in $D^{2,q^{\prime}}({\mathbb{R}}^{N})$, the Rellich-Kondrashov theorem asserts that $v_{k}\to v$ in $L_{\mathrm{loc}}^{r}({\mathbb{R}}^{N})$ for every $r\in[1,p)$. Hence, $\eta_{\varrho}*(v_{k}-v)\to 0$ pointwise in $K$. Moreover, taking $r=p$, we get that $|(\eta_{\varrho}*(v_{k}-v))(x)|\leq C$ for every $x\in K$. So, the dominated convergence theorem yields

[TABLE]

and, consequently,

[TABLE]

Furthermore, since $(f_{k})$ is bounded in $L^{q}(\mathbb{R}^{N})$, we have that

[TABLE]

and

[TABLE]

because, by the Rellich-Kondrashov theorem, $|\nabla(v_{k}-v)|\to 0$ in $L_{\mathrm{loc}}^{r}({\mathbb{R}}^{N})$ for every $r\in[1,(q^{\prime})^{*})$ with $(q^{\prime})^{*}=\frac{Nq^{\prime}}{N-q^{\prime}}>q^{\prime}$. From (A), (A.5), (A.6), and (A.7), we derive

[TABLE]

Thus, by Lemma A.3, $\Delta v_{k}\to\Delta v$ a.e. in $\Theta$, as claimed.

Let $\varphi\in C^{\infty}_{c}(\Theta)^{\phi}$, $X:=\mathrm{supp}(\varphi)$, $f:=|\Delta v|^{q^{\prime}-2}\Delta v$, and $h:=|v|^{p-2}v$. As $(f_{k}-f)\Delta\varphi\to 0$ a.e. in $\Theta$, Egorov’s theorem asserts that, for any $\gamma>0$, there is a subset $Z_{\gamma}$ of $X$ with $|Z_{\gamma}|<\gamma$ such that $(f_{k}-f)\Delta\varphi\to 0$ uniformly in $X\smallsetminus Z_{\gamma}$. Therefore,

[TABLE]

A similar argument shows that

[TABLE]

Since $\gamma$ is arbitrary, we conclude that

[TABLE]

Therefore, $J^{\prime}(v)\varphi=0$ for all $\varphi\in C^{\infty}_{c}(\Theta)^{\phi}$, as claimed. ∎