Bifurcation results for problems with fractional Trudinger-Moser nonlinearity

TL;DR

This paper proves the existence of multiple solutions for a fractional Laplacian problem with critical exponential nonlinearity, extending previous results for the N-Laplacian by employing topological methods and a fractional Trudinger-Moser inequality.

Contribution

It introduces a novel approach combining cohomological linking and a recent fractional Trudinger-Moser inequality to establish bifurcation results for nonlinear problems with exponential growth.

Findings

Multiple solutions established for fractional Laplacian with exponential nonlinearity

Extension of bifurcation results from N-Laplacian to fractional operators

Application of topological methods in fractional nonlinear analysis

Abstract

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends results in the literature for the N-Laplacian operator.

Equations

$$\underset{u\in W_0^{1,N}(\Omega ),\Vert \nabla u{\Vert }_N\le 1}{\sup }{\int }_{\Omega }{e}^{{\alpha }_N|u{|}^{N^{\prime }}}dx<+\infty ,{\alpha }_N=N{\omega }_{N-1}^{1/(N-1)},$$

$$\{\begin{array}{rlrl}-{\Delta }_{N}u& =\lambda |u{|}^{N-2}u{e}^{|u{|}^{N^{\prime }}}& & \text{in }\Omega \\ u& =0& & \text{on }\partial \Omega ,\end{array}$$

$$W_0^{s,N/s}(\Omega ),\text{s∈(0,1), N≥1},$$

$$\|u\|=[u]_{s,N/s}:=\Big{(}\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}\,dxdy\Big{)}^{s/N}.$$

$$\underset{u\in W_0^{s,N/s}(\Omega ),[u{]}_{s,N/s}\le 1}{\sup }{\int }_{\Omega }{e}^{\alpha |u{|}^{N/(N-s)}}dx<+\infty ,$$

$${\alpha }_{N,s}(\Omega )={\alpha }_{N,s}^{\ast }(\Omega ).$$

$$-\frac{C_s}{2}{\int }_{\mathbb{R}}\frac{u(x+y)+u(x-y)-2u(x)}{|y{|}^{1+2s}}dy=f(u),u\in W_0^{1/2,2}(-1,1),$$

$$-\frac{C_s}{2}{\int }_{\mathbb{R}}\frac{u(x+y)+u(x-y)-2u(x)}{|y{|}^{1+2s}}dy+u=f(u),u\in W_0^{1/2,2}(\mathbb{R}),$$

$$\{\begin{array}{rlrl}(-\Delta {)}_{N/s}^{s}u& =\lambda |u{|}^{(N-2s)/s}u{e}^{|u{|}^{N/(N-s)}}& & \text{in }\Omega \\ u& =0& & \text{in }{\mathbb{R}}^N\setminus \Omega ,\end{array}$$

$$(-\Delta {)}_{N/s}^{s}u(x):=2\underset{\epsilon \searrow 0}{\lim }{\int }_{{\mathbb{R}}^N\setminus B_{\epsilon }(x)}\frac{|u(x)-u(y){|}^{(N-2s)/s}(u(x)-u(y))}{|x-y{|}^{2N}}dy,x\in {\mathbb{R}}^N.$$

$$\Psi(u):=\frac{1}{\left|u\right|_{N/s}^{N/s}},\quad u\in{\mathcal{M}},\quad{\mathcal{M}}:=\big{\{}u\in W^{s,N/s}_{0}(\Omega):\left\|u\right\|=1\big{\}}.$$

$${\lambda }_k:=\underset{\begin{array}{c}M\in \mathcal{F}\\ i(M)\ge k\end{array}}{\inf }\underset{u\in M}{\sup }\Psi (u),k\ge 1,({\lambda }_k\to +\infty )$$

$${\mu }_{N,s}(\Omega ):={\alpha }_{N,s}(\Omega {)}^{(N-s)/N}{\left(\frac{N}{s\mathcal{L}(\Omega )}\right)}^{s/N},$$

$$\lambda +{\mu }_{N,s}(\Omega ){\lambda }^{(N-s)/N}>{\lambda }_{k+1},$$

$${\lambda }_k\le \lambda <{\lambda }_{k+1}<\lambda +{\mu }_{N,s}(\Omega ){\lambda }^{(N-s)/N}$$

$$[u{]}_{s,N/s}={\left({\int }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^{N/s}}{|x-y{|}^{2N}}dxdy\right)}^{s/N}.$$

$$W_0^{s,N/s}(\Omega )=\left\{u\in L^{N/s}({\mathbb{R}}^N):[u{]}_{s,N/s}<\infty ,u=0\text{ a.e. in }{\mathbb{R}}^N\setminus \Omega \right\}.$$

$$\begin{gathered} {\int }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^{(N-2s)/s}(u(x)-u(y))(v(x)-v(y))}{|x-y{|}^{2N}}dxdy \\ =\lambda {\int }_{\Omega }|u{|}^{(N-2s)/s}u{e}^{|u{|}^{N/(N-s)}}vdx,\forall v\in W_0^{s,N/s}(\Omega ). \end{gathered}$$

$$\Phi (u)=\frac{s}{N}{\int }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^{N/s}}{|x-y{|}^{2N}}dxdy-\lambda {\int }_{\Omega }F(u)dx,u\in W_0^{s,N/s}(\Omega ),$$

$$\tilde{u}(x,y):=\frac{u(x)-u(y)}{|x-y{|}^{2s}},(x,y)\in {\mathbb{R}}^{2N}.$$

$${\Vert u_j\Vert }^{N/s}-{\Vert u_j-u\Vert }^{N/s}\to {\Vert u\Vert }^{N/s}.$$

$${\tilde{u}}_j(x,y)=\frac{u_j(x)-u_j(y)}{|x-y{|}^{2s}},\tilde{u}(x,y)=\frac{u(x)-u(y)}{|x-y{|}^{2s}},$$

$${\left|{\tilde{u}}_j\right|}_{N/s}^{N/s}-{\left|{\tilde{u}}_j-\tilde{u}\right|}_{N/s}^{N/s}\to {\left|\tilde{u}\right|}_{N/s}^{N/s}$$

$$\sup \left\{{\int }_{\Omega }{e}^{\alpha |u{|}^{N/(N-s)}}dx:u\in W_0^{s,N/s}(\Omega ),[u{]}_{s,N/s}\le 1\right\}<+\infty$$

$$\underset{j\in \mathbb{N}}{\sup }{\int }_{\Omega }{e}^{\alpha |u_j{|}^{N/(N-s)}}dx<+\infty$$

$$|u_j{|}^{N/(N-s)}\le (|u|+|u_j-u|{)}^{N/(N-s)}\le (p|u|{)}^{N/(N-s)}+(q|u_j-u|{)}^{N/(N-s)},$$

$${\int }_{\Omega }{e}^{\alpha |u_j{|}^{N/(N-s)}}dx\le {\left({\int }_{\Omega }{e}^{\alpha \tilde{p}|u{|}^{N/(N-s)}}dx\right)}^{1/p}{\left({\int }_{\Omega }{e}^{\alpha \tilde{q}|u_j-u{|}^{N/(N-s)}}dx\right)}^{1/q}$$

$${\int }_{\Omega }{e}^{\alpha \tilde{q}{\Vert u_j-u\Vert }^{N/(N-s)}|v_j{|}^{N/(N-s)}}dx,$$

$$\alpha \tilde{q}(1-{\Vert u\Vert }^{N/s}{)}^{s/(N-s)}<\beta <{\alpha }_{N,s}(\Omega ).$$

$${\int }_{\Omega }{e}^{\beta |v_j{|}^{N/(N-s)}}dx,$$

Full text

The second author is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

1. Introduction

1.1. Overview

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ with $N\geq 2$ and with Lipschitz boundary $\partial\Omega$. We denote by $\omega_{N-1}$ the measure of the unit sphere in $\mathbb{R}^{N}$ and $N^{\prime}=N/(N-1)$. Since the time when the Trudinger-Moser inequality was first proved (cf. [27, 23, 7])

[TABLE]

existence and multiplicity of solutions for various nonlinear problems with exponential nonlinearity were investigated. For instance, Adimurthi [1] proved the existence of a positive solution to the quasi-linear elliptic problem

[TABLE]

where $\Delta_{N}u:={\rm div}(|\nabla u|^{N-2}\nabla u)$ is the $N$-Laplacian operator for $0<\lambda<\lambda_{1}(N)$, being $\lambda_{1}(N)>0$ the first eigenvalue of $\Delta_{N}$ with Dirichlet boundary conditions, see also [10]. The case $N=2$ was investigated in [8, 9], where the existence of a nontrivial solution was found for $\lambda\geq\lambda_{1}$. Recently, in [28] it was proved that problem (1.1) admits a nontrivial weak solution whenever $\lambda>0$ is not an eigenvalue of $-\Delta_{N}$ in $\Omega$ with Dirichlet boundary conditions. In addition in [28] a bifurcation result for higher (nonlinear) eigenvalues (which are suitably defined via the cohomological index) is also obtained, yielding in turn multiplicity results.

The issue of Trudinger-Moser type embeddings for fractional spaces is rather delicate and only quite recently, Parini and Ruf [25] (see also the refinement obtained in [17]) provided a partial result in the Sobolev-Slobodeckij space

[TABLE]

defined as the completion of $C^{\infty}_{0}(\Omega)$ for the norm

[TABLE]

We also refer the reader to [24, 19, 21, 18] for results in a different functional framework, namely the Bessel potential spaces $H^{s,p}$. In fact, they proved that the supremum $\alpha_{N,s}(\Omega)$ of $\alpha\geq 0$ with

[TABLE]

is positive and finite. Furthermore, they proved the existence of $\alpha_{N,s}^{*}(\Omega)\geq\alpha_{N,s}(\Omega)$ such that the supremum in (1.2) is $+\infty$ for $\alpha>\alpha_{N,s}^{*}(\Omega)$. On the other hand it still remains unknown whether

[TABLE]

The case $N=1$ and $s=1/2$ was earlier considered in [16] (see also [14]), where the authors study the existence of weak solutions to the problem

[TABLE]

where $C_{s}>0$ is a suitable normalization constant. We also mention [11, 12] for other investigations in the one dimensional case on the whole space $\mathbb{R}$, facing the problem of the lack of compactness. In particular in [12], the existence of ground state solutions for the problem

[TABLE]

was proved, where $f$ is a Trudinger–Moser critical growth nonlinearity.

To the authors’ knowledge, in the framework of the Sobolev-Slobodeckij spaces $W^{s,N/s}_{0}(\Omega)$, fractional counterparts of the local quasilinear $N$-Laplacian problem (1.1) were not previously tackled in the literature. This is precisely the goal of this manuscript.

1.2. The main result

Let $N\geq 1$ and $s\in(0,1)$. In the following, the standard norm for the $L^{p}$ space will always be denoted by $|\cdot|_{p}$. For $\lambda>0,$ we consider the quasilinear problem

[TABLE]

where $(-\Delta)_{N/s}^{s}$ is the nonlinear nonlocal operator defined on smooth functions by

[TABLE]

We refer the interested reader to [22] and the references therein for an overview on recent progresses on existence, nonexistence and regularity results for equations involving the fractional $p$-laplacian operator $(-\Delta)^{s}_{p}$, $p>1$. The standard sequence of eigenvalues for $(-\Delta)_{N/s}^{s}$ via the Krasnoselskii genus does not furnish enough information on the structure of sublevels and thus the eigenvalues will be introduced via the cohomological index. We consider critical values of the functional

[TABLE]

Let ${\mathcal{F}}$ be the class of symmetric sets of ${\mathcal{M}}$, $i(M)$ the $\mathbb{Z}_{2}$-cohomological index of a $M\subset{\mathcal{F}}$ and set

[TABLE]

Consider also the positive constant

[TABLE]

being ${\mathcal{L}}$ the Lebesgue measure in $\mathbb{R}^{N}$. The following is our main result

Theorem 1.1**.**

Assume that $\lambda_{k}\leq\lambda<\lambda_{k+1}=\cdots=\lambda_{k+m}<\lambda_{k+m+1}$ for some $k,m\geq 1$ and

[TABLE]

then problem (1.3) has $m$ distinct pairs of nontrivial solutions $\pm\,u^{\lambda}_{j},\,j=1,\dots,m$ such that $u^{\lambda}_{j}\to 0$ as $\lambda\nearrow\lambda_{k+1}$. In particular, if

[TABLE]

for some $k\geq 1$, then problem (1.3) has a nontrivial solution.

This result, which follows from the results in Section 5, is nontrivial since the classical linking arguments of [8, 9] cannot be used in the quasi-linear setting. Instead the abstract machinery developed in [28] will be applied. We also would like to stress that, since the Trudinger-Moser embedding (1.2) still holds with nonoptimal exponent (contrary to the local case), it is not clear how to prove Brezis-Nirenberg type results, namely that problem (1.3) admits a nontrivial weak solution whenever $\lambda>0$ is not an eigenvalue of $(-\Delta)_{N/s}^{s}$.

2. Preliminaries

As anticipated in the introduction, we work in the fractional Sobolev space $W^{s,N/s}_{0}(\Omega)$, defined as the completion of $C^{\infty}_{0}(\Omega)$ with respect to the Gagliardo seminorm

[TABLE]

Furthermore, since $\partial\Omega$ is assumed to be Lipschitz, we have (cf. [5, Proposition B.1])

[TABLE]

A function $u\in W^{s,N/s}_{0}(\Omega)$ is a weak solution of problem (1.3) if

[TABLE]

As proved in [15, Proposition 2.12], a weak solution turns into a poinwise solution if $u\in C^{1,\gamma}_{{\rm loc}}$ for some $\gamma\in(0,1)$ sufficiently close to 1. The integral on the right-hand side is well-defined in view of [25, Proposition 3.2] and the Hölder inequality. Weak solutions coincide with critical points of the $C^{1}$ functional

[TABLE]

where $F(t)=\int_{0}^{t}f(\tau)\,d\tau$ and $f(t)=|t|^{(N-2s)/s}\,t\,e^{\,|t|^{N/(N-s)}}$.

We recall that $W^{s,N/s}_{0}(\Omega)$ is uniformly convex, and hence reflexive. Indeed, for $u\in W^{s,N/s}_{0}(\Omega)$, let

[TABLE]

Then the mapping $u\mapsto\widetilde{u}$ is a linear isometry from $W^{s,N/s}_{0}(\Omega)$ to $L^{N/s}(\mathbb{R}^{2N})$, so the uniform convexity of $L^{N/s}(\mathbb{R}^{2N})$ gives the conclusion.

We also have the following Brézis-Lieb lemma in $W^{s,N/s}_{0}(\Omega)$.

Lemma 2.1**.**

If $\left(u_{j}\right)$ is bounded in $W^{s,N/s}_{0}(\Omega)$ and converges to $u$ a.e.​ in $\Omega$, then

[TABLE]

Proof.

Let

[TABLE]

and note that $\left(\widetilde{u}_{j}\right)$ is bounded in $L^{N/s}(\mathbb{R}^{2N})$ and converges to $\widetilde{u}$ a.e.​ in $\mathbb{R}^{2N}$. Hence

[TABLE]

by the Brézis-Lieb lemma [6], where $\left|\cdot\right|_{N/s}$ denotes the norm in $L^{N/s}(\mathbb{R}^{2N})$, namely the conclusion. ∎

It was shown [25, Theorem 1.1] that the supremum $\alpha_{N,s}(\Omega)$ of all $\alpha\geq 0$ such that

[TABLE]

satisfies $0<\alpha_{N,s}(\Omega)<\infty$. The main result of this section is the following theorem, which is due to P.L. Lions [20] in the local case $s=1$.

Theorem 2.2**.**

If $\left(u_{j}\right)$ is a sequence in $W^{s,N/s}_{0}(\Omega)$ with $\left\|u_{j}\right\|=1$ for all $j\in\mathbb{N}$ and converging a.e.​ to a nonzero function $u$, then

[TABLE]

for all $\alpha<\alpha_{N,s}(\Omega)/(1-\left\|u\right\|^{N/s})^{s/(N-s)}$.

Proof.

We have

[TABLE]

where $1/p+1/q=1$. Then

[TABLE]

by the Hölder inequality, where $\widetilde{p}=p^{(2N-s)/(N-s)}$ and $\widetilde{q}=q^{(2N-s)/(N-s)}$. The first integral on the right-hand side is finite, and the second integral equals

[TABLE]

where $v_{j}=(u_{j}-u)/\left\|u_{j}-u\right\|$. By Lemma 2.1, $\left\|u_{j}-u\right\|^{N/(N-s)}\to(1-\left\|u\right\|^{N/s})^{s/(N-s)}$. Taking $q>1$ sufficiently close to $1$, let

[TABLE]

Then $\alpha\,\widetilde{q}\,\left\|u_{j}-u\right\|^{N/(N-s)}\leq\beta$ and hence the last integral is less than or equal to

[TABLE]

for all sufficiently large $j$, which is bounded since $\beta<\alpha_{N,s}(\Omega)$ and $\left\|v_{j}\right\|=1$. ∎

We close this preliminary section with a technical lemma.

Lemma 2.3**.**

For all $t\in\mathbb{R}$,

1. $(i)$

$F(t)\leq\displaystyle\frac{N-s}{N}\,\frac{tf(t)}{|t|^{N/(N-s)}}$, 2. $(ii)$

$F(t)\leq F(1)+\dfrac{s\,(N-s)}{N^{2}}\;tf(t)$, 3. $(iii)$

$\displaystyle\frac{s}{N}\;tf(t)-F(t)\geq\frac{s^{2}}{N^{2}}\,|t|^{N^{2}/s(N-s)}$, in particular, $tf(t)\geq\dfrac{N}{s}\,F(t)$, 4. $(iv)$

$F(t)\leq\dfrac{s}{N}\,|t|^{N/s}+|t|^{N^{2}/s(N-s)}\,e^{\,|t|^{N/(N-s)}}$, 5. $(v)$

$F(t)\geq\displaystyle\frac{s}{N}\,|t|^{N/s}+\frac{s\,(N-s)}{N^{2}}\,|t|^{N^{2}/s(N-s)}$.

Proof.

Since $f$ is odd, and hence $F$ is even,

[TABLE]

$(i)$ Integrating by parts,

[TABLE]

$(ii)$ For $|t|\leq 1$, $F(t)\leq F(1)$. For $|t|>1$, $F(t)=F(1)+\displaystyle\int_{1}^{|t|}f(\tau)\,d\tau$. Integrating by parts,

[TABLE]

and hence $\displaystyle\int_{1}^{|t|}f(\tau)\,d\tau\leq\frac{s\,(N-s)}{N^{2}}\;tf(t)$.

$(iii)$ integrating by parts,

[TABLE]

$(iv)$ Since $e^{\tau}\leq 1+\tau e^{\tau}$ for all $\tau\geq 0$,

[TABLE]

$(v)$ Since $e^{\tau}\geq 1+\tau$ for all $\tau\geq 0$,

[TABLE]

This concludes the proof. ∎

3. Palais-Smale condition

Recall that $\Phi$ satisfies the $(\text{PS})_{c}$ condition if every sequence $\left(u_{j}\right)$ in $W^{s,N/s}_{0}(\Omega)$ such that $\Phi(u_{j})\to c$ and $\Phi^{\prime}(u_{j})\to 0$, called a $(\text{PS})_{c}$ sequence, has a convergent subsequence. The main result of this section is the following theorem.

Theorem 3.1**.**

$\Phi$* satisfies the $(\text{PS})_{c}$ condition for all $c<\dfrac{s}{N}\;\alpha_{N,s}(\Omega)^{(N-s)/s}$.*

First we prove a lemma.

Lemma 3.2**.**

If $u_{j}$ converges to $u$ weakly in $W^{s,N/s}_{0}(\Omega)$ and a.e.​ in $\Omega$, and

[TABLE]

then

[TABLE]

Proof.

For any $M>0$, write

[TABLE]

By Lemma 2.3 $(i)$ and (3.1), we have

[TABLE]

Hence

[TABLE]

and the desired conclusion follows by letting $j\to\infty$ first and then $M\to\infty$. ∎

We are now ready to prove Theorem 3.1.

Proof of Theorem 3.1.

Let $\left(u_{j}\right)$ be a $(\text{PS})_{c}$ sequence. Then

[TABLE]

and

[TABLE]

Since $s/N>s\,(N-s)/N^{2}$, it follows from Lemma 2.3 $(ii)$, (3.2) and (3.3) that $\left(u_{j}\right)$ is bounded in $W^{s,N/s}_{0}(\Omega)$. Hence a renamed subsequence converges to some $u$ weakly in $W^{s,N/s}_{0}(\Omega)$, strongly in $L^{p}(\Omega)$ for all $p\in[1,\infty)$, and a.e.​ in $\Omega$. Moreover,

[TABLE]

by (3.3), and hence

[TABLE]

by virtue of Lemma 3.2. By Lemma 2.3 $(iii)$, (3.2), and (3.3),

[TABLE]

so

[TABLE]

If $c=0$, then $u=0$ and hence $\displaystyle\int_{\Omega}F(u_{j})\,dx\to 0$ by (3.4), so $\left\|u_{j}\right\|\to 0$ by (3.2).

Now suppose that $0<c<(s/N)\,\alpha_{N,s}(\Omega)^{(N-s)/s}$. We claim that the weak limit $u$ is nonzero. Suppose $u=0$. Then

[TABLE]

by (3.4) and hence

[TABLE]

by (3.2). Let $(Nc/s)^{s/(N-s)}<\alpha<\alpha_{N,s}(\Omega)$. Then $\left\|u_{j}\right\|\leq\alpha^{(N-s)/N}$ for all $j\geq j_{0}$ for some $j_{0}$. Let $1<q<\alpha_{N,s}(\Omega)/\alpha$. By the Hölder inequality,

[TABLE]

where $1/p+1/q=1$. The first integral on the right-hand side converges to zero since $u=0$, while the second integral is bounded for $j\geq j_{0}$ since $q\,|u_{j}|^{N/(N-s)}=q\,\alpha\,|\widetilde{u}_{j}|^{N/(N-s)}$ with $q\,\alpha<\alpha_{N,s}(\Omega)$ and $\widetilde{u}_{j}=u_{j}/\alpha^{(N-s)/N}$ satisfies $\left\|\widetilde{u}_{j}\right\|\leq 1$, so

[TABLE]

Then $u_{j}\to 0$ by (3.3), and hence $c=0$ by (3.2) and (3.5), a contradiction. So $u$ is nonzero.

Since $\Phi^{\prime}(u_{j})\to 0$,

[TABLE]

for all $v\in W^{s,N/s}_{0}(\Omega)$. For $v\in C^{\infty}_{0}(\Omega)$, an argument similar to that in the proof of Lemma 3.2 using the estimate

[TABLE]

shows that $\displaystyle\int_{\Omega}f(u_{j})\,v\,dx\to\displaystyle\int_{\Omega}f(u)\,v\,dx$, so

[TABLE]

Then this holds for all $v\in W^{s,N/s}_{0}(\Omega)$ by density, and taking $v=u$ gives

[TABLE]

Next we claim that

[TABLE]

We have

[TABLE]

where $\widetilde{u}_{j}=u_{j}/\left\|u_{j}\right\|$. By (3.2) and (3.4),

[TABLE]

where $\beta=\displaystyle\int_{\Omega}F(u)\,dx$, so $\widetilde{u}_{j}$ converges a.e.​ to $\widetilde{u}=u/\left[(N/s)\,(c+\lambda\beta)\right]^{s/N}$. Then

[TABLE]

since

[TABLE]

by (3.6) and Lemma 2.3 $(iii)$. Let

[TABLE]

Then $\left\|u_{j}\right\|^{N/(N-s)}\leq\alpha-2\varepsilon$ for all $j\geq j_{0}$ for some $j_{0}$, and

[TABLE]

by Theorem 2.2. For $M>0$ and $j\geq j_{0}$, (3.8) then gives

[TABLE]

The last expression goes to zero as $M\to\infty$ uniformly in $j$ since $\left\|u_{j}\right\|$ is bounded and (3.9) holds, so (3.7) now follows as in the proof of Lemma 3.2. By (3.3), (3.7), and (3.6),

[TABLE]

and hence $\left\|u_{j}\right\|\to\left\|u\right\|$, so $u_{j}\to u$ by the uniform convexity of $W^{s,N/s}_{0}(\Omega)$. ∎

4. Eigenvalue problem

The asymptotic problem associated with (1.3) as $u$ goes to zero is the eigenvalue problem

[TABLE]

The weak formulation of this problem can be written as the operator equation

[TABLE]

where $A$ and $B$ are the nonlinear operators from $W^{s,N/s}_{0}(\Omega)$ to its dual $W^{-s,N/(N-s)}(\Omega)$ defined by setting

[TABLE]

respectively. The operators $A$ and $B$ are homogeneous of degree $(N-s)/s$, odd, and satisfy

[TABLE]

Since $W^{s,N/s}_{0}(\Omega)$ is uniformly convex, then $A$ is of type (S), i.e. every sequence $\left(u_{j}\right)$ in $W^{s,N/s}_{0}(\Omega)$ such that $u_{j}\rightharpoonup u$ and $\langle A(u_{j}),u_{j}-u\rangle\to 0$ as $j\to\infty$ has a subsequence that converges strongly to $u$ (see e.g. [26, Proposition 1.3]). Moreover, $B$ is a compact operator since the embedding

[TABLE]

is compact. Hence, problem (4.2) falls into the abstract framework considered in [26, Ch. 4] and we can construct an increasing and unbounded sequence of eigenvalues as follows.

Eigenvalues of problem (4.1) coincide with critical values of the functional

[TABLE]

Let ${\mathcal{F}}$ denote the class of symmetric subsets of ${\mathcal{M}}$, let $i(M)$ denote the $\mathbb{Z}_{2}$-cohomological index of $M\in{\mathcal{F}}$ (see Fadell and Rabinowitz [13]), and set

[TABLE]

Then

[TABLE]

is the smallest eigenvalue and $\lambda_{k}\nearrow\infty$ is a sequence of eigenvalues (see [26, Proposition 3.52]). Moreover, denoting by

[TABLE]

the sub- and superlevel sets of $\Psi$, respectively, we have

[TABLE]

whenever $\lambda_{k}<\lambda_{k+1}$ (see [26, Proposition 3.53]). The main result of this section is the following.

Theorem 4.1**.**

If $\lambda_{k}<\lambda_{k+1}$, then the sublevel set $\Psi^{\lambda_{k}}$ contains a compact symmetric subset of index $k$.

First a couple of lemmas.

Lemma 4.2**.**

The operator $A$ is strictly monotone, i.e.,

[TABLE]

for all $u\neq v$ in $W^{s,N/s}_{0}(\Omega)$.

Proof.

By [26, Lemma 6.3], it suffices to show that

[TABLE]

and the equality holds if and only if $\alpha u=\beta v$ for some $\alpha,\beta\geq 0$, not both zero. We have

[TABLE]

by the Hölder inequality. Clearly, equality holds throughout if $\alpha u=\beta v$ for some $\alpha,\beta\geq 0$, not both zero. Conversely, if $\langle A(u),v\rangle=\left\|u\right\|^{(N-s)/s}\left\|v\right\|$, equality holds in both inequalities. The equality in the second inequality gives

[TABLE]

for some $\alpha,\beta\geq 0$, not both zero, and then the equality in the first inequality gives

[TABLE]

Since $u$ and $v$ vanish a.e.​ in $\mathbb{R}^{N}\setminus\Omega$, it follows that $\alpha u=\beta v$ a.e.​ in $\Omega$. ∎

Lemma 4.3**.**

For each $w\in L^{N/s}(\Omega)$, the problem

[TABLE]

has a unique weak solution $u\in W^{s,N/s}_{0}(\Omega)$. Moreover, the map

[TABLE]

is continuous, homogeneous of degree $(N-s)/s$, and satisfies

[TABLE]

for all $w\neq 0$ in $L^{N/s}(\Omega)$.

Proof.

The existence follows from a standard minimization argument and the uniqueness from Lemma 4.2. Clearly, $J$ is homogeneous of degree $(N-s)/s$. To see that it is continuous, let $w_{j}\to w$ in $L^{N/s}(\Omega)$ and let $u_{j}=J(w_{j})$, so

[TABLE]

Testing with $v=u_{j}$ gives

[TABLE]

by the Hölder inequality, which together with the imbedding $W^{s,N/s}_{0}(\Omega)\hookrightarrow L^{N/s}(\Omega)$ shows that $\left(u_{j}\right)$ is bounded. Therefore, a renamed subsequence of $\left(u_{j}\right)$ converges to some $u$ weakly, strongly in $L^{N/s}(\Omega)$ and a.e.​ in $\Omega$. Then $u$ is a weak solution of problem (4.4) as in the proof of Theorem 3.1, so $u=J(w)$. Testing (4.6) with $u_{j}-u$ gives

[TABLE]

so $u_{j}\to u$ for a further subsequence since the operator $A$ is of type (S). Finally, testing

[TABLE]

with $v=u,w$ and using the Hölder inequality gives

[TABLE]

from which (4.5) follows. ∎

We are now ready to prove Theorem 4.1.

Proof of Theorem 4.1.

Let

[TABLE]

be the radial projections onto ${\mathcal{M}}$ and

[TABLE]

respectively, let $i$ be the imbedding $W^{s,N/s}_{0}(\Omega)\hookrightarrow L^{N/s}(\Omega)$, let $J$ be the map defined in Lemma 4.3, and let $\varphi:\Psi^{\lambda_{k}}\to{\mathcal{M}}$ be the composition of the maps

[TABLE]

Since $i$ is compact,

[TABLE]

is compact in $L^{N/s}(\Omega)$, and hence $K_{0}=\varphi(\Psi^{\lambda_{k}})$ is compact in $W^{s,N/s}_{0}(\Omega)$. Since $\varphi$ is an odd continuous map, $i(K_{0})\geq i(\Psi^{\lambda_{k}})$. For $u\in\Psi^{\lambda_{k}}$, $\varphi(u)=J(u)/\left\|J(u)\right\|$ since $J$ is homogeneous, so

[TABLE]

by (4.5), and hence $K_{0}\subset\Psi^{\lambda_{k}}$. Then $i(K_{0})\leq i(\Psi^{\lambda_{k}})$ by the monotonicity of the index, so $i(K_{0})=i(\Psi^{\lambda_{k}})=k$ by (4.3). ∎

5. Bifurcation and multiplicity

In this section we prove the following bifurcation and multiplicity results for problem (1.3), in which the constant

[TABLE]

plays an important role, where ${\mathcal{L}}$ denotes the Lebesgue measure in $\mathbb{R}^{N}$.

Theorem 5.1**.**

If

[TABLE]

then problem (1.3) has a pair of nontrivial solutions $\pm\,u^{\lambda}$ such that $u^{\lambda}\to 0$ as $\lambda\nearrow\lambda_{1}$.

Theorem 5.2**.**

If $\lambda_{k}\leq\lambda<\lambda_{k+1}=\cdots=\lambda_{k+m}<\lambda_{k+m+1}$ for some $k,m\geq 1$ and

[TABLE]

then problem (1.3) has $m$ distinct pairs of nontrivial solutions $\pm\,u^{\lambda}_{j},\,j=1,\dots,m$ such that $u^{\lambda}_{j}\to 0$ as $\lambda\nearrow\lambda_{k+1}$.

In particular, we have the following existence result.

Corollary 5.1**.**

If

[TABLE]

for some $k\geq 1$, then problem (1.3) has a nontrivial solution.

Remark 5.3*.*

Since $\lambda\geq\lambda_{k}$ in Theorem 5.2, (5.1) holds if

[TABLE]

or if

[TABLE]

We only give the proof of Theorem 5.2. The proof of Theorem 5.1 is similar and simpler. The proof will be based on an abstract critical point theorem proved in Yang and Perera [28] that generalizes Bartolo et al. [3, Theorem 2.4].

Let $\Phi$ be an even $C^{1}$-functional on a Banach space $W$. Let ${\mathcal{A}}^{\ast}$ denote the class of symmetric subsets of $W$, let $r>0$, let $S_{r}=\left\{u\in W:\left\|u\right\|=r\right\}$, let $0<b\leq\infty$, and let $\Gamma$ denote the group of odd homeomorphisms of $W$ that are the identity outside $\Phi^{-1}(0,b)$. The pseudo-index of $M\in{\mathcal{A}}^{\ast}$ related to $i$, $S_{r}$, and $\Gamma$ is defined by

[TABLE]

(see Benci [4]).

Theorem 5.4** ([28, Theorem 2.4]).**

Let $K_{0}$ and $B_{0}$ be symmetric subsets of ${\mathcal{M}}=\left\{u\in W:\left\|u\right\|=1\right\}$ such that $K_{0}$ is compact, $B_{0}$ is closed, and

[TABLE]

for some $k\geq 0$ and $m\geq 1$. Assume that there exists $R>r$ such that

[TABLE]

where $K=\left\{Ru:u\in K_{0}\right\}$, $B=\left\{ru:u\in B_{0}\right\}$, and $X=\left\{tu:u\in K,\,0\leq t\leq 1\right\}$. For $j=k+1,\dots,k+m$, let

[TABLE]

and set

[TABLE]

Then

[TABLE]

in particular, $0<c_{j}^{\ast}<b$. If, in addition, $\Phi$ satisfies the $(\text{PS})_{c}$ condition for all $c\in(0,b)$, then each $c_{j}^{\ast}$ is a critical value of $\Phi$ and there are $m$ distinct pairs of associated critical points.

We are now ready to prove Theorem 5.2.

Proof of Theorem 5.2.

In view of Theorem 3.1, we apply Theorem 5.4 with

[TABLE]

By Theorem 4.1, the sublevel set $\Psi^{\lambda_{k+m}}$ has a compact symmetric subset $K_{0}$ with

[TABLE]

We take $B_{0}:=\Psi_{\lambda_{k+1}}$, so that

[TABLE]

by (4.3). Let $R>r>0$ and let $K$, $B$, and $X$ be as in Theorem 5.4. By Lemma 2.3 $(iv)$,

[TABLE]

so for $u\in\Psi_{\lambda_{k+1}}$,

[TABLE]

The first integral in the last expression is bounded since $W^{s,N/s}_{0}(\Omega)\hookrightarrow L^{2N^{2}/s(N-s)}(\Omega)$, and the second integral is also bounded if $2r^{N/(N-s)}<\alpha_{N,s}(\Omega)$. Since $\lambda<\lambda_{k+1}$, it follows that $\inf\Phi(B)>0$ if $r$ is sufficiently small. By Lemma 2.3 $(v)$ and the Hölder inequality,

[TABLE]

so for $u\in K_{0}\subset\Psi^{\lambda_{k+1}}$,

[TABLE]

It follows that $\Phi\leq 0$ on $K$ if $R$ is sufficiently large. By Lemma 2.3 $(v)$ and the Hölder inequality,

[TABLE]

so for $u\in X$,

[TABLE]

So

[TABLE]

by (5.1). Thus, problem (1.3) has $m$ distinct pairs of nontrivial solutions $\pm\,u^{\lambda}_{j},\,j=1,\dots,m$ such that

[TABLE]

by Theorem 5.4. To prove that $u^{\lambda}_{j}\to 0$ as $\lambda\nearrow\lambda_{k+1}$, it suffices to show that for every sequence $\nu_{n}\nearrow\lambda_{k+1}$, a subsequence of $v_{n}:=u^{\nu_{n}}_{j}$ converges to zero. We have

[TABLE]

by (5.2) and

[TABLE]

Since $s/N>s\,(N-s)/N^{2}$, it follows from Lemma 2.3 $(ii)$, (5.3), and (5.4) that $\left(v_{n}\right)$ is bounded in $W^{s,N/s}_{0}(\Omega)$. Hence a renamed subsequence converges to some $v$ weakly in $W^{s,N/s}_{0}(\Omega)$, strongly in $L^{p}(\Omega)$ for all $p\in[1,\infty)$, and a.e.​ in $\Omega$. By Lemma 2.3 $(iii)$, (5.3), and (5.4),

[TABLE]

so

[TABLE]

and hence $v=0$. Since $\int_{\Omega}v_{n}\,f(v_{n})\,dx$ is bounded by (5.4), then

[TABLE]

by Lemma 3.2, so $\left\|v_{n}\right\|\to 0$ by (5.3). ∎