Uniqueness of minimal energy solutions for a semilinear problem involving the fractional laplacian

TL;DR

This paper investigates the uniqueness of minimal energy solutions for a fractional Laplacian semilinear problem, particularly in small domains, extending classical Neumann problem concepts to fractional settings.

Contribution

It establishes the uniqueness of minimal energy solutions for fractional Laplacian problems in small domains, a novel extension of classical Neumann problem results.

Findings

Uniqueness of minimal energy solutions in small domains

Extension of classical Neumann problem concepts to fractional Laplacian

Results applicable to fractional semilinear problems

Abstract

In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains.

Equations

$$(-\Delta {)}^{s}u(x)=\text{p.v.}{\int }_{\Omega }\frac{u(x)-u(y)}{|x-y{|}^{n+2s}}dy,$$

$$(-\Delta {)}^{s}u+u=\lambda |u{|}^{q-2}u,\text{in }\Omega .$$

$$\{\begin{array}{ll}-\Delta u+u=\lambda |u{|}^{q-2}u& \text{in }\Omega \\ \frac{\partial u}{\partial \nu }=0& \text{on }\partial \Omega .\end{array}$$

$$1\le q<2_s^{\ast }:=\frac{2n}{n-2s}.$$

$$\mathcal{F}(u):=\frac{\frac{1}{2}[u{]}_{s;\Omega }^2+\Vert u{\Vert }_{2;\Omega }^2}{\Vert u{\Vert }_{q;\Omega }^2},$$

$$[u{]}_{s;\Omega }:={\left({\iint }_{\Omega \times \Omega }\frac{(u(x)-u(y){)}^2}{|x-y{|}^{n+2s}}dxdy\right)}^{\frac{1}{2}}$$

$$\mathcal{F}(u)=\underset{v\in H^s(\Omega )}{\inf }\mathcal{F}(v).$$

$${\Omega }_{\epsilon }:=\epsilon \cdot \Omega =\{\epsilon x:x\in \Omega \}\text{as }\epsilon \downarrow 0$$

$$H^s(\Omega ):=\left\{u\in L^2(\Omega ):\frac{u(x)-u(y)}{|x-y{|}^{\frac{n}{2}+s}}\in L^2(\Omega \times \Omega )\right\}.$$

$$\Vert u{\Vert }_{s;\Omega }:=([u{]}_{s;\Omega }^2+\Vert u{\Vert }_{2;\Omega }^2{)}^{\frac{1}{2}}.$$

$$2_s^{\ast }:=\{\begin{array}{ll}\frac{2n}{n-2s}& \text{if }2s<n,\\ \infty & \text{otherwise}\end{array}$$

$$S(\Omega )=S_{s,q}(\Omega ):=\underset{u\in H^s(\Omega )}{\inf }\frac{\Vert u{\Vert }_{s;\Omega }^2}{\Vert u{\Vert }_{q;\Omega }^2}.$$

$$⟨(-\Delta {)}^{s}u,v⟩=\frac{1}{2}{\iint }_{\Omega \times \Omega }\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y{|}^{n+2s}}dxdy.$$

$$\frac{1}{2}{\iint }_{\Omega \times \Omega }\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y{|}^{n+2s}}dxdy+{\int }_{\Omega }uvdx=\lambda {\int }_{\Omega }|u{|}^{q-2}uvdx,$$

$$S({\Omega }_{\epsilon })\le |{\Omega }_{\epsilon }{|}^{1-\frac{2}{q}}={\epsilon }^{n(1-\frac{2}{q})}|\Omega {|}^{1-\frac{2}{q}}.$$

$$\underset{\epsilon \downarrow 0}{\lim }\frac{S({\Omega }_{\epsilon })}{{\epsilon }^{n(1-\frac{2}{q})}}=|\Omega {|}^{1-\frac{2}{q}}.$$

$${\bar{u}}_{\epsilon }\to |\Omega {|}^{-\frac{1}{q}}\text{strongly in }{H}^s(\Omega ).$$

$$\frac{\Vert v{\Vert }_{s;{\Omega }_{\epsilon }}^2}{\Vert v{\Vert }_{q;{\Omega }_{\epsilon }}^2}={\epsilon }^{n(1-\frac{2}{q})}\frac{{\epsilon }^{-2s}[\bar{v}{]}_{s;\Omega }^2+\Vert \bar{v}{\Vert }_{2;\Omega }^2}{\Vert \bar{v}{\Vert }_{q;\Omega }^2}.$$

$$S({\Omega }_{\epsilon })={\epsilon }^{n(1-\frac{2}{q})}\frac{{\epsilon }^{-2s}[{\bar{u}}_{\epsilon }{]}_{s;\Omega }^2+\Vert {\bar{u}}_{\epsilon }{\Vert }_{2;\Omega }^2}{\Vert {\bar{u}}_{\epsilon }{\Vert }_{q;\Omega }^2}.$$

$$\frac{{\epsilon }^{-2s}[{\bar{u}}_{\epsilon }{]}_{s;\Omega }^2+\Vert {\bar{u}}_{\epsilon }{\Vert }_{2;\Omega }^2}{\Vert {\bar{u}}_{\epsilon }{\Vert }_{q;\Omega }^2}\le |\Omega {|}^{1-\frac{2}{q}}.$$

$${\bar{u}}_{\epsilon }\rightharpoonup \bar{u}\text{weakly in }{H}^s(\Omega ),$$

$${\bar{u}}_{\epsilon }\to \bar{u}\text{strongly in }{L}^r(\Omega )\text{ for any }1\le r<2_s^{\ast }.$$

$$[\bar{u}{]}_{s;\Omega }\le \underset{\epsilon \downarrow 0}{\mathrm{liminf}}[{\bar{u}}_{\epsilon }{]}_{s;\Omega }=0,$$

$$|\Omega {|}^{1-\frac{2}{q}}$$

$$=\underset{\epsilon \downarrow 0}{\mathrm{liminf}}\frac{S({\Omega }_{\epsilon })}{{\epsilon }^{n(1-\frac{2}{q})}}\le \underset{\epsilon \downarrow 0}{\mathrm{limsup}}\frac{S({\Omega }_{\epsilon })}{{\epsilon }^{n(1-\frac{2}{q})}}\le |\Omega {|}^{1-\frac{2}{q}}.$$

$$(-\Delta {)}^{s}u+{\epsilon }^{2s}u={\epsilon }^{2s}{\lambda }_{\epsilon }|u{|}^{q-2}u\text{in }\Omega ,$$

$$\mathcal{E}:=\{u\in H^s(\Omega ):\Vert u{\Vert }_{q;\Omega }=1\}.$$

$$⟨F(u,\epsilon ),v⟩:=$$

$$+{\epsilon }^{2s}{\int }_{\Omega }uvdx-{\epsilon }^{2s}{\lambda }_{\epsilon }{\int }_{\Omega }|u{|}^{q-2}uvdx.$$

$$\varphi (0)=u_0\text{and}F(\varphi (\epsilon ),\epsilon )=0\text{for every }0\le \epsilon <\delta ,$$

$$T_{u_0}\mathcal{E}=\left\{v\in H^s(\Omega ):{\int }_{\Omega }vdx=0\right\}.$$

$${\int }_{\Omega }|\alpha (t){|}^{q}dx=1\text{for every }t\in (-1,1).$$

Full text

1. Introduction

In recent years there has been an increasing amount of attention to problems involving nonlocal diffusion operators. These problems are so vast that is impossible to give a comprehensive list of references, just to cite a few, see [11, 12, 14, 17, 22, 27] for some physical models, [2, 18, 24] for some applications in finances, [6] for appications in fluid dynamics, [16, 21, 23] for application in ecology and [15] for some applications in image processing.

Among these applications, one operator that is of particular importance is the fractional laplacian that is defined (up to some normalization constant) as

[TABLE]

where p.v. stands for in principal value and $\Omega\subset{\mathbb{R}}^{n}$ is a bounded domain.

This operator is classical and have been studied by several authors. See for instance [1, 4, 5, 7, 9, 8, 20, 25], etc.

Recall that this operator is to the so called regional fractional laplacian that corresponds to Levy processes where the long jumps are restricted to be inside $\Omega$. See, for instance, [10] for a discussion on this.

This operator is commonly used as a fractional version of the Neumann laplacian. In the literature, there exists several forms of fractional version of the Neumann laplacian. We refer the interested reader again to the article [10] for more on this.

In this paper, we address the following semilinear problem associated to $(-\Delta)^{s}$

[TABLE]

This is the fractional counterpart of the classical Neumann problem

[TABLE]

Here, $q$ is a subcritical exponent in the sense of the Sobolev embeddings. That is,

[TABLE]

The problem can be separated into three different types of behavior: sublinear ($1\leq q<2$); linear ($q=2$) and superlinear ($q>2$).

The linear case is by now well understood as an eigenvalue problem and will not be considered here.

For the sublinear and superlinear cases, the parameter $\lambda$ is superfluous since if one gets a solution for some particular value of $\lambda$, then taking a suitable multiple of the solution the value of $\lambda$ can be taken to be 1.

It is fairly easy to see that any solution to (1.1) is a critical point of the functional

[TABLE]

where

[TABLE]

is the so-called Gagliardo seminorm of $u$ and, as usual, $\|u\|_{r;\Omega}$ denotes the $L^{r}(\Omega)-$norm.

By standard variational methods, one can see that there exists minimal energy solutions to (1.1). That is, functions $u\in H^{s}(\Omega)$ such that

[TABLE]

Moreover, by a direct application of the Ljusternik – Schnirelmann method, one can construct a sequence $\lambda_{k}\uparrow\infty$ of critical energy levels and a sequence of critical points $\{u_{k}\}_{k\in{\mathbb{N}}}$ of ${\mathcal{F}}$ associated to $\{\lambda_{k}\}_{k\in{\mathbb{N}}}$.

Therefore, there exist infinitely many solutions to problem (1.1).

In this paper, we focus on minimal energy solutions to (1.1). In particular to the multiplicity problem of such solutions.

To this end, inspired by the results of [13], given a domain $\Omega$ we consider the family of contracted domains

[TABLE]

and look for the asymptotic behavior of minimal energy solutions in $\Omega_{\varepsilon}$ as $\varepsilon\downarrow 0$.

We first show that the asymptotic behavior of every minimal energy solution is the same and, using this asymptotic behavior, we are able to conclude the uniqueness of minimal energy solution for contracted domains.

Finally, we give an estimate on the contraction parameter in order to obtain the uniqueness result.

To end this introduction, we want to remark that the same ideas can be used to deal with the Neumann problem (1.2). The changes needed are easy and are left to the interested reader.

We want to recall that uniqueness results for problem (1.2) has been considered in the literature before. See for instance [19].

This result is a first step in an investigation which might be pursued in various directions: for instance, taking different Neumann fractional operators, as the one considered in [10], or show if the known uniqueness results for the classical Neumann laplacian hold in this situation. We leave these questions for further investigation.

2. Preliminaries.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded smooth domain and $0<s<1$. The fractional order Sobolev space is defined as

[TABLE]

This space is endowed with the norm

[TABLE]

It is well known (see, for instance, [9]) that there exists a critical exponent

[TABLE]

such that for any $1\leq q<2^{*}_{s}$ the embedding $H^{s}(\Omega)\subset L^{q}(\Omega)$ is compact.

We define the Sobolev constant as the number

[TABLE]

It is easy to see, as a consequence of the compactness of the embedding, that there exists an extremal for $S(\Omega)$. That is a function $u\in H^{s}(\Omega)$ where the above infimum is attained.

Also, any extremal for $S(\Omega)$ is a minimal energy (weak) solution to (1.1).

The constant $\lambda$ in (1.1) depends on the normalization of the extremal. For instance, if the extremal $u$ is normalized as $\|u\|_{q;\Omega}=1$ then $\lambda=S(\Omega)$.

Recall that the operator $(-\Delta)^{s}$ is a bounded operator between the Sobolev space $H^{s}(\Omega)$ and its dual $(H^{s}(\Omega))^{\prime}$ and can be computed by

[TABLE]

Therefore, for a solution to (1.1) we mean a function $u\in H^{s}(\Omega)$ such that

[TABLE]

for every $v\in H^{s}(\Omega)$.

3. Asymptotic behavior in thin domains

Throughout this section we fix the exponent $q\in[1,2^{*}_{s})$, $q\neq 2$.

Our objective in this section is to study the asymptotic behavior of the constant $S(\Omega_{\varepsilon})$ as $\varepsilon\downarrow 0$, where the contracted domains $\Omega_{\varepsilon}$ are given by (1.3).

We begin with a simple estimate.

Lemma 3.1**.**

Under the above notations, we have that

[TABLE]

Proof.

The lemma follows just by taking $u=1$ as a test function in the definition of $S(\Omega_{\varepsilon})$. ∎

Now we want to be more precise. We need to show that the asymptotic behavior of $S(\Omega_{\varepsilon})$ is given precisely by $\varepsilon^{n(1-\frac{2}{q})}$ and also to study the behavior of the associated extremals.

Lemma 3.2**.**

Under the above notations, we have that

[TABLE]

Moreover if $u_{\varepsilon}\in H^{s}(\Omega_{\varepsilon})$ is an extremal for $S(\Omega_{\varepsilon})$, the rescaled extremals $\bar{u}_{\varepsilon}(x):=u_{\varepsilon}(\varepsilon x)$ normalized such that $\|\bar{u}_{\varepsilon}\|_{q;\Omega}=1$ verify that

[TABLE]

Proof.

First, observe that for $v\in H^{s}(\Omega_{\varepsilon})$, if we denote $\bar{v}(x)=v(\varepsilon x)$, then $\bar{v}\in H^{s}(\Omega)$. Moreover, $[v]_{s;\Omega_{\varepsilon}}=\varepsilon^{\frac{n}{2}-s}[\bar{v}]_{s;\Omega}$ and $\|v\|_{r;\Omega_{\varepsilon}}=\varepsilon^{\frac{n}{r}}\|\bar{v}\|_{r;\Omega}$ for $1\leq r<2^{*}_{s}$. Therefore

[TABLE]

Now, let $u_{\varepsilon}\in H^{s}(\Omega_{\varepsilon})$ be an extremal for $S(\Omega_{\varepsilon})$ and let $\bar{u}_{\varepsilon}(x)=u_{\varepsilon}(\varepsilon x)$. Then,

[TABLE]

Now, by Lemma 3.1, it follows that

[TABLE]

Let us now fix the normalization of the extremal $u_{\varepsilon}$ such that $\|\bar{u}_{\varepsilon}\|_{q;\Omega}=1$, and by (3.1), we obtain that $\bar{u}_{\varepsilon}$ is bounded in $H^{s}(\Omega)$ uniformly on $\varepsilon>0$. So, there exists $\bar{u}\in H^{s}(\Omega)$ such that (up to some sequence $\varepsilon_{k}\to 0$),

[TABLE]

Also, from (3.1) and (3.2), we have that

[TABLE]

therefore $\bar{u}$ is constant and, since $\|\bar{u}_{\varepsilon}\|_{q;\Omega}=1$, from (3.3) we obtain that $\|\bar{u}\|_{q;\Omega}=1$.

All these together imply that $\bar{u}=|\Omega|^{-\frac{1}{q}}$.

From these estimates, one easily concludes that

[TABLE]

The proof is complete. ∎

4. Uniqueness of extremals for small domains

In this section we show the uniqueness of extremals if the domain is contracted enough.

For that purpose, observe that if $u_{\varepsilon}$ is an extremal for $S(\Omega_{\varepsilon})$ and $\bar{u}_{\varepsilon}$ is the rescaled extremal normalized as $\|\bar{u}_{\varepsilon}\|_{q;\Omega}=1$, then $\bar{u}_{\varepsilon}$ is a weak solution of the problem

[TABLE]

where $\lambda_{\varepsilon}=S(\Omega_{\varepsilon})\varepsilon^{-n(1-\frac{2}{q})}$. Recall also, that $0\leq\lambda_{\varepsilon}\leq|\Omega|^{1-\frac{2}{q}}$ (c.f. Lemma 3.1).

So, we define the space

[TABLE]

It is easy to see that ${\mathcal{E}}$ is a $C^{1}$ manifold.

We then define $F\colon{\mathcal{E}}\times[0,1)\to(H^{s}(\Omega))^{\prime}$ as

[TABLE]

Denote $u_{0}=|\Omega|^{-\frac{1}{q}}\in{\mathcal{E}}$ and observe that $F(u_{0},0)=0$.

Following the ideas of [13] we want to use the Implicit Function Theorem (IFT) to show the existence of a small number $\delta>0$ and a curve $\phi\colon[0,\delta)\to{\mathcal{E}}$ such that

[TABLE]

and if $(u,\varepsilon)\in{\mathcal{E}}\times[0,\delta)$ is such that $F(u,\varepsilon)=0$ and $u$ is close to $u_{0}$ then $u=\phi(\varepsilon)$.

Observe that if we can apply the IFT then, combining this with Lemma 3.2, automatically we obtain the uniqueness of extremals of $S(\Omega_{\varepsilon})$ for $\varepsilon$ small.

In order to be able to apply the IFT we need to check that $d_{u}F|_{(u_{0},0)}$ is invertible (see [3] or [26]). Recall that since $F$ is define on a manifold, the derivative is defined on the tangent space of ${\mathcal{E}}$ at the point $u_{0}$.

Let us begin with a couple of lemmas.

Lemma 4.1**.**

The tangent space of ${\mathcal{E}}$ at $u_{0}$, that we denote by $T_{u_{0}}{\mathcal{E}}$ is given by

[TABLE]

Proof.

Let $v\in T_{u_{0}}{\mathcal{E}}$. Then, there exists a differentiable curve, $\alpha\colon(-1,1)\to{\mathcal{E}}$ such that $\alpha(0)=u_{0}$ and $\dot{\alpha}(0)=v$.

But, since $\alpha(t)\in{\mathcal{E}}$ for every $t\in(-1,1)$ it follows that

[TABLE]

Differentiating both sides of the equality gives

[TABLE]

So, if we evaluate at $t=0$ and recall that $u_{0}$ is constant, we obtain that

[TABLE]

On the other hand, if $v\in H^{s}(\Omega)$ verifies (4.2), we construct the curve $\alpha\colon(-1,1)\to{\mathcal{E}}$ as

[TABLE]

Straightforward computations show that $\alpha(0)=u_{0}$ and $\dot{\alpha}(0)=v$. ∎

Now, we denote ${\mathcal{A}}=(\text{span}\{1\})^{\perp}=\{f\in(H^{s}(\Omega))^{\prime}\colon\langle f,1\rangle=0\}$.

Lemma 4.2**.**

We have that

[TABLE]

Moreover, the following expression holds

[TABLE]

Proof.

To prove the Lemma, first observe that

[TABLE]

for every $u\in{\mathcal{E}}$.

From this expression the lemma follows. ∎

It remains to see that $d_{u}F|_{(u_{0},0)}$ has a continuous inverse.

Lemma 4.3**.**

The derivative $d_{u}F|_{(u_{0},0)}\colon T_{u_{0}}{\mathcal{E}}\to{\mathcal{A}}$ has a continuous inverse.

Proof.

First observe that $T_{u_{0}}{\mathcal{E}}$ is a Hilbert space with inner product given by

[TABLE]

So, the Lemma follows from the Riesz representation Theorem. ∎

Combining Lemmas 4.1, 4.2, 4.3 we are in position to apply the IFT and conclude the main result of the section

Theorem 4.4**.**

Given $\Omega\subset{\mathbb{R}}^{n}$ smooth and of finite measure and $1<q<2^{*}_{s}$, there exists $\delta>0$ such that $S(\Omega_{\varepsilon})$ has a unique extremal for $0<\varepsilon<\delta$.

Proof.

At this point the proof is a direct consequence of the IFT and the remarks made at the beginning of the section. ∎

5. Estimates for the contraction parameter

Let us first define

[TABLE]

From the results of the previous section, we know that $\varepsilon_{0}>0$. We now want to find a lower bound for $\varepsilon_{0}$.

Lemma 5.1**.**

There exists $u_{0}\in H^{s}(\Omega_{\varepsilon_{0}})$ an extremal for $S(\Omega_{\varepsilon_{0}})$, such that the rescaled function $\bar{u}_{0}(x)=u_{0}(\varepsilon_{0}x)$ normalized as $\|\bar{u}_{0}\|_{q;\Omega}=1$ verifies that $d_{u}F|_{(\bar{u}_{0},\varepsilon_{0})}$ is not invertible.

Proof.

Assume the oposite.

We first claim that there is a unique extremal for $S(\Omega_{\varepsilon_{0}})$. Otherwise, if $u_{0}\neq u_{1}$ are extremals such that the rescaled functions $\bar{u}_{i}(x)=u_{i}(\varepsilon_{0}x)$, $i=0,1$ normalized as $\|\bar{u}_{i}\|_{q;\Omega}=1$, $i=0,1$, verify that $d_{u}F|_{(\bar{u}_{i},\varepsilon_{0})}$ is invertible for $i=0,1$. But then, by the IFT, there exists $\delta>0$ and two curves $\phi_{i}\colon(\varepsilon_{0}-\delta,\varepsilon_{0}+\delta)\to{\mathcal{E}}$ such that

[TABLE]

But this contradicts the uniqueness of extremals for $\varepsilon<\varepsilon_{0}$.

Now, let $\varepsilon_{k}>\varepsilon_{0}$ be such that $\varepsilon_{k}\to\varepsilon_{0}$ as $k\to\infty$ and let $u_{k}$ be an extremal for $S(\Omega_{\varepsilon_{k}})$. We normalized these extremals so that the rescaled functions $\bar{u}_{k}(x)=u_{k}(\varepsilon_{k}x)$ verify $\|\bar{u}_{k}\|_{q;\Omega}=1$.

We want to see that $\{\bar{u}_{k}\}_{k\in{\mathbb{N}}}$ converges to $\bar{u}_{0}$ that is the rescaled function of the unique extremal $u_{0}$ for $S(\Omega_{\varepsilon_{0}})$.

In fact, it is immediate to see that $\sup_{k\in{\mathbb{N}}}\|\bar{u}_{k}\|_{s;\Omega}<\infty$, and so, up to a subsequence, there exists $\bar{w}\in H^{s}(\Omega)$ such that

[TABLE]

From these convergence results it follows that

[TABLE]

and

[TABLE]

Now, let $u_{0}\in H^{s}(\Omega_{\varepsilon_{0}})$ be the unique extremal for $S(\Omega_{\varepsilon_{0}})$ normalized such that the rescaled function $\bar{u}_{0}$ satisfies $\|\bar{u}_{0}\|_{q;\Omega}=1$.

Then,

[TABLE]

These two inequalities combined imply that $w(x)=\bar{w}(\varepsilon_{0}^{-1}x)$ is an extremal for $S(\Omega_{\varepsilon_{0}})$ and so $w=u_{0}$.

Now, since we are assuming that $d_{u}F|_{(u_{0},\varepsilon_{0})}$ is invertible, we can apply the IFT as in the proof of Theorem 4.4 to conclude that for some $\delta>0$ there is a unique extremal for $S(\Omega_{\varepsilon})$ for $\varepsilon<\varepsilon_{0}+\delta$. But this contradicts the definition of $\varepsilon_{0}$. ∎

Remark 5.2*.*

By a simple application of the Fredholm’s alternative, it follows that $d_{u}F|_{(u_{0},\varepsilon_{0})}$ is not invertible if and only if it has a nontrivial kernel.

The following Poincaré-type inequality plays an important role in the bound of $\varepsilon_{0}$

Lemma 5.3**.**

Let $\Omega\subset{\mathbb{R}}^{n}$ be an open, smooth and of finite measure. Let $0<s<1$ and $1\leq q<2^{*}_{s}$. Then, there exists $c>0$, that depends on $q,s$ and $\Omega$, such that

[TABLE]

for every $w\in H^{s}(\Omega)$ such that $\int_{\Omega}w\,dx=0$.

Proof.

The proof follows by a standard compactness argument and is omitted. ∎

We are ready to prove the main result of the section.

Theorem 5.4**.**

Under the notations and assumptions of the section, we have that

[TABLE]

where $c>0$ is the constant in the Poincaré-type inequality of Lemma 5.3.

Proof.

By Lemma 5.1, there exists $u_{0}$ an extremal for $S(\Omega_{\varepsilon_{0}})$ such that the rescaled function $\bar{u}_{0}$ normalized such that $\|\bar{u}_{0}\|_{q;\Omega}=1$ verifies that $d_{u}F|_{(\bar{u}_{0},\varepsilon_{0})}$ is not invertible.

Moreover, by Remark 5.2, $d_{u}F|_{(\bar{u}_{0},\varepsilon_{0})}$ has a nontrivial kernel.

Let $0\neq z\in\ker(d_{u}F|_{(\bar{u}_{0},\varepsilon_{0})})$, then $z$ is a nontrivial weak solution to the problem

[TABLE]

Using $z$ as a test function in the weak formulation of (5.1) gives

[TABLE]

Now, we use the Poincaré-type inequality of Lemma 5.3, Lemma 3.1 and Hölder’s inequality to deduce from (5.2) that

[TABLE]

Finally, recalling that $\|\bar{u}_{0}\|_{q;\Omega}=1$ and that $z\neq 0$ we arrive that the desired result. ∎

Acknowledgements

This paper was partially supported by grants UBACyT 20020130100283BA, CONICET PIP 11220150100032CO and ANPCyT PICT 2012-0153.

J. Fernández Bonder and A. Silva are members of CONICET.

This work started while JFB was visiting the National University of San Luis. He wants to thank the Math Department and the IMASL for the hospitality that make the visit so enjoyable.