Fractional Laplacian equations with critical Sobolev exponent

Abstract

In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator $\mathcal {L}_K$

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \mathcal {L}_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 &{} \hbox {in } \Omega \\ u=0 &{} \hbox {in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$

that was started by Servadei and Valdinoci (Commun Pure Appl Anal 12(6):2445–2464, 2013). Here $s\in (0,1),\, \Omega$ is an open bounded set of ${\mathbb {R}}^n,\, n>2s$, with continuous boundary, $\lambda$ is a positive real parameter, $2^*=2n/(n-2s)$ is a fractional critical Sobolev exponent and $f$ is a lower order perturbation of the critical power $|u|^{2^*-2}u$, while $\mathcal {L}_K$ is the integrodifferential operator defined as

$$\begin{aligned} \mathcal {L}_Ku(x)= \int _{{\mathbb {R}}^n}\left( u(x+y)+u(x-y)-2u(x)\right) K(y)\,dy, \quad x\in {\mathbb {R}}^n. \end{aligned}$$

Under suitable growth condition on $f$, we show that this problem admits non-trivial solutions for any positive parameter $\lambda$. This existence theorem extends some results obtained in [15, 19, 20]. In the model case, that is when $K(x)=|x|^{-(n+2s)}$ (this gives rise to the fractional Laplace operator $-(-\Delta )^s$), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities.

Appendix A: The functional setting

Here we recall the definition of the functional analytic spaces $X$ and $X_0$ we work in.

In our framework the functional space $X$ denotes the linear space of Lebesgue measurable functions from ${\mathbb {R}}^n$ to ${\mathbb {R}}$ such that the restriction to $\Omega$ of any function $g$ in $X$ belongs to $L^2(\Omega )$ and

$$\begin{aligned}\hbox {the map}\quad (x,y)\mapsto (g(x)-g(y))\sqrt{K(x-y)}\quad \hbox {is in}\quad L^2\left( {\mathbb {R}}^{2n} \setminus ({\mathcal {C}}\Omega \times {\mathcal {C}}\Omega ), dxdy\right) , \end{aligned}$$

where ${\mathcal {C}}\Omega :={\mathbb {R}}^n {\setminus }\Omega$, while

$$\begin{aligned} X_0=\{g\in X : g=0\,\, \hbox {a.e. in}\,\, {\mathbb {R}}^n{\setminus } \Omega \}. \end{aligned}$$

Note that $X$ and $X_0$ are non-empty, since $C^2_0 (\Omega )\subseteq X_0$ by [16], Lemma 5.1] (for this we need condition (1.6)).

The space $X$ is endowed with the norm defined as

$$\begin{aligned} \Vert g\Vert _X=\Vert g\Vert _{L^2(\Omega )}+\left( \int _Q |g(x)-g(y)|^2K(x-y)dx\,dy\right) ^{1/2}, \end{aligned}$$

(3.1)

where $Q={\mathbb {R}}^{2n}{\setminus } \mathcal {O}$ and ${\mathcal {O}}=({\mathcal {C}}\Omega )\times ({\mathcal {C}}\Omega ) \subset {\mathbb {R}}^{2n}$. It is easily seen that $\Vert \cdot \Vert _X$ is a norm on $X$ (see, for instance, [17] for a proof). By [17], Lemmas 6and7] as a norm on $X_0$ we can consider the function

$$\begin{aligned} X_0\ni g\mapsto \Vert g\Vert _{X_0}=\left( \int _Q|g(x)-g(y)|^2K(x-y)\,dx\,dy\right) ^{1/2}. \end{aligned}$$

(3.2)

Along the paper we also denote by $H^s(\Omega )$ the usual fractional Sobolev space endowed with the norm (the so-called Gagliardo norm)

$$\begin{aligned} \Vert g\Vert _{H^s(\Omega )}=\Vert g\Vert _{L^2(\Omega )}+ \left( \int _{\Omega \times \Omega }\frac{|g(x)-g(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\right) ^{1/2}. \end{aligned}$$

(3.3)

We remark that, even in the model case in which $K(x)=|x|^{-(n+2s)}$, the norms in (3.1) and (3.3) are not the same, because $\Omega \times \Omega$ is strictly contained in $Q$ : this makes the classical fractional Sobolev space approach not sufficient for studying the problem.

For further details on the fractional Sobolev spaces we refer to [10] and to the references therein.

Some relations between the space $X_0$ and the usual fractional Sobolev space $H^s({\mathbb {R}}^n)$ were proved in [17], Lemma 5-$b)$] and in [19], Lemma 7], while, as for the embeddings of $X_0$ and $H^s({\mathbb {R}}^n)$ into the classical Lebesgue spaces, many properties were proved in [10, 17, 19]. We would like to note that, with respect to such embeddings, the fractional Sobolev space $H^s({\mathbb {R}}^n)$ behaves like the usual Sobolev space $H^1({\mathbb {R}}^n)$, while $X_0$ as $H^1_0(\Omega )$. This is due to the fact that the functions $v\in X_0$ are such that $v=0$ a.e. in ${\mathbb {R}}^n{\setminus } \Omega$ and so $X_0$ may be seen as a space of functions defined in the bounded set $\Omega$.