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.