Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent

Abstract

In the present paper, we study the weak lower semicontinuity of the functional \begin{align*} \Phi_{\lambda, \gamma}(u): & =\frac 1 2 \int_{\mathbb R ^n\times\mathbb R ^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dx\,dy -\frac \lambda 2 \int_\Omega |u(x)|^2 dx \\ & -\frac\gamma 2 \Big(\int_\Omega |u(x)|^{2^*} dx\Big)^{2/2^*}, \end{align*} where $\Omega$ is an open bounded subset of $\mathbb R ^n$, $n >2s$, $s\in (0,1)$, with continuous boundary, $\lambda$ and $\gamma$ are real parameters and $2^*:=2n/(n-2s)$ is the fractional critical Sobolev exponent. As a consequence of this regularity result for $\Phi_{\lambda, \gamma}$, we prove the existence of a nontrivial weak solution for two different nonlocal critical equations driven by the fractional Laplace operator $(-\Delta)^{s}$ which, up to normalization factors, may be defined as $$ -(-\Delta)^s u(x):= \int_{\mathbb{R}^{n}}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \mathbb R ^n. $$ These two existence results were obtained using, respectively, the direct method in the calculus of variations and critical points theory.