Giovanni Molica Bisci and Raffaella Servadei
Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent
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ABSTRACT.  In the present paper we study the weak lower semicontinuity of the functional 
$$\Phi_{\lambda,\,\gamma}(u):=rac 1 2 \int_{\RR^n	imes\RR^n}rac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy-rac \lambda 2 \int_\Omega |u(x)|^2\,dx-rac\gamma 2\Big(\int_\Omega |u(x)|^{2^*}\,dx\Big)^{2/2^*},$$ 
where $\Omega$ 
is an open bounded subset of $\RR^n$, $n>2s$, $s\in (0,1)$\,, with Lipschitz 
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 
$$-(-\Delta)^s u(x):= 
\int_{\mathbb{R}^{n}}rac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \RR^n.$$ 
These two existence results were obtained using, respectively, the direct method in the calculus of variations and critical points theory.