 1324 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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Mar 13, 13

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Abstract. In the present paper we study the weak lower semicontinuity of the functional
$$\Phi_{\lambda,\,\gamma}(u):=rac 1 2 \int_{\RR^n imes\RR^n}rac{u(x)u(y)^2}{xy^{n+2s}}\,dx\,dyrac \lambda 2 \int_\Omega u(x)^2\,dxrac\gamma 2\Big(\int_\Omega u(x)^{2^*}\,dx\Big)^{2/2^*},$$
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 $2^*:=2n/(n2s)$ 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
$$(\Delta)^s u(x):=
\int_{\mathbb{R}^{n}}rac{u(x+y)+u(xy)2u(x)}{y^{n+2s}}\,dy, \quad x\in \RR^n.$$
These two existence results were obtained using, respectively, the direct method in the calculus of variations and critical points theory.
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