Alessio Fiscella, Giovanni Molica Bisci, Raffaella Servadei
Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems
(490K, pdf)
ABSTRACT. In this paper we consider the following critical nonlocal problem
$$\left\{
egin{array}{ll}
-\mathcal L_K u=\lambda u+|u|^{2^*-2}u & \mbox{in } \Omega\
u=0 & \mbox{in } \RR^n\setminus \Omega\,,
\end{array}
ight.$$
where $s\in (0,1)$, $\Omega$ is an open bounded subset of $\RR^n$, $n>2s$, with Lipschitz boundary, $\lambda$ is a positive real parameter, $2^*:=2n/(n-2s)$ is the fractional critical Sobolev exponent, while $\mathcal L_K$ is the nonlocal integrodifferential operator
$$\mathcal L_Ku(x):=
\int_{\RR^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy\,,
\,\,\,\,\, x\in \RR^n\,,$$
whose model is given by the fractional Laplacian~$-(-\Delta)^s$\,.
Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of $-\mathcal L_K$ (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in