Bifurcation results for a fractional elliptic equation with critical exponent in R^n

Abstract

In this paper we study some nonlinear elliptic equations in $\R^n$ obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is $$ (-\Delta)^s u = \epsilon\,h\,u^q + u^p \ {{in}}\R^n,$$ where $s\in(0,1)$, $n>4s$, $\epsilon>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $0<q<p$ and $h$ is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov-Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case $0<q<1$ is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.