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.