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

Abstract

In this paper we study some nonlinear elliptic equations in ${\mathbb{R}}^n$ obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is $$(-\Delta {)}^{s}u=\epsilon h{u}^q+u^{p}in{\mathbb{R}}^n,$$ 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.

References in corpus (1)

• Concentration phenomenon for fractional nonlinear Schrödinger equations

Cited by in corpus (3)

• Fractional elliptic problems with critical growth in the whole of ${\mathbb{R}}^n$
• Spectrum of the fractional $p-$Laplacian in ${\mathbb{R}}^N$ and decay estimate for positive solutions of a Schrödinger equation
• A fractional elliptic problem in ${\mathbb{R}}^n$ with critical growth and convex nonlinearities