A Morse lemma at infinity for nonlinear elliptic fractional equations

Abstract

In this paper, we consider the nonlinear fractional critical equation with zero Dirichlet boundary condition $A_{s}u=K{u}^{\frac{n+2s}{n-2s}}$, $u>0$ in $\Omega$ and $u=0$ on $\partial \Omega$, where $K$ is a positive function, $\Omega$ is a regular bounded domain of ${\mathbb{R}}^n$, $n\ge 2$ and $A_s$, $s\in (0,1)$ represents the spectral fractional Laplacian operator $(-\Delta {)}^s$ in $\Omega$ with zero Dirichlet boundary condition. We prove a version of Morse lemmas at infinity for this problem. We also exhibit a relevant application of our novel result. More precisely, we characterize the critical points at infinity of the associated variational problem and we prove an existence result for $s=\frac{1}{2}$ and $n=3$.