A Morse lemma at infinity for nonlinear elliptic fractional equations
Summary: 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 \(\mathbb{R}^n\), \(n\geq 2\) 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 \(n=3\).
semilinear equation with fractional Laplacian
scientific article; zbMATH DE number 7473027
scientific article