JournalsrsmupVol. 146pp. 1–42

A Morse lemma at infinity for nonlinear elliptic fractional equations

  • Wael Abdelhedi

    Qassim University, Buraidah, Saudi Arabia
  • Hichem Hajaiej

    California State University, Los Angeles, USA
  • Zeinab Mhamdi

    Sfax University, Tunisia
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Abstract

In this paper, we consider the nonlinear fractional critical equation with zero Dirichlet boundary condition Asu=Kun+2sn2sA_s u= K u^\frac{n+2s}{n-2s}, u>0u>0 in Ω\Omega and u=0u=0 on Ω\partial\Omega, where KK is a positive function, Ω\Omega is a regular bounded domain of Rn\mathbb{R}^n, n2n\geq 2 and AsA_s, s(0,1)s\in (0,1) represents the spectral fractional Laplacian operator (Δ)s(-\Delta)^s in Ω\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=12s=\frac{1}{2} and n=3n=3.

Cite this article

Wael Abdelhedi, Hichem Hajaiej, Zeinab Mhamdi, A Morse lemma at infinity for nonlinear elliptic fractional equations. Rend. Sem. Mat. Univ. Padova 146 (2021), pp. 1–42

DOI 10.4171/RSMUP/82