A Morse lemma at infinity for nonlinear elliptic fractional equations

Abdelhedi, Wael; Hajaiej, Hichem; Mhamdi, Zeinab

doi:10.4171/rsmup/82

Abdelhedi, Wael; Hajaiej, Hichem; Mhamdi, Zeinab

Rendiconti del Seminario Matematico della Università di Padova, Volume 146 (2021), pp. 1-42.

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Abstract

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

Published online: 2021-05-25

MR Zbl

DOI: 10.4171/rsmup/82

Classification: 35, 58

@article{RSMUP_2021__146__1_0,
     author = {Abdelhedi, Wael and Hajaiej, Hichem and Mhamdi, Zeinab},
     title = {A {Morse} lemma at infinity for nonlinear elliptic fractional equations},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {1--42},
     volume = {146},
     year = {2021},
     doi = {10.4171/rsmup/82},
     mrnumber = {4349650},
     zbl = {1484.35211},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/rsmup/82/}
}

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Abdelhedi, Wael; Hajaiej, Hichem; Mhamdi, Zeinab. A Morse lemma at infinity for nonlinear elliptic fractional equations. Rendiconti del Seminario Matematico della Università di Padova, Volume 146 (2021), pp. 1-42. doi : 10.4171/rsmup/82. http://www.numdam.org/articles/10.4171/rsmup/82/

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