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, Tome 146 (2021), pp. 1-42.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

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 $n = 3$ .

Publié le : 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/}
}

TY  - JOUR
AU  - Abdelhedi, Wael
AU  - Hajaiej, Hichem
AU  - Mhamdi, Zeinab
TI  - A Morse lemma at infinity for nonlinear elliptic fractional equations
JO  - Rendiconti del Seminario Matematico della Università di Padova
PY  - 2021
SP  - 1
EP  - 42
VL  - 146
UR  - http://www.numdam.org/articles/10.4171/rsmup/82/
DO  - 10.4171/rsmup/82
LA  - en
ID  - RSMUP_2021__146__1_0
ER  -

%0 Journal Article
%A Abdelhedi, Wael
%A Hajaiej, Hichem
%A Mhamdi, Zeinab
%T A Morse lemma at infinity for nonlinear elliptic fractional equations
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2021
%P 1-42
%V 146
%U http://www.numdam.org/articles/10.4171/rsmup/82/
%R 10.4171/rsmup/82
%G en
%F RSMUP_2021__146__1_0

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

Cité par Sources :