An elliptic problem involving a potential with exponential growth

Oussama Dammak¹ ; Brahim Dridi² ; Rached Jaidane³

¹ Department of Mathematics Al-Lith University College Umm Al-Qura University Holly Makkah 21955, Saudi Arabia <a href="https://orcid.org/0000-0001-8210-9357">ORCID: 0000-0001-8210-9357</a>
² Department of Mathematics Faculty of Sciences Umm Al-Qura University Holly Makkah 21955, Saudi Arabia <a href="https://orcid.org/0000-0001-5863-029X">ORCID: 0000-0001-5863-029X</a>
³ Department of Mathematics Faculty of Science of Tunis University of Tunis El Manar Tunis, Tunisia <a href="https://orcid.org/0000-0001-7241-6847">ORCID: 0000-0001-7241-6847</a>

Applicationes Mathematicae, Tome 51 (2024) no. 1, pp. 47-73.

Résumé

We study the nonlinear weighted elliptic problem $$-\nabla . (w_{\beta }(x)|\nabla u|^{N-2} \nabla u)+V(x)|u|^{N-2}u = f(x,u), \quad u\in W_{0}^{1,N}(B,w_{\beta }) , $$ where $B$ is the unit ball of $\mathbb R^{N}$, $N \gt 2$, $w_{\beta }(x)=(1-\log |x|)^{\beta (N-1)}$, $\beta \in [0,1)$, is the singular logarithmic weight with the limiting exponent $N-1$ in the Trudinger–Moser embedding, and $V$ is a continuous positive potential. The nonlinearities critical or subcritical growth in view of Trudinger–Moser inequalities. We prove the existence of nontrivial solutions via critical point theory. In the critical case, the associated energy functional does not satisfy the compactness condition. We give a new growth condition and we point out its importance for checking the Palais–Smale compactness condition.

Détail

DOI : 10.4064/am2501-5-2024

Keywords: study nonlinear weighted elliptic problem nabla beta nabla n nabla n quad beta where unit ball mathbb beta log beta n beta singular logarithmic weight limiting exponent n trudinger moser embedding continuous positive potential nonlinearities critical subcritical growth view trudinger moser inequalities prove existence nontrivial solutions via critical point theory critical associated energy functional does satisfy compactness condition growth condition point out its importance checking palais smale compactness condition