A+ CATEGORY SCIENTIFIC UNIT

An elliptic problem involving a potential with exponential growth

Volume 51 / 2024

Oussama Dammak, Brahim Dridi, Rached Jaidane Applicationes Mathematicae 51 (2024), 47-73 MSC: Primary 35J20; Secondary 35J30, 35K57, 35J60 DOI: 10.4064/am2501-5-2024 Published online: 10 July 2024

Abstract

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.

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