Elliptic problem involving logarithmic weight under exponential nonlinearities growth

Brahim Dridi

Mathematische Nachrichten, 2023, vol. 296, issue 12, 5551-5568

Abstract: In this paper, we study the nonlinear weighted elliptic problem −∇.(w(x)|∇u|N−2∇u)=f(x,u),u∈W01,N(B,w),$$\begin{equation*}\hskip5pc -\nabla .(w(x)|\nabla u|^{N-2} \nabla u) = f(x,u),\nobreakspace \nobreakspace u\in W_{0}^{1,N}(B,w),\hskip-5pc \end{equation*}$$where B is the unit ball of RN$\mathbb {R^{N}}$, N≥2$N\ge 2$, and w(x)=log1|x|β(N−1)$ w(x)=\left(\log \frac{1}{|x|}\right)^{\beta (N-1)}$ the singular logarithm weight with the limiting exponent N−1$N-1$ in the Trudinger–Moser embedding. The nonlinearities are critical or subcritical growth in view of Trudinger–Moser inequalities. We prove the existence of nontrivial solutions via the 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.

Date: 2023
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https://doi.org/10.1002/mana.202100601

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