Nonlocal problem with critical exponential nonlinearity of convolution type: A non-resonant case

TL;DR

This paper proves the existence of nontrivial solutions for a class of weighted Choquard equations with critical exponential nonlinearity, excluding resonant parameter values, using variational methods in a nonlocal PDE setting.

Contribution

It establishes existence results for weighted Choquard equations with critical exponential growth, addressing non-resonant cases and allowing singular weights at zero.

Findings

Existence of solutions for non-resonant parameter values.

Handling of singular weights in the equation.

Application of variational methods to nonlocal PDEs with critical growth.

Abstract

In this paper, we study the following class of weighted Choquard equations \begin{align*} -\Delta u =\lambda u + \Bigg(\displaystyle\int\limits_\Omega \frac{Q(|y|)F(u(y))}{|x-y|^\mu}dy\Bigg) Q(|x|)f(u) ~~\textrm{in}~~ \Omega~~ \text{and}~~ u=0~~ \textrm{on}~~ \partial \Omega, \end{align*} where $\Omega \subset \mathbb{R}^2$ is a bounded domain with smooth boundary, $\mu \in (0,2)$ and $\lambda >0$ is a parameter. We assume that $f$ is a real valued continuous function satisfying critical exponential growth in the Trudinger-Moser sense, and $F$ is the primitive of $f$. Let $Q$ be a positive real valued continuous weight, which can be singular at zero. Our main goal is to prove the existence of a nontrivial solution for all parameter values except the resonant case, i.e., when $\lambda$ coincides with any of the eigenvalues of the operator $(-\Delta, H^1_0(\Omega))$.