Nonlocal elliptic equation with critical exponential growth and resonance in high-order eigenvalues

Abstract

In this paper we are interested in the following nonlocal elliptic equation$$\begin{cases}\displaystyle -\Delta u=\lambda_{k}u+\bigg[\frac{1}{|x|^{\mu}}\ast G(x,u)\bigg]g(x,u)&\mbox{in } \Omega,\\ u=0 &\mbox{on } \partial\Omega,\end{cases}$$where an open $\Omega\subset\mathbb{R}^{2}$ is bounded with smooth boundary. The nonlinearity $g(x,s)$ has the critical exponential growth in the sense of the Trudinger-Moser inequality and $\lambda_{k}$ denotes the $k$th eigenvalue of $(-\Delta,H_{0}^{1}(\Omega))$, $k\geq2$. Employing variational methods we prove the existence of a nontrivial solution for this nonlocal elliptic problem.