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

In this paper, we study the following class of weighted Choquard equations \begin{align*} -Δu =λu + \Bigg(\displaystyle\int\limits_Ω\frac{Q(|y|)F(u(y))}{|x-y|^μ}dy\Bigg) Q(|x|)f(u) $\text{in}$ Ω~~ $\text{and}$~~ u=0~~ $\text{on}$~~ \partial Ω, \end{align*} where $\mathrm{\Omega }\subset {\mathbb{R}}^2$ 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 $(-\mathrm{\Delta },H_0^1(\mathrm{\Omega }))$.