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

In this paper, we study the following class of weighted Choquard equations: −Δu=λu+∫ΩQ(|y|)F(u(y))|x−y|μdyQ(|x|)f(u)inΩandu=0on∂Ω,$$\begin{align*} -\Delta u =\lambda u + {\left(\int \limits _\Omega \frac{Q(|y|)F(u(y))}{|x-y|^\mu }dy\right)} Q(|x|)f(u) \nobreakspace \nobreakspace \textrm {in}\nobreakspace \nobreakspace \Omega \nobreakspace \nobreakspace \text{and}\nobreakspace \nobreakspace u=0\nobreakspace \nobreakspace \textrm {on}\nobreakspace \nobreakspace \partial \Omega, \end{align*}$$where Ω⊂R2$\Omega \subset \mathbb {R}^2$ is a bounded domain with smooth boundary, μ∈(0,2)$\mu \in (0,2)$ and λ>0$\lambda >0$ is a parameter. We assume that f$f$ is a real‐valued continuous function satisfying critical exponential growth in the Trudinger–Moser sense, and F$F$ is the primitive of f$f$. Let Q$Q$ 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 when λ$\lambda$ coincides with any of the eigenvalues of the operator (−Δ,H01(Ω))$(-\Delta, H^1_0(\Omega))$.