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 {align*} -Δ u =λ u + (∫_Ω Q(|y|)F(u(y))/|x-y|^) Q(|x|)f(u) ~~in~~ Ω~~ and~~ u=0~~ on~~ ∂ Ω, {align*} where Ω ⊂ R² is a bounded domain with smooth boundary, μ ∈ (0,2) and λ >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 λ coincides with any of the eigenvalues of the operator (-Δ, H¹₀(Ω)).