Generalized Choquard equation with potential vanishing at infinity

Abstract

In this paper we investigate the existence of solution for the following generalized Choquard equation $$-\Delta_{\Phi}u+V(x)\phi(|u|)u=\left(\int_{\mathbb{R}^{N}} \dfrac{K(y)F(u(y))}{|x-y|^{\lambda}}dy\right)K(x)f(u(x)), \;\;x\in \mathbb{R}^{N}$$ where $N\geq 3$, $\lambda>0$ is a positive parameter, $V,K\in C(\mathbb R^N,[0,\infty))$ are nonnegative functions that may vanish at infinity, the function $f\in C(\mathbb{R}, \mathbb R)$ is quasicritical and $F(t)=\int_{0}^{t}f(s)ds$. This work incorporates the reflexive and non-reflexive cases taking into account from Orlicz-Sobolev framework. The non-reflexive case occurs when the $N$-function $\tilde{\Phi}$ does not verify the $\Delta_{2}$-condition. In order to prove our main results we employ variational methods and regularity results