A Generalized Choquard equation with weighted anisotropic Stein-Weiss potential on nonreflexive Orlicz-Sobolev Spaces

Abstract

In this paper we investigate the existence of solution for the following nonlocal problem with anisotropic Stein-Weiss convolution term $$ -\Delta_{\Phi} u+V(x)\phi(|u|)u=\dfrac{1}{|x|^\alpha}\left(\int_{\mathbb{R}^{N}} \dfrac{K(y)F(u(y))}{|x-y|^{\lambda}|y|^\alpha}dy\right)K(x)f(u(x)),\;\;x\in \mathbb{R}^{N} $$ where $\alpha\geq 0$, $ N \geq 2$, $\lambda>0$ 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 $F(t)=\int_{0}^{t}f(s)ds$. To establish our existence and regularity results, we use the Hardy-type inequalities for Orlicz-Sobolev Space and the Stein-weiss inequality together with a variational technique based on the mountain pass theorem for a functional that is not necessarily in $C^1$. Furthermore, we also prove the existence of a ground state solution by the method of Nehari manifold in the case where the strict monotonicity condition on $f$ is not required. This work incorporates the case where the $N$-function $\tilde{\Phi }$ does not verify the $\Delta_{2}$-condition.