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 $$-{\Delta }_{\Phi }u+V(x)\varphi (|u|)u=\frac{1}{|x{|}^{\alpha }}\left({\int }_{{\mathbb{R}}^N}\frac{K(y)F(u(y))}{|x-y{|}^{\lambda }|y{|}^{\alpha }}dy\right)K(x)f(u(x)),x\in {\mathbb{R}}^N$$ where $\alpha \ge 0$, $N\ge 2$, $\lambda >0$ 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.

arXiv admin note: substantial text overlap with arXiv:2205.12298