In this paper we investigate the existence of solution for the following nonlocal problem with Stein-Weiss convolution term $\begin{equation*} -\Delta_{\Phi}u+V(x)\phi(|u|)u=\dfrac{1}{|x|^\alpha}\left(\int_{\mathbb{R}^{N}} \dfrac{K(y)F(u(y))}{|x-y|^{\lambda}|y|^\alpha}{\rm d}y\right)K(x)f(u(x)),\;\;x\in \mathbb{R}^{N}, \end{equation*}$ where $\alpha\geq 0$, $ N \geq 2$, $\lambda>0$ is a positive parameter, $V,K\in {C}(\mathbb R^N,[0,\infty))$ are nonne-gative 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){\rm d}s$. 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 $\mathcal{N}$-function $\tilde{\Phi }$ does not verify the $\Delta_{2}$-condition.
Lucas DA SILVA
,
Marco A. S. SOUTO
. A GENERALIZED CHOQUARD EQUATION WITH WEIGHTED ANISOTROPIC STEIN-WEISS POTENTIAL ON A NONREFLEXIVE ORLICZ-SOBOLEV SPACES[J]. Acta mathematica scientia, Series B, 2025
, 45(2)
: 569
-601
.
DOI: 10.1007/s10473-025-0217-4
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