非自反Orlicz-Sobolev空间上带加权各向异性Stein-Weiss势的广义Choquard方程
A Generalized Choquard equation with weighted anisotropic Stein-Weiss potential on nonreflexive Orlicz-Sobolev Spaces
AI总结：
本文研究非自反Orlicz-Sobolev空间中带各向异性Stein-Weiss势的广义Choquard方程，利用Hardy不等式、Stein-Weiss不等式及山路引理和Nehari流形方法，证明了解与基态解的存在性，并涵盖N-函数不满足Δ2条件的情形。
摘要：
本文研究以下带各向异性Stein-Weiss卷积项的非局部问题解的存在性：$$-Δ_Φ u+V(x)ϕ(|u|)u=\dfrac{1}{|x|^α}\left(\int_{\mathbb{R}^{N}} \dfrac{K(y)F(u(y))}{|x-y|^λ|y|^α}dy\right)K(x)f(u(x)),\\;\\;x\in \mathbb{R}^{N}$$ 其中$α\geq 0$，$N \geq 2$，$λ>0$为正参数，$V,K\in {C}(\mathbb R^N,[0,\infty))$是可能在无穷远处消失的非负函数，函数$f\in C (\mathbb{R}, \mathbb R)$是拟临界的，且$F(t)=\int_{0}^{t}f(s)ds$。为建立存在性和正则性结果，我们利用Orlicz-Sobolev空间的Hardy型不等式和Stein-Weiss不等式，结合基于山路引理的变分技术，该技术适用于不一定属于$C^1$的泛函。此外，在不需要$f$的严格单调性条件的情况下，我们还通过Nehari流形方法证明了基态解的存在性。本工作包含了$N$-函数$\tilde{Φ}$不满足$Δ_{2}$条件的情形。
英文摘要：
In this paper we investigate the existence of solution for the following nonlocal problem with anisotropic Stein-Weiss convolution term $$ -Δ_Φ u+V(x)ϕ(|u|)u=\dfrac{1}{|x|^α}\left(\int_{\mathbb{R}^{N}} \dfrac{K(y)F(u(y))}{|x-y|^λ|y|^α}dy\right)K(x)f(u(x)),\;\;x\in \mathbb{R}^{N} $$ where $α\geq 0$, $ N \geq 2$, $λ>0$ 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 $\tilde{Φ}$ does not verify the $Δ_{2}$-condition.
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