$\mathbb{R}^3$ 中具有非常数位势的 Kirchhoff--Choquard 系统基态解的存在性

中文摘要 AI 辅助

本文研究 $\mathbb{R}^3$ 中如下线性耦合的 Kirchhoff-Choquard 系统：\begin{align*}\left\{\begin{array}{l} &-\left(a_1+b_1\int_{\mathbb{R}^3}|\nabla u|^2\\,dx\right)\Delta u+V_1(x)u=\mu(I_{\alpha}*|u|^p)|u|^{p-2}u+\lambda v,\quad x \in \mathbb{R}^3,\\\\ &-\left(a_2+b_2\int_{\mathbb{R}^3}|\nabla v|^2\\,dx\right)\Delta v+V_2(x)v=\nu(I_{\alpha}*|v|^q)|v|^{q-2}v+\lambda u,\quad x \in \mathbb{R}^3,\\\\ &u, v \in H^1(\mathbb{R}^3), \end{array}\right. \end{align*} 其中 $a_1, a_2, b_1, b_2, \lambda, \mu,$ 和 $\nu$ 均为正常数。当位势为常数函数时，作者先前利用 Nehari-Pohozaev 流形方法（NoDEA Nonlinear Differential Equations Appl.33(2026)）证明了以下情形中正基态解的存在性：非临界情形 $\frac{3+\alpha}{3}<p\le q<3+\alpha$、上半临界情形 $\frac{3+\alpha}{3}<p<q=3+\alpha$ 以及下半临界情形 $\frac{3+\alpha}{3}=p<q<3+\alpha$。在本文中，我们将这些结果推广到位势为非常数函数的情形。在对 $V_1(x)$、$V_2(x)$ 和 $\lambda$ 的适当假设下，我们证明了非平凡基态解的存在性。在非临界和上半临界情形中，主要工具是 Jeanjean 的单调性技巧和全局紧性引理。对于这些情形，我们通过将原点处的标准增长假设从 $o(|t|)$ 放宽到最优的 $o(|t|^{\alpha/3})$，建立了分裂引理的改进版本，从而显著拓宽了可适用的非线性类别。相比之下，对于下半临界情形 $p=\frac{3+\alpha}{3}$，分裂引理不再有效。为克服这一本质困难，我们通过对位势施加稍强的条件，直接在 Nehari-Pohozaev 流形上取极小元而获得基态解。

英文摘要

In this paper, we study the following linearly coupled Kirchhoff-Choquard system in $\mathbb{R}^3$: \begin{align*}\left\{\begin{array}{l} &-\left(a_1+b_1\int_{\mathbb{R}^3}|\nabla u|^2\,dx\right)Δu+V_1(x)u=μ(I_α*|u|^p)|u|^{p-2}u+λv,\quad x \in \mathbb{R}^3,\cr &-\left(a_2+b_2\int_{\mathbb{R}^3}|\nabla v|^2\,dx\right)Δv+V_2(x)v=ν(I_α*|v|^q)|v|^{q-2}v+λu,\quad x \in \mathbb{R}^3,\cr &u, v \in H^1(\mathbb{R}^3), \end{array}\right. \end{align*} where $a_1, a_2, b_1, b_2, λ, μ,$ and $ν$ are positive constants. When the potentials are constant functions, the author previously proved the existence of positive ground state solutions in the following cases: the noncritical case $\frac{3+α}{3}<p\le q<3+α$, the upper half critical case $\frac{3+α}{3}<p<q=3+α$, and the lower half critical case $\frac{3+α}{3}=p<q<3+α$, by using the Nehari-Pohozaev manifold method (NoDEA Nonlinear Differential Equations Appl.33(2026)). In the present paper, we extend these results to the case of nonconstant potentials. Under suitable assumptions on $V_1(x)$, $V_2(x)$, and $λ$, we prove the existence of nontrivial ground state solutions. In the noncritical and upper half critical cases, the main tools are Jeanjean's monotonicity trick and a global compactness lemma. For these cases, we establish a refined version of the splitting lemma by relaxing the standard growth assumption at the origin from $o(|t|)$ to the optimal $o(|t|^{α/3})$, thereby significantly broadening the applicable class of nonlinearities. In contrast, for the lower half critical case $p=\frac{3+α}{3}$, the splitting lemma is no longer valid. To overcome this essential difficulty, we obtain a ground state solution directly as a minimizer on the Nehari-Pohozaev manifold by imposing a slightly stronger condition on the potentials.

发表机构

• Kanagawa University（神奈川大学）

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