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We study the nonlocal Schrödinger-Poisson-Slater type equation \( -\Delta u+\omega u+\lambda (I_\alpha \star u)=|u|^{p-2}u, \) where \(u \in \dot{H}^1(\mathbb {R}^N)\bigcap L^p(\mathbb R^N)\) , \(p>2\) , \(\omega \in \mathbb R\) , \(I_\alpha \) with \(0 < \alpha < N\) is the Riesz transform, and \(\lambda >0\) . We consider the cases when \(\omega =0\) , \(\omega =1\) and \(\lambda <0\) and \(\lambda >0\) . We prove existence results of radial groundstates in some cases and our method is based on the Nehari method.

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Ground States of Schrödinger-Poisson-Slatter Equations with Linear Non-local Terms

  • 会议论文

  • 发表日期：

    2025年

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  • Li Ma

摘要

We study the nonlocal Schrödinger-Poisson-Slater type equation \( -\Delta u+\omega u+\lambda (I_\alpha \star u)=|u|^{p-2}u, \) where \(u \in \dot{H}^1(\mathbb {R}^N)\bigcap L^p(\mathbb R^N)\) , \(p>2\) , \(\omega \in \mathbb R\) , \(I_\alpha \) with \(0 < \alpha < N\) is the Riesz transform, and \(\lambda >0\) . We consider the cases when \(\omega =0\) , \(\omega =1\) and \(\lambda <0\) and \(\lambda >0\) . We prove existence results of radial groundstates in some cases and our method is based on the Nehari method.