Existence and asymptotic behavior of solutions to eigenvalue problems for Schrodinger-Bopp-Podolsky equations

Pub Date : 2023-10-13 DOI:10.58997/ejde.2023.66

Lorena Soriano Hernandez, Gaetano Siciliano

引用次数: 1

Abstract

We study the existence and multiplicity of solutions for the Schrodinger-Bopp-Podolsky system $$\displaylines{ -\Delta u + \phi u = \omega u \quad\text{ in } \Omega \cr a^2\Delta^2\phi-\Delta \phi = u^2 \quad\text{ in } \Omega \cr u=\phi=\Delta\phi=0\quad\text{ on } \partial\Omega \cr \int_{\Omega} u^2\,dx =1 }$$ where $\Omega$ is an open bounded and smooth domain in $\mathbb R^{3}$, $a>0 $ is the Bopp-Podolsky parameter. The unknowns are $u,\phi:\Omega\to \mathbb R$ and $\omega\in\mathbb R$. By using variational methods we show that for any $a>0$ there are infinitely many solutions with diverging energy and divergent in norm. We show that ground states solutions converge to a ground state solution of the related classical Schrodinger-Poisson system, as $a\to 0$. For more information see https://ejde.math.txstate.edu/Volumes/2023/66/abstr.html

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薛定谔-波普-波多尔斯基方程特征值问题解的存在性和渐近性

我们研究了Schrodinger-Bopp-Podolsky系统$$\displaylines{ -\Delta u + \phi u = \omega u \quad\text{ in } \Omega \cr a^2\Delta^2\phi-\Delta \phi = u^2 \quad\text{ in } \Omega \cr u=\phi=\Delta\phi=0\quad\text{ on } \partial\Omega \cr \int_{\Omega} u^2\,dx =1 }$$的解的存在性和多重性，其中$\Omega$为$\mathbb R^{3}$中的开有界光滑域，$a>0 $为Bopp-Podolsky参数。未知数是$u,\phi:\Omega\to \mathbb R$和$\omega\in\mathbb R$。利用变分方法证明了对于任意$a>0$存在无穷多个能量发散且范数发散的解。我们证明了基态解收敛于相关的经典薛定谔-泊松系统的基态解，如$a\to 0$。欲了解更多信息，请参阅https://ejde.math.txstate.edu/Volumes/2023/66/abstr.html

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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