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Ground state sign-changing solutions for nonlinear Schr\"{o}dinger-Poisson system with indefinite potentials

Shubin Yu|Ziheng Zhang
대한수학회 (The Korean Mathematical Society)|2022.10|대한수학회논문집 (Communications of the KMS)|제37권 제4호|pp. 1269-1284|DOI 10.4134/CKMS.c210337

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키워드

Schr\"{o}dinger-Poisson systemnonlocal termsign-changing solution
주제분류수학

초록

This paper is concerned with the following Schr\"{o}dinger-\linebreak Poisson system $$ \left\{\begin{array}{ll} -{\Delta}u+V(x)u+K(x){\phi}u=a(x)|u|^{p-2}u &\mbox{in}\ \mathbb{R}^3, \\[0.1cm] -{\Delta}{\phi}=K(x)u^{2}&\mbox{in}\ \mathbb{R}^3, \\[0.1cm] \end{array} \right. $$ where $4<p<6$. For the case that $K$ is nonnegative, $V$ and $a$ are indefinite, we prove the above problem possesses one ground state sign-changing solution with exactly two nodal domains by constraint variational method and quantitative deformation lemma. Moreover, we show that the energy of sign-changing solutions is larger than that of the ground state solutions. The novelty of this paper is that the potential $a$ is indefinite and allowed to vanish at infinity. In this sense, we complement the existing results obtained by Batista and Furtado \cite{BF18}.

Abstract

This paper is concerned with the following Schr\"{o}dinger-\linebreak Poisson system $$ \left\{\begin{array}{ll} -{\Delta}u+V(x)u+K(x){\phi}u=a(x)|u|^{p-2}u &\mbox{in}\ \mathbb{R}^3, \\[0.1cm] -{\Delta}{\phi}=K(x)u^{2}&\mbox{in}\ \mathbb{R}^3, \\[0.1cm] \end{array} \right. $$ where $4<p<6$. For the case that $K$ is nonnegative, $V$ and $a$ are indefinite, we prove the above problem possesses one ground state sign-changing solution with exactly two nodal domains by constraint variational method and quantitative deformation lemma. Moreover, we show that the energy of sign-changing solutions is larger than that of the ground state solutions. The novelty of this paper is that the potential $a$ is indefinite and allowed to vanish at infinity. In this sense, we complement the existing results obtained by Batista and Furtado \cite{BF18}.

참고문헌 (31)

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