Electron. J. Differential Equations, Vol. 2017 (2017), No. 282, pp. 1-13.

Least energy sign-changing solutions for the nonlinear Schrodinger-Poisson system

Chao Ji, Fei Fang, Binlin Zhang

Abstract:
This article concerns the existence of the least energy sign-changing solutions for the Schrodinger-Poisson system
$$\displaylines{ -\Delta u+V(x)u+\lambda\phi(x)u=f(u),\quad \text{in } \mathbb{R}^3,\cr -\Delta\phi=u^2,\quad \text{in } \mathbb{R}^3. }$$
Because the so-called nonlocal term $\lambda\phi(x)u$ is involved in the system, the variational functional of the above system has totally different properties from the case of $\lambda=0$. By constraint variational method and quantitative deformation lemma, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $\lambda>0$, we show that the energy of a sign-changing solution is strictly larger than twice of the ground state energy. Finally, we consider $\lambda$ as a parameter and study the convergence property of the least energy sign-changing solutions as $\lambda\searrow 0$.

Submitted September 9, 2017. Published November 13, 2017.
Math Subject Classifications: 35J20, 35J60.
Key Words: Schrodinger-Poisson system; sign-changing solutions; constraint variational method; quantitative deformation lemma.

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Chao Ji
Department of Mathematics
East China University of Science and Technology
Shanghai 200237, China
email: jichao@ecust.edu.cn
Fei Fang
Department of Mathematics
Beijing Technology and Business University
Beijing 100048, China
email: fangfei68@163.com
Binlin Zhang
Department of Mathematics
Heilongjiang Institute of Technology
Harbin 150050, China
email: zhangbinlin2012@163.com

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