Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger–Poisson system in ${\mathbb{R}^3}$

Abstract

We are interested in the existence and asymptotic behavior of sign-changing solutions to the following nonlinear Schrödinger–Poisson system

$$\left\{\begin{array}{ll}-\Delta u+V(x)u+\lambda \phi(x)u =f(u), \ &\quad x \in \mathbb{R}^3,\\ -\Delta \phi=u^2, \ &\quad x \in \mathbb{R}^3,\end{array}\right.$$

where V(x) is a smooth function and λ is a positive parameter. Because the so-called nonlocal term ${\lambda \phi_u(x)u}$ is involving in the equation, the variational functional of the equation has totally different properties from the case of ${\lambda=0}$. Under suitable conditions, combining constraint variational method and quantitative deformation lemma, we prove that the problem possesses one sign-changing solution ${u_\lambda}$. Moreover, we show that any sign-changing solution of the problem has an energy exceeding twice the least energy, and for any sequence ${\{\lambda_n\} \rightarrow 0^+(n \rightarrow \infty)}$, there is a subsequence $\{\lambda_{n_k}\}$, such that ${u_{\lambda_{n_k}}}$ converges in ${H^1(\mathbb{R}^3)}$ to ${u_0}$ as ${k\rightarrow \infty}$, where ${u_0}$ is a sign-changing solution of the following equation

$$-\Delta u+V(x)u=f(u),\quad \ x \in \mathbb{R}^3$$

.