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

Abstract

We are interested in the existence and asymptotic behavior of sign-changing solutions to the following nonlinear Schrodinger–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 $$-\Delta u+V(x)u=f(u),\quad \ x \in \mathbb{R}^3$$ .