Existence and asymptotic behavior of ground state sign-changing solutions for a nonlinear Schrödinger–Poisson–Kirchhoff system in ℝ3

Abstract

In this paper, we consider the existence and asymptotic behavior of ground state sign-changing solutions for the following nonlinear Schrödinger–Poisson–Kirchhoff system:

Under some mild assumptions, by using a series of constructive Nehari manifolds and a quantitative deformation lemma, we prove that the Schrödinger–Poisson–Kirchhoff system possesses at least one ground state sign-changing solution $u_b$ for all . Moreover, for any sequence $\left\{b_n\right\}\to 0^{+}$ $\left(n\to \infty \right)$, there exists a subsequence $\left\{b_{n_k}\right\}$, such that $\left\{u_{b_{n_k}}\right\}$ converges to $u_0$, where $u_0$ is a sign-changing solution of the equation $-a△u+V\left(x\right)u=\lambda f\left(x\right)u+g\left(x,u\right)$, $x\in {ℝ}^3$.