Ground state sign-changing solutions for a class of Schrödinger–Poisson type problems in $${\mathbb{R}^{3}}$$ R 3

Abstract

This paper is dedicated to studying the following Schrodinger–Poisson system $${\left\{ \begin{array}{ll}-\triangle u+V(x)u+\lambda\phi u=K(x)f(u),& \quad x\in \mathbb{R}^{3},\\-\triangle\phi= u^2,\quad x\in \mathbb{R}^{3},\end{array}\right.}$$ where V,K are positive continuous potentials, f is a continuous function and \({\lambda}\) is a positive parameter. We develop a direct approach to establish the existence of one ground state sign-changing solution \({u_\lambda}\) with precisely two nodal domains, by introducing a weaker condition that there exists \({\theta_0\in (0,1)}\) such that $$K(x)\left[\frac{f(\tau)}{\tau^3}-\frac{f(t\tau)}{(t\tau)^3} \right]\mathrm{sign}(1-t)+\theta_0V(x)\frac{|1-t^2|}{(t\tau)^2} \geq 0, \quad \forall x \in\mathbb{R}^3, t > 0, \tau\ne 0$$ than the usual increasing condition on \({f(t)/|t|^3}\). Under the above condition, we also prove that the energy of any sign-changing solution is strictly larger than two times the least energy, and give a convergence property of \({u_\lambda}\) as \({\lambda\searrow 0}\).