Radial ground state sign-changing solutions for a class of asymptotically cubic or super-cubic Schrödinger–Poisson type problems

Abstract

This paper is dedicated to studying the following Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{ll} -\triangle u+V(|x|)u+\lambda \phi u=f(|x|,u), &{}\quad x\in \mathbb {R}^3,\\ -\triangle \phi = u^2,&{}\quad x\in \mathbb {R}^3, \end{array} \right. \end{aligned}$$ where $$\lambda $$ is a positive parameter, $$V\in {\mathcal {C}}(\mathbb {R}^{3}, (0,\infty ))$$ and $$f\in {\mathcal {C}}({\mathbb {R}}^{3}\times \mathbb {R}, \mathbb {R})$$ . Using weaker conditions $$\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s){\mathrm {d}}s}{|t|^3}=\infty $$ uniformly in $$x\in \mathbb {R}^3$$ , and $$\begin{aligned} \left[ \frac{f(x,\tau )}{\tau ^3}-\frac{f(x,t\tau )}{(t\tau )^3}\right] {\mathrm {sign}}(1-t) +\theta _0V(x)\frac{|1-t^2|}{(t\tau )^2}\ge 0, \quad \forall \;\; x\in \mathbb {R}^3,\ t>0, \;\; \tau \ne 0 \end{aligned}$$ with a constant $$\theta _0\in (0,1)$$ , instead of the usual super-cubic condition $$\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s){\mathrm {d}}s}{|t|^4}=\infty $$ uniformly in $$x\in \mathbb {R}^3$$ , and the Nehari type monotonic condition on $$f(x,t)/|t|^3$$ , we establish the existence of one radial ground state sign-changing solution $$u_\lambda $$ with precisely two nodal domains. Under the same conditions, we also prove that the energy of any radial sign-changing solution is strictly larger than two times the least energy; and give a convergence property of $$u_\lambda $$ as $$\lambda \searrow 0$$ . Our result unifies both asymptotically cubic and super-cubic cases, which extends the existing ones in the literature.