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

Research paper by **Sitong Chen, Xianhua Tang**

Indexed on: **01 Mar '18**Published on: **25 Jan '18**Published in: **Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas**

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#### Abstract

This paper is dedicated to studying the following Schrödinger–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.