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



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.