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

Research paper by Sitong Chen, Xianhua Tang

Indexed on: 13 Aug '16Published on: 01 Aug '16Published in: Zeitschrift für angewandte Mathematik und Physik



Abstract

Abstract This paper is dedicated to studying the following Schrödinger–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 \({\lambda\searrow 0}\) .AbstractThis paper is dedicated to studying the following Schrödinger–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 \({\lambda\searrow 0}\) . $${\left\{ \begin{array}{ll}-\triangle u+V(x)u+\lambda\phi u=K(x)f(u),& \quad x\in \mathbb{R}^{3},\\-\triangle\phi= u^2,\quad x\in \mathbb{R}^{3},\end{array}\right.}$$ $${\left\{ \begin{array}{ll}-\triangle u+V(x)u+\lambda\phi u=K(x)f(u),& \quad x\in \mathbb{R}^{3},\\-\triangle\phi= u^2,\quad x\in \mathbb{R}^{3},\end{array}\right.}$$VKf \({\lambda}\) \({\lambda}\) \({u_\lambda}\) \({u_\lambda}\) \({\theta_0\in (0,1)}\) \({\theta_0\in (0,1)}\) $$K(x)\left[\frac{f(\tau)}{\tau^3}-\frac{f(t\tau)}{(t\tau)^3} \right]\mathrm{sign}(1-t)+\theta_0V(x)\frac{ 1-t^2 }{(t\tau)^2} \geq 0, \quad \forall x \in\mathbb{R}^3, t > 0, \tau\ne 0$$ $$K(x)\left[\frac{f(\tau)}{\tau^3}-\frac{f(t\tau)}{(t\tau)^3} \right]\mathrm{sign}(1-t)+\theta_0V(x)\frac{ 1-t^2 }{(t\tau)^2} \geq 0, \quad \forall x \in\mathbb{R}^3, t > 0, \tau\ne 0$$ \({f(t)/ t ^3}\) \({f(t)/ t ^3}\) \({u_\lambda}\) \({u_\lambda}\) \({\lambda\searrow 0}\) \({\lambda\searrow 0}\)