Positive solutions for Schrödinger-Poisson-Slater system with coercive potential
- 20 February 2021
- journal article
- research article
- Published by Uniwersytet Mikolaja Kopernika/Nicolaus Copernicus University in Topological Methods in Nonlinear Analysis
- Vol. 57 (2), 1-13
- https://doi.org/10.12775/tmna.2020.041
Abstract
We study the following Schrödinger-Poisson-Slater type system: \begin{equation}\label{eq:0.1} \begin{cases} -\Delta u + V(x) u + \lambda\phi (x) u =|u|^{p-1}u,& x\in \mathbb{R}^3, \\ -\Delta\phi = u^2, \quad \lim\limits_{|x|\to +\infty}\phi (x)=0, \end{cases} \end{equation} where $\lambda> 0$ is a parameter, $p\in (1,2)$, $V\in C(\mathbb{R}^N,\mathbb{R}_0^+)$. If $V$ is a constant, or $p \in (2,5)$ with $V\in L^{\infty}(\mathbb{R}^3)$, the above system has been considered in many papers. In this paper, we are interested in the case: $p\in (1,2)$ and $V$ being a coercive potential, i.e., $\lim\limits_{|x|\to +\infty} V(x)=\infty$. We prove that there exists $\lambda_0> 0$ such that system (\ref{eq:0.1}) has at least two positive solutions $u^0_\lambda$ and $u^1_\lambda$ for any $\lambda \in (0, \lambda_0)$. Moreover, $u^0_\lambda$ is a ground state (i.e., the least energy solution) which must blow up as $\lambda \rightarrow 0$. Particularly, when $p\in ({11}/{7},2)$ and $\lambda> 0$ is small enough, we show that the ground state of (\ref{eq:0.1}) must be non-radially symmetric even if $V(x)=V(|x|)$, such as $V(x)=|x|^2$.
Keywords
This publication has 17 references indexed in Scilit:
- On the radiality of constrained minimizers to the Schrödinger–Poisson–Slater energyAnnales de l'Institut Henri Poincaré C, Analyse non linéaire, 2012
- Scaling properties of functionals and existence of constrained minimizersJournal of Functional Analysis, 2011
- Schrödinger–Poisson system with steep potential wellJournal of Differential Equations, 2011
- INFINITELY MANY POSITIVE SOLUTIONS FOR THE NONLINEAR SCHRÖDINGER–POISSON SYSTEMCommunications in Contemporary Mathematics, 2010
- On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial CasesArchive for Rational Mechanics and Analysis, 2010
- On Schrödinger-Poisson SystemsMilan Journal of Mathematics, 2008
- Ground state solutions for the nonlinear Schrödinger–Maxwell equationsJournal of Mathematical Analysis and Applications, 2008
- On the existence of solutions for the Schrödinger–Poisson equationsJournal of Mathematical Analysis and Applications, 2008
- The Schrödinger–Poisson equation under the effect of a nonlinear local termJournal of Functional Analysis, 2006
- Energy bounds for entire nodal solutions of autonomous superlinear equationsCalculus of Variations and Partial Differential Equations, 2006