Ricerche di Matematica

, Volume 60, Issue 2, pp 263–297

Ground states for Schrödinger–Poisson type systems

Article

DOI: 10.1007/s11587-011-0109-x

Cite this article as:
Vaira, G. Ricerche mat. (2011) 60: 263. doi:10.1007/s11587-011-0109-x

Abstract

In this paper we consider the following elliptic system in \({\mathbb{R}^3}\)
$$\qquad\left\{\begin{array}{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\end{array}\right.$$
where λ is a real parameter, \({p\in (1, 5)}\) if λ < 0 while \({p\in (3, 5)}\) if λ > 0 and K(x), a(x) are non-negative real functions defined on \({\mathbb{R}^3}\) . Assuming that \({\lim_{|x|\rightarrow+\infty}K(x)=K_{\infty} >0 }\) and \({\lim_{|x|\rightarrow+\infty}a(x)=a_{\infty} >0 }\) and satisfying suitable assumptions, but not requiring any symmetry property on them, we prove the existence of positive ground states, namely the existence of positive solutions with minimal energy.

Keywords

Non-autonomous Schrödinger–Poisson system Lack of compactness Variational methods 

Mathematics Subject Classification (2000)

35J05 35J10 35J50 35J60 

Copyright information

© Università degli Studi di Napoli "Federico II" 2011

Authors and Affiliations

  1. 1.SISSATriesteItaly

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