Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in

• Published in: Calculus of variations and partial differential equations Vol. 48; no. 1-2; pp. 243 - 273
• Main Authors: Wang, Jun, Tian, Lixin, Xu, Junxiang, Zhang, Fubao
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2013
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; 35J60
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-012-0548-6

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger–Poisson system where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that a ( x ) has at least one minimum and b ( x ) has at least one maximum. We first prove the existence of least energy solution ( u ε , φ ε ) for λ ≠ 0 and ε > 0 sufficiently small. Then we show that u ε converges to the least energy solution of the associated limit problem and concentrates to some set. At the same time, some properties for the least energy solution are also considered. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.