Electron. J. Differential Equations, Vol. 2019 (2019), No. 28, pp. 1-23.

Ground state, bound states and bifurcation properties for a Schrodinger-Poisson system with critical exponent

Jianqing Chen, Lirong Huang, Eugenio M. Rocha

This article concerns the existence of ground state and bound states, and the study of their bifurcation properties for the Schrodinger-Poisson system
 -\Delta u +u + \phi u = |u|^4u +\mu h(x)u, \quad
 -\Delta \phi = u^2 \quad\text{in }\mathbb{R}^3.
Under suitable assumptions on the coefficient h(x), we prove that the ground state must bifurcate from zero, and that another bound state bifurcates from a solution, when $\mu=\mu_1$ is the first eigenvalue of $-\Delta u +u = \mu h(x)u$ in $H^1(\mathbb{R}^3)$.

Submitted April 22, 2018. Published February 18, 2019.
Math Subject Classifications: 35J20, 35J70.
Key Words: Ground state and bound states; bifurcation properties; Schrodinger-Poisson system; variational method.

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Jianqing Chen
College of Mathematics and Informatics & FJKLMAA
Fujian Normal University
Qishan Campus, Fuzhou 350108, China
email: jqchen@fjnu.edu.cn
Lirong Huang
College of Mathematics and Physics
Fujian Jiangxia University
Fuzhou 350108, China
email: lrhuang515@126.com
Eugénio M. Rocha
Department of Mathematics
University of Aveiro, 3810-193
Aveiro, Portugal
email: eugenio@ua.pt

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