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

Abstract:
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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