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

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 $H^1(\mathbb{R}^3)$.

Keywords

• Ground state and bound states
• bifurcation properties
• Schrodinger-Poisson system
• variational method