Bifurcation analysis of the Schrödinger-Poisson-Slater equation with critical growth

Abstract

This paper deals with bifurcation results for (weak) solutions of the Schrödinger–Poisson–Slater equation involving the Coulomb potential and critical nonlinearity, modeled by

$$-\mathrm{\Delta }u+(|x{|}^{\alpha -N}\ast |u{|}^2)u=\mathrm{\lambda g}(x)u+|u{|}^{2^{\ast }-2}u\mathrm{in}{\mathrm{ℝ}}^N,$$

where $\lambda \in \mathrm{ℝ}$ , $N\ge 5$ , $0<\alpha <N-4$ , $2^{\ast }=2N∕(N-2)$ and g is a weight function. Using the global bifurcation theorem due to Rabinowitz, the existence of unbounded components and a bifurcation point of positive (weak) solutions of the nonlinear Schrödinger–Poisson–Slater equation are proved.