Global bifurcation branches for radially symmetric Schrödinger equations

Abstract

We prove a new result on bifurcating branches of bound states for the nonlinear radially symmetric Schrödinger equation $$ -\Delta u =w(|x|)|u|^{\sigma}u -\lambda^2u \ \ \text{ on ${\mathbb{R}}^N$.} $$ We show that, under suitable assumptions on $w$ and $\sigma$, there exist infinitely many continua of nontrivial bound states $u_\lambda$ which emanate from the trivial solution branch at $\lambda=0$. These continua reach arbitrarily large values of $\lambda$, and they are distinguished by the number of nodal domains of the corresponding solutions $u_\lambda$.