Nonradial normalized solutions for nonlinear scalar field equations

Abstract

We study the following nonlinear scalar field equation $$-\Delta u=f(u)-\mu u,u\in H^1({\mathbb{R}}^N)\text{with}\Vert u{\Vert }_{L^2({\mathbb{R}}^N)}^2=m.$$ Here $f\in C(\mathbb{R},\mathbb{R})$, $m>0$ is a given constant and $\mu \in \mathbb{R}$ is a Lagrange multiplier. In a mass subcritical case but under general assumptions on the nonlinearity $f$, we show the existence of one nonradial solution for any $N\ge 4$, and obtain multiple (sometimes infinitely many) nonradial solutions when $N=4$ or $N\ge 6$. In particular, all these solutions are sign-changing.

The version correspond to the one published in Nonlinearity