Nonradial solutions of nonlinear scalar field equations

TL;DR

This paper proves the existence of nonradial, sign-changing solutions to nonlinear scalar field equations in higher dimensions, extending previous results and introducing a new critical point theory on a topological manifold.

Contribution

It establishes the existence of nonradial solutions for scalar field equations under general conditions and develops a novel critical point framework for elliptic problems.

Findings

Existence of at least one nonradial solution in all dimensions N≥4.

Infinitely many nonradial solutions when N≠5.

Solutions are sign-changing and minimize energy on Pohozaev constraint.

Abstract

We prove new results concerning the nonlinear scalar field equation \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = g(u)&\quad \hbox{in }\mathbb{R}^N,\; N\geq 3, u\in H^1(\mathbb{R}^N)& \end{array} \right. \end{equation*} with a nonlinearity $g$ satisfying the general assumptions due to Berestycki and Lions. In particular, we find at least one nonradial solution for any $N\geq 4$ minimizing the energy functional on the Pohozaev constraint in a subspace of $H^1(\mathbb{R}^N)$ consisting of nonradial functions. If in addition $N\neq 5$, then there are infinitely many nonradial solutions. These solutions are sign-changing. The results give a positive answer to a question posed by Berestycki and Lions in [5,6]. Moreover, we build a critical point theory on a topological manifold, which enables us to solve the above equation as well as to treat new elliptic problems.