Nonlinear scalar field equations with general nonlinearity

TL;DR

This paper develops an elementary approach to establish the existence and multiplicity of solutions for nonlinear scalar field equations with general nonlinearity, extending recent critical point theory results.

Contribution

It introduces a simpler method using an extended monotonicity trick and a new decomposition for Palais-Smale sequences, recovering recent advanced results.

Findings

Existence of positive ground states established.

Multiple radial and nonradial solutions demonstrated.

An elementary approach successfully reproduces recent complex results.

Abstract

Consider the nonlinear scalar field equation \begin{equation} \label{a1} -\Delta{u}= f(u)\quad\text{in}~\mathbb{R}^N,\qquad u\in H^1(\mathbb{R}^N), \end{equation} where $N\geq3$ and $f$ satisfies the general Berestycki-Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski [30] made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski's results on the scalar field equation. The keys to our approach are an extension to the symmetric mountain pass setting of the monotonicity trick, and a new decomposition result for bounded Palais-Smale sequences.