Infinitely many positive solutions to nonlinear scalar field equation with nonsmooth nonlinearity

TL;DR

This paper proves the existence of infinitely many positive solutions for a nonsmooth logarithmic scalar field equation using variational methods, addressing challenges posed by the non-smoothness of the energy functional.

Contribution

It introduces a novel application of localized variational methods to nonsmooth functionals in the context of logarithmic nonlinearities.

Findings

Existence of infinitely many positive solutions for the scalar field equation.

Existence of normalized multi-bump solutions with finite bumps.

First successful application of localized variational method to nonsmooth functionals.

Abstract

This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation \begin{equation} \tag{$P$} \label{equ1} -\Delta u+ V(x) u= u\log u^2, \quad u\in H^1(\mathbb{R}^N), \end{equation} and its counterpart with prescribed $L^2$-norms \begin{align}\label{equ2} \tag{$P_N$} & -\Delta u+ V(x) u +\lambda u= u\log u^2, \quad u\in H^1(\mathbb{R}^N), &\int_{\mathbb{R}^N} u^2 ~\mathrm{d}x=a^2>0, \end{align} which come from physically relevant situations. Here, $N\geq 2$, $V:\mathbb{R}^N\to \mathbb{R}$ is a non-symmetric and non-periodic potential satisfying certain decay conditions, $ a $ is prescribed constant, and $\lambda$ arises as an unknown Lagrange multipliers. For problem \eqref{equ1}, using purely variational methods, we establish the existence of multi-bump positive solutions with either finitely or infinitely many bumps. For normalized problem \eqref{equ2}, we prove the existence of normalized multi-bump positive solutions with a finite number of bumps. The main difficulty comes from the nonsmooth nature of logarithmic nonlinearity, which introduces some challenges to the variational framework. In particular, the corresponding energy functional is not of class $C^1$ on $H^1(\mathbb{R}^N)$, which prevents the direct application of standard critical point theory for $C^1$ functional or any reduction methods for $C^{1+\sigma}$ nonlinearity. The main ingredients in this paper are nonsmooth critical point theory, localized variational methods and a max-min argument. To the best of our knowledge, this paper appears to be the first successful application of the localized variational method to nonsmooth functionals.