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

TL;DR

The paper addresses the existence of infinitely many positive solutions to the logarithmic scalar field equation $-\Delta u+ V(x)u= u\log u^2$ on $\mathbb{R}^N$ and its $L^2$-normalized variant, for non-symmetric, non-periodic decaying potentials. It develops a nonsmooth variational framework by decomposing the energy into a $C^1$ part and a convex, lower semicontinuous part, and then applies a localized max-min scheme with a localized Nehari constraint to construct multi-bump solutions, including infinitely many bumps, under small local perturbations of the limit potential $V_\infty$. Key elements include a careful analysis of submerged/emerging parts, a precise characterization of the limit Gaussson ground state $U$, and a rigorous energy-inequality structure that prevents bump escape, leading to both finite-bump normalized solutions and an infinite-bump, infinite-energy solution. The results extend multiplicity theory for nonlinear scalar field equations to nonsmooth, non-symmetric, non-periodic settings and provide detailed asymptotic behavior and decay (Gaussian-type) consistent with the logarithmic nonlinearity.

Abstract

This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation \begin{equation} \tag{$P$} \label{equ1} -Δu+ 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{$P_N$} & -Δu+ V(x) u +λu= 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 $λ$ 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 $C^{1+σ}$ 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.

Theorems & Definitions (60)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.2
• Theorem 1.4
• Lemma 2.1
• Definition 2.1
• Lemma 2.2
• Lemma 2.3