This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation and its counterpart with prescribed <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norms which come from physically relevant situations. Here, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$V:\mathbb {R}^N \rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is a non-symmetric and non-periodic potential satisfying certain decay conditions, a is a prescribed constant, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> arises as an unknown Lagrange multiplier. For problem (1), using purely variational methods, we establish the existence of multi-bump positive solutions with either finitely or infinitely many bumps. For the normalized problem (2), we prove the existence of normalized multi-bump positive solutions with a finite number of bumps. The main difficulty comes from the nonsmooth nature of logarithmic nonlinearity, which introduces some challenges to the variational framework. In particular, the corresponding energy functional is not of class <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$H^1(\mathbb {R}^N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which prevents the direct application of standard critical point theory for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> functionals or any reduction methods for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$C^{1+\sigma }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>σ</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> nonlinearities. 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.

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

期刊论文
发表日期：

2026年4月1日

查看全文

Tianhao Liu,
Juncheng Wei,
Wenming Zou

摘要

This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation and its counterpart with prescribed $L^2$ $L^{2}$ -norms which come from physically relevant situations. Here, $N \ge 2$ $N \geq 2$ , $V:\mathbb {R}^N \rightarrow \mathbb {R}$ $V : R^{N} \to R$ is a non-symmetric and non-periodic potential satisfying certain decay conditions, a is a prescribed constant, and $\lambda $ $λ$ arises as an unknown Lagrange multiplier. For problem (1), using purely variational methods, we establish the existence of multi-bump positive solutions with either finitely or infinitely many bumps. For the normalized problem (2), 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 $C^1$ $C^{1}$ on $H^1(\mathbb {R}^N)$ $H^{1} (R^{N})$ , which prevents the direct application of standard critical point theory for $C^1$ $C^{1}$ functionals or any reduction methods for $C^{1+\sigma }$ $C^{1 + σ}$ nonlinearities. 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.