This paper investigates periodic solutions for semilinear partial differential equations in the whole space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(n\ge 3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and develops an application to a singular quasilinear elliptic problem. Motivated by the influence of temporal periodicity on nonlinear dynamics, we introduce a framework based on Lyapunov-type ordered barriers and a monotone time-T iteration for the associated parabolic evolution. We first implement the method in the three-dimensional setting <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>, where we establish the existence of time-periodic solutions under general structural assumptions on the nonlinearity. We then extend the construction to higher dimensions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n>3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, demonstrating that the approach is robust with respect to the spatial dimension. As an application, we study a singular quasilinear elliptic problem of p-Laplacian type with a parametric reaction term and a singular contribution. We embed the elliptic equation into a corresponding autonomous parabolic flow and exploit an energy dissipation identity to relate time-T periodic solutions of the flow to equilibria, thereby obtaining at least one positive weak solution. We further establish multiplicity of solutions via a variational argument based on a Nehari-type decomposition. These results provide insight into the existence and multiplicity mechanisms for nonlinear PDEs with singular structures.

Periodic Solutions of Semilinear PDEs: High-Dimensional Extensions and an Application to a Singular Quasilinear Elliptic Problem

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发表日期：

2026年3月14日

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Anyao Wang

摘要

This paper investigates periodic solutions for semilinear partial differential equations in the whole space $\mathbb {R}^n$ $R^{n}$ $(n\ge 3)$ $(n \geq 3)$ and develops an application to a singular quasilinear elliptic problem. Motivated by the influence of temporal periodicity on nonlinear dynamics, we introduce a framework based on Lyapunov-type ordered barriers and a monotone time-T iteration for the associated parabolic evolution. We first implement the method in the three-dimensional setting $\mathbb {R}^3$ $R^{3}$ , where we establish the existence of time-periodic solutions under general structural assumptions on the nonlinearity. We then extend the construction to higher dimensions $\mathbb {R}^n$ $R^{n}$ with $n>3$ $n > 3$ , demonstrating that the approach is robust with respect to the spatial dimension. As an application, we study a singular quasilinear elliptic problem of p-Laplacian type with a parametric reaction term and a singular contribution. We embed the elliptic equation into a corresponding autonomous parabolic flow and exploit an energy dissipation identity to relate time-T periodic solutions of the flow to equilibria, thereby obtaining at least one positive weak solution. We further establish multiplicity of solutions via a variational argument based on a Nehari-type decomposition. These results provide insight into the existence and multiplicity mechanisms for nonlinear PDEs with singular structures.