A Unified Truncation Method for Infinitely Many Solutions Without Symmetry

TL;DR

This paper introduces a unified truncation method to prove the existence of infinitely many solutions in various nonlinear PDEs without symmetry, including variational, non-variational, and dynamical systems.

Contribution

It develops a novel truncation technique applicable across different PDE classes and overcomes longstanding challenges in non-symmetric, gradient-dependent, and infinite domain problems.

Findings

Established existence of infinitely many solutions for semilinear elliptic PDEs.

Proved multiple solutions for non-variational elliptic PDEs with gradient dependence.

Showed multiplicity of solutions persists on the entire real line for periodic Hamiltonian systems.

Abstract

This paper establishes the existence of infinitely many solutions for nonlinear problems without any symmetry, achieving three major advances. First, in the setting of semilinear elliptic PDEs, we introduce a refined variational truncation method that yields infinite sequences of positive as well as negative solutions. Second and most notably, we resolve a long-standing and difficult problem for nonvariational elliptic PDEs with gradient dependence. By combining our truncation method with an iterative scheme, we prove, for the first time, the existence of infinitely many solutions for this class of PDEs. Third, we overcome a central difficulty for periodic Hamiltonian systems on the real line: we show that the multiplicity of solutions, constructed on a sequence of finite intervals, survives in the limit; in other words, no collapse occurs, and we obtain multiple distinct solutions on the whole real line. The core novelty lies in a carefully designed truncation methodology that systematically separates solutions and remains effective across variational and non-variational PDEs as well as infinite dimensional dynamical systems. This unified perspective provides a robust and versatile tool for addressing multiplicity problems in the absence of symmetry.