A Unified Truncation Method for Infinitely Many Solutions Without Symmetry
Anouar Bahrouni
TL;DR
This work tackles the problem of obtaining infinitely many solutions without symmetry across three increasingly challenging settings: a variational semilinear PDE, a non-variational elliptic PDE with gradient dependence, and periodic Hamiltonian systems on the real line. It introduces a unified truncation framework that localizes the nonlinearity around carefully chosen zeros, enabling a constructive multiplicity approach even when standard variational tools fail. For the variational problem, it yields two infinite sequences of positive and negative solutions; for the non-variational case, it combines a truncated variational scheme with an iterative method to produce two infinite disjoint solution families; for periodic Hamiltonian systems, it proves multiplicity persistence in the limit from finite intervals to the real line. Together, these results extend multiplicity theory beyond symmetry-related methods and offer a robust, cross-context truncation strategy.
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.