Infinitely many solutions and asymptotics for resonant oscillatory problems

TL;DR

This paper proves the existence of infinitely many solutions for certain resonant semilinear PDEs on standard domains, analyzes their asymptotic behavior, and validates findings with detailed numerical computations.

Contribution

It establishes the existence of infinitely many solutions without restrictions on the first harmonic and derives asymptotic formulas, including numerical validation.

Findings

Existence of infinitely many solutions for resonant PDEs.

Asymptotic formulas accurately describe solutions.

Numerical methods effectively validate theoretical results.

Abstract

For a class of oscillatory resonant problems, involving Dirichlet problems for semilinear PDE's on balls and rectangles in $R^n$, we show the existence of infinitely many solutions, and study the global solution set. The first harmonic of the right hand side is not required to be zero, or small. We also derive asymptotic formulas in terms of the first harmonic of solutions, and illustrate their accuracy by numerical computations. The numerical method is explained in detail.