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Infinitely many solutions and asymptotics for resonant oscillatory problems

Philip Korman, Dieter S. Schmidt

TL;DR

The paper investigates semilinear elliptic PDEs at resonance on balls and rectangles, proving the existence of infinitely many solutions by tracking a global solution curve tied to the first eigenfunction. It provides asymptotic formulas for the first-harmonic parameter μ1 along these curves and complements the theory with detailed numerical computations. The results extend Landesman-Lazer-type frameworks and reveal oscillatory solution sets in resonant oscillatory problems, including radial cases in higher dimensions. Altogether, the work advances understanding of how resonance and oscillatory nonlinearities shape solution multiplicity and asymptotics in bounded domains.

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.

Infinitely many solutions and asymptotics for resonant oscillatory problems

TL;DR

The paper investigates semilinear elliptic PDEs at resonance on balls and rectangles, proving the existence of infinitely many solutions by tracking a global solution curve tied to the first eigenfunction. It provides asymptotic formulas for the first-harmonic parameter μ1 along these curves and complements the theory with detailed numerical computations. The results extend Landesman-Lazer-type frameworks and reveal oscillatory solution sets in resonant oscillatory problems, including radial cases in higher dimensions. Altogether, the work advances understanding of how resonance and oscillatory nonlinearities shape solution multiplicity and asymptotics in bounded domains.

Abstract

For a class of oscillatory resonant problems, involving Dirichlet problems for semilinear PDE's on balls and rectangles in , 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.
Paper Structure (5 sections, 6 theorems, 56 equations, 6 figures)

This paper contains 5 sections, 6 theorems, 56 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Global solution set for a ball in $R^2$
  3. Asymptotic formula in case of a rectangle
  4. Numerical computations
  5. Asymptotics and numerics for radial solutions

Key Result

Theorem 2.1

For the problem (intro1) assume that $h(u) \in C^2(R)$, $f(x) \in L^2(\Omega)$, and Then the solution set of (intro1) consists of a continuous curve $(u(x), \mu _1)(\xi _1)$ parameterized by $\xi _1 \in R$.

Figures (6)

  • Figure 1: The solution curve $\mu _1= \mu _1(\xi _1)$ of (\ref{['0']}), oscillating to $\pm \infty$. Values with $|\mu_1|<1$ are not shown.
  • Figure 2: The solution curve $\mu _1= \mu _1(\xi _1)$ of (\ref{['0']}) for $\xi _1 \in (0,1000)$
  • Figure 3: The solution curve $\mu _1= \mu _1(\xi _1)$ of (\ref{['1ex']}), compared with (\ref{['5']})
  • Figure 4: The solution curve $\mu _1= \mu _1(\xi _1)$ of (\ref{['1exa']}), compared with (\ref{['5']})
  • Figure 5: The solution curve $\mu _1= \mu _1(\xi _1)$ of (\ref{['r3']}), compared with (\ref{['r2']})
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 2.1
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 3.2