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Infinitely many solutions for a class of resonant problems

Philip Korman

TL;DR

The paper addresses resonance phenomena for semilinear elliptic problems on the unit ball, focusing on radial solutions and the structure of solution curves as the first harmonic parameter varies. It develops a robust framework of global solution curves and employs stationary-phase techniques to analyze oscillatory radial integrals, revealing sign-change and multiplicity patterns that depend critically on the dimension $n$ and on higher-order harmonics of the nonlinearity $g$. Through detailed one-, higher-dimensional analyses and a 1D generalization, the authors establish when infinitely many solutions occur and when they are finite, tying these results to precise asymptotics and symmetry properties. The work extends prior radial results, provides computational avenues, and yields criteria for oscillatory bifurcation behavior in resonant problems with mean-zero periodic nonlinearities.

Abstract

We consider radially symmetric solutions for a class of resonant problems on a unit ball $B \subset R^n$ around the origin \[ Δu+\la _1 u +g(u)=f(r) \s \mbox{for $x \in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function $g(u)$ is periodic of mean zero, $x \in R^n$, $r=|x|$, $\la _1$ is the principal eigenvalue of $Δ$ on $B$. The problem has either infinitely many or finitely many solutions depending on the space dimension $n$. The situation turns out to be different for each of the following cases: $1 \leq n \leq 3$, $n=4$, $n=5$, $n=6$, and $n \geq 7$.

Infinitely many solutions for a class of resonant problems

TL;DR

The paper addresses resonance phenomena for semilinear elliptic problems on the unit ball, focusing on radial solutions and the structure of solution curves as the first harmonic parameter varies. It develops a robust framework of global solution curves and employs stationary-phase techniques to analyze oscillatory radial integrals, revealing sign-change and multiplicity patterns that depend critically on the dimension and on higher-order harmonics of the nonlinearity . Through detailed one-, higher-dimensional analyses and a 1D generalization, the authors establish when infinitely many solutions occur and when they are finite, tying these results to precise asymptotics and symmetry properties. The work extends prior radial results, provides computational avenues, and yields criteria for oscillatory bifurcation behavior in resonant problems with mean-zero periodic nonlinearities.

Abstract

We consider radially symmetric solutions for a class of resonant problems on a unit ball around the origin Here the function is periodic of mean zero, , , is the principal eigenvalue of on . The problem has either infinitely many or finitely many solutions depending on the space dimension . The situation turns out to be different for each of the following cases: , , , , and .

Paper Structure

This paper contains 4 sections, 11 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Radially symmetric oscillatory integrals
  3. Oscillations of the solution curve
  4. A more general result in one dimension

Key Result

Lemma 2.1

Assume that the functions $f(x)$ and $\varphi (x)>0$ are of class $C^2[0,1]$, and satisfy Then, as $\xi \to \infty$,

Figures (3)

  • Figure 1: The integral in (\ref{['14']}), and its asymptotic approximation
  • Figure 2: The solution curve $\mu _1= \mu _1(\xi _1)$ of the problem (\ref{['1d5']}), oscillating around the $\xi _1$-axis.
  • Figure 3: The solution curve $\mu _1= \mu _1(\xi _1)$ of the problem (\ref{['1d6']}), oscillating near $\mu _1=0$

Theorems & Definitions (11)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Theorem 4.2
  • ...and 1 more