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Necessary and sufficient condition for existence at resonance for eigenvalues of multiplicity two

Philip Korman

TL;DR

The paper addresses existence of solutions to semilinear Dirichlet problems at resonance for eigenvalues with two-dimensional eigenspaces. It extends the Landesman–Lazer framework by deriving a necessary and sufficient condition expressed through projections of the forcing term onto the two-dimensional eigenspace and a universal quantity $J_{n,m}$ that depends on the jumps $g(\infty) - g(-\infty)$. In the disc setting $B_a \subset \mathbb{R}^2$, where all nonprincipal eigenvalues have multiplicity two, the authors obtain an explicit criterion: $\sqrt{A_k^2 + B_k^2} < J_{n,m}(g(\infty) - g(-\infty))$, with $A_k,B_k$ the appropriate projections, and $J_{n,m}$ computed from the eigenfunction radial structure. They illustrate with a unit-disc example and apply the results to prove unboundedness of all steady states for the corresponding semilinear heat equation, highlighting a PDE analogue of the Lazer–Leach theorem.

Abstract

We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation.

Necessary and sufficient condition for existence at resonance for eigenvalues of multiplicity two

TL;DR

The paper addresses existence of solutions to semilinear Dirichlet problems at resonance for eigenvalues with two-dimensional eigenspaces. It extends the Landesman–Lazer framework by deriving a necessary and sufficient condition expressed through projections of the forcing term onto the two-dimensional eigenspace and a universal quantity that depends on the jumps . In the disc setting , where all nonprincipal eigenvalues have multiplicity two, the authors obtain an explicit criterion: , with the appropriate projections, and computed from the eigenfunction radial structure. They illustrate with a unit-disc example and apply the results to prove unboundedness of all steady states for the corresponding semilinear heat equation, highlighting a PDE analogue of the Lazer–Leach theorem.

Abstract

We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation.
Paper Structure (2 sections, 6 theorems, 37 equations, 1 figure)

This paper contains 2 sections, 6 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Resonance for a two-dimensional disc

Key Result

Theorem 1.1

(L,wi) Assume that $g(u)$ satisfies (1), $f(x) \in L^2(D)$, while for any $w(x)$ belonging to the eigenspace of $\lambda _k$ Then the problem (0) has a solution $u(x) \in W^{2,2}(D) \cap W^{1,2}_0(D)$. Condition (2) is also necessary for the existence of solutions.

Figures (1)

  • Figure 1: The graph of $J_1 \left( \alpha _{_{1,2}} r \right)$ on the interval $(0,1)$

Theorems & Definitions (6)

  • Theorem 1.1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 2.1
  • Theorem 2.2