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Global solution curves in harmonic parameters, and multiplicity of solutions

Philip Korman

TL;DR

This work develops a global continuation framework for semilinear Dirichlet problems by decomposing data and solutions into harmonic components and tracking how the first harmonic drives solution multiplicity. It proves the existence of a principal global solution curve μ1(ξ1) that exhausts the solution set under a spectral gap condition g'(u)<λ2, and extends the approach to higher harmonics and resonant oscillatory cases, with precise asymptotics for μk(ξk) and 1D stationary-phase formulas. The method yields sharp multiplicity results (e.g., at least two solutions when μ1>μ0) and supports efficient curve-following numerical algorithms, illustrated by numerous one-dimensional examples. Overall, harmonic parameter decomposition provides a unifying, computable view of solution branches and multiplicity in both nonresonant and resonant semilinear elliptic problems.

Abstract

\[ Δu+g(u)=f(x) \s \mbox{for $x \in Ω$}, \s u=0 \s \mbox{on $\partial Ω$} \] decompose $f(x)=μ_1 \p _1+e(x)$, where $\p _1$ is the principal eigenfunction of the Laplacian with zero boundary conditions, and $e(x) \perp \p _1$ in $L^2(Ω)$, and similarly write $u(x)= ξ_1 \p _i+U (x)$, with $ U \perp \p _1$ in $L^2(Ω)$. We study properties of the solution curve $(u(x),μ_1)(ξ_1)$, and in particular its section $μ_1=μ_1(ξ_1)$, which governs the multiplicity of solutions. We consider both general nonlinearities, and some important classes of equations, and obtain detailed description of solution curves under the assumption $g'(u)<\la _2$. We obtain particularly detailed results in case of one dimension. This approach is well suited for numerical computations, which we perform to illustrate our results.

Global solution curves in harmonic parameters, and multiplicity of solutions

TL;DR

This work develops a global continuation framework for semilinear Dirichlet problems by decomposing data and solutions into harmonic components and tracking how the first harmonic drives solution multiplicity. It proves the existence of a principal global solution curve μ1(ξ1) that exhausts the solution set under a spectral gap condition g'(u)<λ2, and extends the approach to higher harmonics and resonant oscillatory cases, with precise asymptotics for μk(ξk) and 1D stationary-phase formulas. The method yields sharp multiplicity results (e.g., at least two solutions when μ1>μ0) and supports efficient curve-following numerical algorithms, illustrated by numerous one-dimensional examples. Overall, harmonic parameter decomposition provides a unifying, computable view of solution branches and multiplicity in both nonresonant and resonant semilinear elliptic problems.

Abstract

decompose , where is the principal eigenfunction of the Laplacian with zero boundary conditions, and in , and similarly write , with in . We study properties of the solution curve , and in particular its section , which governs the multiplicity of solutions. We consider both general nonlinearities, and some important classes of equations, and obtain detailed description of solution curves under the assumption . We obtain particularly detailed results in case of one dimension. This approach is well suited for numerical computations, which we perform to illustrate our results.
Paper Structure (6 sections, 28 theorems, 131 equations, 3 figures)

This paper contains 6 sections, 28 theorems, 131 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Continuation of solutions
  4. The principal global solution curve
  5. Two important classes of equations
  6. Two classes of oscillatory equations at resonance

Key Result

Lemma 2.1

Assume that $u(x) \in L^2(\Omega)$, and $u(x)=\sum _{k=n+1}^{\infty} \xi _k \varphi _k$. Then

Figures (3)

  • Figure 1: The solution curve $\mu _1= \mu _1(\xi _1)$ of (\ref{['o2aaa']})
  • Figure 2: The solution curve $\mu _1= \mu _1(\xi _1)$ of (\ref{['o2a']}), compared with (\ref{['o2']})
  • Figure 3: The solution curve $\mu _7= \mu _7(\xi _7)$ of (\ref{['o7a']}), compared with (\ref{['o6']})

Theorems & Definitions (32)

  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Corollary 2.3
  • Theorem 3.1
  • Remark 3.1
  • Theorem 3.2
  • Proposition 3.1
  • ...and 22 more