Global solution curves in harmonic parameters, and multiplicity of solutions

TL;DR

This paper analyzes the structure of solution curves for a class of nonlinear elliptic boundary value problems, revealing how the solutions depend on harmonic parameters and establishing conditions for solution multiplicity, with detailed results especially in one dimension.

Contribution

It introduces a detailed analysis of solution curves in harmonic parameters for nonlinear elliptic equations, including numerical illustrations and conditions for multiple solutions.

Findings

Solution curves are characterized by the parameter $mbda_1$ and variable ta_1.

Under the condition g'(u)<, detailed properties of solution curves are established.

Numerical computations support the theoretical results, especially in one-dimensional cases.

Abstract

\[ \Delta u+g(u)=f(x) \s \mbox{for $x \in \Omega$}, \s u=0 \s \mbox{on $\partial \Omega$} \] decompose $f(x)=\mu _1 \p _1+e(x)$, where $\p _1$ is the principal eigenfunction of the Laplacian with zero boundary conditions, and $e(x) \perp \p _1$ in $L^2(\Omega)$, and similarly write $u(x)= \xi _1 \p _i+U (x)$, with $ U \perp \p _1$ in $L^2(\Omega)$. We study properties of the solution curve $(u(x),\mu _1)(\xi _1)$, and in particular its section $\mu _1=\mu _1(\xi _1)$, 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.