Global solution curves in harmonic parameters, and multiplicity of solutions

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.