Global solution curves in harmonic parameters, and multiplicity of solutions

Δu+g(u)=f(x) \s for $x \in Ω$, \s u=0 \s on $\partial Ω$ decompose $f(x)={\mu }_1{\text{p}}_1+e(x)$, where ${\text{p}}_1$ is the principal eigenfunction of the Laplacian with zero boundary conditions, and $e(x)\perp {\text{p}}_1$ in $L^2(\mathrm{\Omega })$, and similarly write $u(x)={\xi }_1{\text{p}}_i+U(x)$, with $U\perp {\text{p}}_1$ in $L^2(\mathrm{\Omega })$. 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 $g^{\mathrm{\prime }}(u)<{\text{la}}_2$. 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.