Exact multiplicity of solutions to superlinear and sublinear problems
The author considers the equation \(\Delta u+\lambda f(u)=0\) in a bounded smooth domain \(\Omega\subset\mathbb R^n\) with Dirichlet boundary conditions. It is required that all Dirichlet eigenvalues \(\lambda_1<\lambda_2<\lambda_3<\dots\) of \(-\Delta\) on \(\Omega\) are simple. The nonlinearity \(f\in{\mathcal{C}}^1(\mathbb R)\) satisfies \(f(0)=0\), \(f'>0\), and \(f(u)/|u|\) is either decreasing or increasing for \(u\in\mathbb R\setminus\{0\}\). Moreover, the limits \(f(u)/u\rightarrow f\pm>0\) as \(u\rightarrow \pm\infty\) are assumed to exist. The main results of the paper state the existence of precisely \(0,1,2\) or \(4\) solutions for \(\lambda\) lying in certain parameter ranges depending on the quotients \(\lambda_k/f'(0)\), \(\lambda_k/f_+\), \(\lambda_k/f_-\). There are also informations on the Morse indices and the signs of the solutions. The proofs are based on bifurcation theorems and on degree arguments. The conditions on \(f\) imply that solutions are nondegenerate and that their Morse indices are determined by \(\lambda\).
scientific article; zbMATH DE number 1788034
scientific article