Nonlinear elliptic problems in annular domains
This paper deals with positive radially symmetric solutions of \(\Delta u+f(u)=0\) in \(R_ 1<| x| <R_ 0\), \(x\in R^ N\), \(N\geq 3\) with Dirichlet boundary conditions, or with Neumann boundary conditions on one boundary sphere and Dirichlet conditions on the other. The nonlinearity f satisfies \(f(0)=0\), f is nondecreasing in (0,\(\infty)\), \(\lim_{t\to 0}f(t)/t=0,\lim_{t\to \infty}f(t)/t=\infty.\) Various existence results are obtained by a shooting method combined with comparison theorems and estimates for eigenvalues. We also prove uniqueness results which yield additional information on the solutions. Moreover we formulate a conjecture and prove the first half at the final section.
scientific article; zbMATH DE number 4002627
scientific article