Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem. (English) Zbl 0729.35046

The authors examine solutions of the semilinear boundary value problem (*) \(-\Delta u=g(x)f(u)\) in D, \(Bu=0\) on \(\partial D\), where D is a smooth bounded domain in \({\mathbb{R}}^ n\) and B denotes Dirichlet, Neumann, or Robin boundary conditions. Suppose also that f is strictly concave, that \(f(0)=0\), and that f is sublinear at infinity or vanishes at some finite number. Under suitable conditions on g, which allow g to change sign, the authors determine the number of positive solutions of (*); this number will be zero or one.


35J65 Nonlinear boundary value problems for linear elliptic equations
35B35 Stability in context of PDEs