## 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.

### MSC:

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