Positive solutions for some semi-positone problems via bifurcation theory. (English) Zbl 0808.35030

The existence of positive solutions of Dirichlet boundary value problems \[ -\Delta u= \lambda f(x,u) \quad \text{on } \Omega \qquad u=0 \quad \text{ on } \partial\Omega \] is studied, where \(\Omega\) is a bounded domain with a smooth boundary \(\partial\Omega\) in \(\mathbb{R}^ N\), \(f:\Omega\times \mathbb{R}^ +\to \mathbb{R}\). It is always supposed that \(f(x,0)<0\) for all \(x\in \Omega\) (the so-called semi-positone problem). It is shown that the bifurcation theory can be used for the study of these problems. Asymptotically linear, superlinear and sublinear cases are successively considered and the existence of positive solutions for some positive parameters \(\lambda\) is proved. Particularly, in the superlinear and sublinear case, the existence of a positive solution is shown for all positive \(\lambda\) small enough and large enough, respectively.
Reviewer: M.Kučera (Praha)


35J65 Nonlinear boundary value problems for linear elliptic equations
35B32 Bifurcations in context of PDEs