Positive solution curves of an infinite semipositone problem

Abstract

In this article we consider the infinite semipositone problem $-\Delta u =\lambda f(u)$ in $\Omega$, a smooth bounded domain in $\mathbb{R}^N$, and $u=0$ on $\partial\Omega$, where $f(t) = t^q-t^{-\beta}$ and $0 < q$, $\beta <1$. Using stability analysis we prove the existence of a connected branch of maximal solutions emanating from infinity. Under certain additional hypothesis on the extremal solution at $\lambda=\Lambda$ we prove a version of Crandall-Rabinowitz bifurcation theorem which provides a multiplicity result for $\lambda\in (\Lambda,\Lambda+\epsilon)$.

Keywords

• Semipositone problems
• topological methods
• bifurcation theory