Electron. J. Differential Equations, Vol. 2018 (2018), No. 178, pp. 1-14.

Positive solution curves of an infinite semipositone problem

Rajendran Dhanya

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 $\lambda\in (\Lambda,\Lambda+\epsilon)$.

Submitted May 3, 2018. Published November 1, 2018.
Math Subject Classifications: 35J25, 35J61, 35J75.
Key Words: Semipositone problems; topological methods; bifurcation theory.

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Rajendran Dhanya
School of Mathematics and Computer Science
Indian Institute of Technology
Goa 403401, India
email: dhanya.tr@gmail.com

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