Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 49, pp. 1-32.

Title: Regularity of the lower positive branch for singular elliptic bifurcation problems

Authors: Tomas Godoy    (Univ. Nacional de Cordoba, Argentina)
         Alfredo Guerin (Univ. Nacional de Cordoba, Argentina)
 
Abstract:
We consider  the problem
$$\displaylines{
 -\Delta u=au^{-\alpha}+f(\lambda,\cdot,u) \quad\text{in }\Omega,\cr
 u=0\quad \text{on }\partial\Omega, \cr
 u>0 \quad \text{in }\Omega,
}$$
 where $\Omega$ is a  bounded domain in $\mathbb{R}^n$, $\lambda\geq 0$,
 $0\leq a\in L^{\infty}(\Omega) $, and $0<\alpha<3$.
 It is known that, under suitable assumptions on f, there exists
 $\Lambda>0$ such that this problem has at least one weak solution in
 $H_0^1(\Omega)\cap C(\overline{\Omega}) $ if and only if
 $\lambda\in[0,\Lambda] $; and that, for $0<\lambda<\Lambda$,
 at least two such solutions exist. Under additional hypothesis on a
 and f, we prove regularity properties of the branch formed by the minimal
 weak solutions of the above problem. As a byproduct of the method used,
 we obtain the uniqueness of the positive solution when $\lambda=\Lambda$.

Submitted August 7, 2018. Published April 12, 2019.
Math Subject Classifications: 35J75, 35D30, 35J20.
Key Words: Singular elliptic problems; positive solutions; bifurcation problems;
           implicit function theorem; sub and super solutions.