Regularity of the lower positive branch for singular elliptic bifurcation problems

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 $00$ 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 $\lambda=\Lambda$.

Keywords

• Singular elliptic problems
• positive solutions
• bifurcation problems
• implicit function theorem
• sub and super solutions