Electron. J. Differential Equations, Vol. 2019 (2019), No. 49, pp. 1-32.

Regularity of the lower positive branch for singular elliptic bifurcation problems

Tomas Godoy, Alfredo Guerin

We consider the problem
 -\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.

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Tomas Godoy
FaMAF, Universidad Nacional de Córdoba
(5000) Córdoba, Argentina
email: godoy@mate.uncor.edu
  Alfredo Guerin
FaMAF, Universidad Nacional de Córdoba
(5000) Córdoba, Argentina
email: guerin.alfredojose@gmail.com

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