Some Existence Results for a Singular Elliptic Problem via Bifurcation Theory

TL;DR

This paper investigates a singular elliptic boundary value problem with a negative nonlinear term, establishing local bifurcation results and analyzing global bifurcation branches in the radial case.

Contribution

It introduces a variational approach to a singular elliptic problem with a negative nonlinear term and characterizes bifurcation phenomena.

Findings

No solutions for Dirichlet problem due to singularity.

Existence of bifurcating solutions from constant solutions.

Identification of a global unbounded bifurcation branch in the radial case.

Abstract

We study a semilinear elliptic problem with a singular nonlinear term of the type $g(u)=-u^{-1}$, using a variational approach. Note that the minus sign is important since the corresponding term in the Euler-Lagrange functional is concave. Contrary to the convex case there are no solutions for the Dirichlet problem, due to the power being $-1$. We therefore study the Neumann problem and prove a local existence result for solutions bifurcating from constant solutions. In the radial case we show that one of the two bifurcation branches is global and unbounded, and we find its asympotic behaviour.