Some Existence Results for a Singular Elliptic Problem via Bifurcation Theory

2020/12/26 by Saccon, Claudio
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.13743

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.

Related

• Bifurcation and Asymptotics for Elliptic Problems with Singular Nonlinearity
• Bifurcation results for semilinear elliptic problems in R^N
• Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities
• Non-negative solutions of a sublinear elliptic problem
• Singular solutions and bifurcation diagram of semilinear elliptic equations with general nonlinearity in two dimensions