PDE Seminar Abstracts

Elliptic problems with sign-changing weights and boundary blow-up

Jorge García-Melián
Universidad de La Laguna, Spain
Mon 30 May 2011 2-3pm, Mills Lecture Room 202

Abstract

We consider the elliptic boundary blow-up problem

Δu=(a+(x)-εa-(x))upin Ω,u=on Ω,

where Ω is a smooth bounded domain of N, a+, a- are positive continuous functions supported in disjoint subdomains Ω+, Ω- of Ω, respectively, p>1 and ε>0 is a parameter. We show that there exists ε*>0 such that no positive solutions exist when ε>ε*, while a minimal positive solution exists for every ε(0,ε*). Under the additional hypotheses that Ω¯+ and Ω¯- intersect along a smooth (N-1)-dimensional manifold Γ and a+, a- have a convenient decay near Γ, we show that a second positive solution exists for every ε(0,ε*) if p<N*=(N+2)(N-2). Our proofs are mainly based on continuation methods.