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


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.