Multiplicity of positive solutions to boundary blow‐up problem with variable exponent and sign‐changing weights

Abstract

In this paper, we consider the elliptic boundary blow‐up problem urn:x-wiley:mma:media:mma4119:mma4119-math-0001 where Ω is a bounded smooth domain of are positive continuous functions supported in disjoint subdomains Ω+,Ω− of Ω, respectively, a+ vanishes on the boundary of satisfies p(x)≥1 in Ω,p(x) > 1 on ∂Ω and , and ε is a parameter. We show that there exists ε∗>0 such that no positive solutions exist when ε > ε∗, while a minimal positive solution uε exists for every ε∈(0,ε∗). Under the additional hypotheses that is a smooth N − 1‐dimensional manifold and that a+,a− have a convenient decay near Γ, we show that a second positive solution vε exists for every ε∈(0,ε∗) if , where N∗=(N + 2)/(N − 2) if N > 2 and if N = 2. Our results extend that of Jorge Garcá‐Melián in 2011, where the case that p > 1 is a constant and a+>0 on ∂Ω is considered. Copyright © 2016 John Wiley & Sons, Ltd.