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 normalΔu=(a+(x)−εa−(x))u(x)p(x)in0.3em0.3emnormalΩ,u=∞on0.3em0.3em∂normalΩ, where Ω is a bounded smooth domain of double-struckRN,a+,a− are positive continuous functions supported in disjoint subdomains Ω+,Ω− of Ω, respectively, a+ vanishes on the boundary of normalΩ,p(x)∈C2(truenormalΩ̄) satisfies p(x)≥1 in Ω,p(x) > 1 on ∂Ω and supx∈normalΩ+p(x)≤infx∈normalΩ−p(x), 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 normalΓ=truenormalΩ̄+∩truenormalΩ̄− 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 supx∈normalΩp(x)<N∗, where N∗=(N + 2)/(N − 2) if N > 2 and N∗=∞ 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.