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

In this paper, we consider the elliptic boundary blow‐up problemΔ u = ( a + ( x ) − ε a − ( x ) ) u ( x ) p ( x )inΩ ,u = ∞ on∂ Ω ,where Ω is a bounded smooth domain ofR N , a + , a −are positive continuous functions supported in disjoint subdomains Ω + ,Ω − of Ω, respectively, a + vanishes on the boundary of Ω , p ( x ) ∈ C 2 ( Ω ̄ ) satisfies p ( x )≥1 in Ω, p ( x ) > 1 on ∂ Ω andsup x ∈ Ω +p ( x ) ≤ inf x ∈ Ω −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 u ε exists for every ε ∈(0, ε ∗ ). Under the additional hypotheses that Γ =Ω ̄+ ∩Ω ̄−is a smooth N − 1‐dimensional manifold and that a + , a − have a convenient decay near Γ, we show that a second positive solution v ε exists for every ε ∈(0, ε ∗ ) ifsup x ∈ Ω p ( x ) < N ∗ , where N ∗ =( N + 2)/( N − 2) if N > 2 andN ∗ = ∞ 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.