Blow-up boundary solutions for a class of nonhomogeneous logistic equations

Abstract

In this paper we will be concerned with the equations Δ_p(x)u = g(x)f(u), where Ω is a bounded domain, g is a non-negative continuous function on Ω which is allowed to be unbounded on Ω and non-linearity f is a non-negative non-decreasing functions. We show that the equation Δ_p(x)u = g(x)f(u) admits a non-negative local weak solution u ∈ W^1,p(x) _loc (Ω) ∩ C(Ω) such that u(x) → ∞  as x → ∂Ω if Δ_p(x)w = -g(x) in the weak sense for some w ∈ W^1,p(x)₀ (Ω) and f satisfies a generalized Keller-Osserman condition.

2000 Mathematics Subject Classification. Primary 35J60; Secondary 58E05.

Key words and phrases. elliptic equation, blow-up solutions, p(x)-Laplacian.