### 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.