Taiwanese Journal of Mathematics
In this paper we investigate boundary blow-up solutions of the problem
⎧⎩⎨⎪⎪−△p(x)u+f(x,u)=ρ(x,u)+K(|x|)|∇u|δ(|x|) in Ω, u(x)→+∞ as d(x, ∂Ω)→0,
where −△p(x)u=−div(|∇u|p(x)−2∇u) is called p(x)-Laplacian. The existence of boundary blow-up solutions is proved and the singularity of boundary blow-up solution is also given for several cases including the case of ρ(x,u) being a large perturbation (namely, ρ(x,u(x))f(x,u(x))→1 as x→∂Ω). In particular, we do not have the comparison principle.
Liang, Yuan, Qihu Zhang, Chunshan Zhao.
"On the Boundary Blow-Up Solutions of p(x)-Laplacian Equations with Gradient Terms."
Taiwanese Journal of Mathematics, 18 (2): 599-632.