Existence and blow-up rate of large solutions of p(x)-Laplacian equations with gradient terms

In this paper we investigate boundary blow-up solutions of the problem{−Δp(x)u+f(x,u)=±K(x)|∇u|m(x) in Ω,u(x)→+∞as d(x,∂Ω)→0, where Δp(x)u=div(|∇u|p(x)−2∇u) is called the p(x)-Laplacian. Our results extend the previous work [25] of Y. Liang, Q.H. Zhang and C.S. Zhao from the radial case to the non-radial setting, and [46] due to Q.H. Zhang and D. Motreanu from the assumption that K(x)|∇u(x)|m(x) is a small perturbation, to the case in which ±K(x)|∇u|m(x) is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of d(x,∂Ω) and in terms of the growth of the exponents. Furthermore, the comparison principle is no longer applicable in our context, since f(x,⋅) is not assumed to be monotone in this paper.