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 {-delta(p(x))u + f(x,u) = +/- K(x)vertical bar DEL;u vertical bar(m(x)) in omega, u(x) -> +infinity as d(x, PARTIAL;omega -> 0, where delta(p(x))u = div (vertical bar DEL;u vertical bar(DEL)-D-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)vertical bar DEL;u{x)vertical bar(m(x)) is a small perturbation, to the case in which +/- K(x)vertical bar DEL;u vertical bar(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, PARTIAL;omega) and in terms of the growth of the exponents. Furthermore, the comparison principle is no longer applicable in our context, since f(x, center dot) is not assumed to be monotone in this paper. (c) 2017 Published by Elsevier Inc.