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 p(x)-Laplacian equations with gradient terms Our results extend the previous work of Y. Liang, Q.H. Zhang and C.S. Zhao, published in 2014, from the radial case to the non-radial setting, and of Q.H. Zhang and D. Motreanu, appeared in 2016, from the assumption that the term involving the gradient is a small perturbation, to the harder case in which it is a large perturbation.We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of of the distance of a point x to the boundary of the domain,say d(x). and in terms of the growth of the exponents. Furthermore, the comparison principle is no longer applicable in our context, since the forcing term f is not assumed to be monotone in this paper.