LAUSR

Abstract In this paper we investigate boundary blow-up solutions of the problem { − Δ p ( x ) u + f ( x , u ) = ± K ( x ) | ∇ u | m ( x )  in  Ω , u ( x ) → + ∞ as  d ( x , ∂ Ω ) → 0 , where Δ p ( 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 Furthermore, the comparison principle is no longer applicable in our context, since f ( x , ⋅ ) is not assumed to be monotone in this paper.