Existence, nonexistence and asymptotic behavior of boundary blow-up solutions to p(x)p(x)-Laplacian problems with singular coefficient

This paper investigates the problem {−Δp(x)u+ρ(x)f(x,u)=0in Ω,u(x)→+∞as d(x,∂Ω)→0, where −Δp(x)u=−div(|∇u|p(x)−2∇u)−Δp(x)u=−div(|∇u|p(x)−2∇u) is called the p(x)p(x)-Laplacian, and ρ(x)ρ(x) is a singular coefficient. The existence and nonexistence of boundary blow-up solutions is discussed, and the asymptotic behavior of boundary blow-up solutions is given. In particular, we do not assume radial symmetric conditions, and the pointwise different exact blow-up rate of solutions has been discussed.