Existence of large solutions for a quasilinear elliptic problem via explosive sub-supersolutions

We consider the boundary blow-up quasilinear elliptic problems,div(|∇u|m-2∇u)±λ|∇u|q(m-1)=k(x)g(u)div(|∇u|m-2∇u)±λ|∇u|q(m-1)=k(x)g(u)in a C2C2 bounded domain with boundary condition u|∂Ω=+∞u|∂Ω=+∞, where m>1,q∈[0,m/(m-1)]m>1,q∈[0,m/(m-1)] and λ⩾0λ⩾0. Under suitable growth assumptions on k near the boundary and on g   both at zero and at infinity, we show the existence of at least one solution in C1(Ω)C1(Ω). Our proof is based on the method of explosive sub-supersolutions, which permits positive weights k(x)k(x) which are unbounded and/or oscillatory near the boundary.