Weak solutions of quasilinear problems with nonlinear boundary condition

The growing attention for the study of the p-Laplacian operator Δp in the last few decades is motivated by the fact that it arises in various applications. Weak solutions for a quasilinear problems with nonlinear boundary condition are presented. A nonlinear elliptic boundary value problem: -div(a(x)|▽u|p-2▽u) = λ(1+|x|)α(1)|u|p-2u +(1+|x|)α(2)|u|q-2u in Ω, a(x)|▽u|p-2▽u·n+b(x) ·|u|p-2u = g(x,u) on Γ. It is assume throughout that 1<p<N, p<q<p* = Np/(N-p), -N<α1<-p, -N<α2<q·(N-p)/p-N, 0<a0≤a∈L∞(Ω) and b:Γ→R is a continuous function satisfying c/(1+|x|)p-1≤b(x) ≤C/(1+|x|)p-1, for constants 0<c≤C. Some computations that prove these equations are presented.