On a quasilinear elliptic problem with convection term and nonlinear boundary condition

Abstract

The first part of this paper deals with existence of solutions to the quasilinear elliptic problem (P)−diva(x,∇u)=f(x,u,∇u)inΩ,a(x,∇u)⋅ν=g(x,u)−ζ|u|p−2uon∂Ω,involving a general nonhomogeneous differential operator, namely diva, and Carathéodory functions f:Ω×R×RN→R and g:∂Ω×R→R. Under appropriate conditions on the perturbations, we show that (P) possesses a bounded solution. In the second part, we consider the special case when diva is the (p,q)-Laplacian with a parameter μ>0, and study the asymptotic behavior of solutions as μ goes to zero or to infinity. A uniqueness result is also provided.