Existence of multiple solutions for singular elliptic problems with nonlinear boundary conditions

In this paper, the existence and multiplicity results of solutions for the following quasilinear elliptic problem(0.1) { − div ( | x | − a p | ∇ u | p − 2 ∇ u ) + h ( x ) | u | p − 2 u = g ( x ) | u | r − 2 u , x ∈ Ω , | x | − a p | ∇ u | p − 2 ∂ u ∂ ν = λ f ( x ) | u | q − 2 u , x ∈ ∂ Ω are established, where Ω is an exterior domain in R N with the compact and smooth boundary ∂Ω. By the variational methods, we prove that the problem (0.1) has at least two positive solutions. At the last part of the paper, we also consider the critical case and prove the existence of solution by concentration-compactness principle.