Existence of solutions for quasilinear elliptic exterior problem with the concave-convex nonlinearities and the nonlinear boundary conditions

In this paper, we consider the following quasilinear elliptic exterior problem { − div ( a ( x ) | ∇ u | p − 2 ∇ u ) + g ( x ) | u | q − 2 u = h ( x ) | u | s − 2 u + λ H ( x ) | u | r − 2 u , x ∈ Ω , a ( x ) | ∇ u | p − 2 ∂ u ∂ ν + b ( x ) | u | p − 2 u = 0 , x ∈ Γ = ∂ Ω where Ω is a smooth exterior domain in R N , and ν is the unit vector of the outward normal on the boundary Γ, 1 < p < N , 1 < s < p < r < p ⁎ = N p / ( N − p ) . By the variational principle and the Mountain Pass Theorem, we establish the existence of infinitely many solutions if q > r and at least one solution if 1 < q < s .