Multiple solutions for nonhomogeneous p-Laplacian equations with nonlinear boundary conditions on R+N

In this paper, we study the existence of multiple solutions for the nonlinear boundary value problem (0.1){−div(|∇u|p−2∇u)+V(x)|u|p−2u=h(x),x∈R+N,|∇u|p−2∂u∂ν=λh1(x)|u|q−2u+h2(x)|u|r−2u,x∈∂R+N, where R+N={(x′,xN)∈RN−1×R+} is an upper half space in RN and 10, and 10 such that problem (0.1) admits at least two solutions provided that λ∈(0,λ0) and ‖h‖p′≤m00 is independent of λ>0. On the other hand, if h2=0, we prove that problem (0.1) admits at least one solution for any λ>0 and h∈Lp′(R+N).