Multiple solutions for nonhomogeneous -Laplacian equations with nonlinear boundary conditions on

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 − 2 u = h ( x ) , x ∈ R + N , | ∇ u | p − 2 ∂ u ∂ ν = λ h 1 ( x ) | u | q − 2 u + h 2 ( x ) | u | r − 2 u , x ∈ ∂ R + N , where R + N = { ( x ′ , x N ) ∈ R N − 1 × R + } is an upper half space in R N and 1 < p < N , λ > 0 , and 1 < q < p < r < p ∗ = p ( N − 1 ) N − p , and ν denotes the unit outward normal to the boundary ∂ R + N . The functions V ( x ) , h ( x ) , h 1 ( x ) , and h 2 ( x ) satisfy some suitable conditions. Using the mountain pass theorem and Ekeland’s variational principle, we prove that there exist λ 0 , m 0 > 0 such that problem (0.1) admits at least two solutions provided that λ ∈ ( 0 , λ 0 ) and ‖ h ‖ p ′ ≤ m 0 < c 1 λ ( p − 1 ) / ( r − q ) , where the constant c 1 > 0 is independent of λ > 0 . On the other hand, if h 2 = 0 , we prove that problem (0.1) admits at least one solution for any λ > 0 and h ∈ L p ′ ( R + N ) .