Ambrosetti–Prodi type results for a Neumann problem for the p-Laplacian operator

We study the Neumann boundary value problem aligned -Δ ₚ u&= f(x,u) + tφ (x) + h(x) in Ω ,\\ |∇ u|ᵖ⁻² ∂ u/∂ ν &= 0 on ∂ Ω , aligned - Δ p u = f ( x , u ) + t φ ( x ) + h ( x ) in Ω , | ∇ u | p - 2 ∂ u ∂ ν = 0 on ∂ Ω , where t is a parameter, Ω ⊂ RN Ω ⊂ R N is a smooth bounded domain, and Δ ₚ Δ p denotes the $$p-$$ p - Laplace operator with $$p>1$$ p > 1 . The functions φ φ and h are bounded, with φ φ nonnegative and nontrivial. Assuming an Ambrosetti–Prodi type asymmetric behavior for the nonlinearity f , we prove the existence of τ ₁≤ τ ₂ τ 1 ≤ τ 2 such that: if t> τ ₂