Multiple solution results for elliptic Neumann problems involving set-valued nonlinearities

The main goal of this paper is to present multiple solution results for elliptic inclusions of Clarkeʹs gradient type under nonlinear Neumann boundary conditions involving the p-Laplacian and set-valued nonlinearities. To be more precise, we study the inclusion − Δ p u ∈ ∂ F ( x , u ) − | u | p − 2 u in Ω with the boundary condition | ∇ u | p − 2 ∂ u ∂ ν ∈ a ( u + ) p − 1 − b ( u − ) p − 1 + ∂ G ( x , u ) on ∂ Ω . We prove the existence of two constant-sign solutions and one sign-changing solution depending on the parameters a and b. Our approach is based on truncation techniques and comparison principles for elliptic inclusions along with variational tools like the nonsmooth Mountain-Pass Theorem, the Second Deformation Lemma for locally Lipschitz functionals as well as comparison results of local C 1 ( Ω ¯ ) -minimizers and local W 1 , p ( Ω ) -minimizers of nonsmooth functionals.