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

Abstract

The main goal of this paper is to present multiple solution results for elliptic inclusions of Clarke's gradient type under nonlinear Neumann boundary conditions involving the p-Laplacian and set-valued nonlinearities. To be more precise, we study the inclusion−Δpu∈∂F(x,u)−|u|p−2uin 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 C1(Ω¯)-minimizers and local W1,p(Ω)-minimizers of nonsmooth functionals.