By using linking and ∇-theorems in this paper we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet–Neumann boundary data, (Formular Presented) where (−Δ)s, s ∈ (1/2, 1), is the spectral fractional Laplacian operator, Ω ⊂ RN, N > 2s, is a smooth bounded domain, λ > 0 is a real parameter, ν is the outward normal to ∂Ω, ΣD, ΣN are smooth (N − 1)-dimensional submanifolds of ∂Ω such that ΣD ∪ ΣN = ∂Ω, ΣD ∩ ΣN = ∅ and ΣD ∩ ΣN = Γ is a smooth (N − 2)-dimensional submanifold of ∂Ω.
