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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, (− Δ) s u = λ u + f (x , u) in  Ω , u = 0 on  Σ , ∂ u ∂ ν = 0 on  Σ , where (− Δ) s , s ∈ (1 / 2 , 1) , is the spectral fractional Laplacian operator, Ω ⊂ ℝ N , N > 2 s , is a smooth bounded domain, λ > 0 is a real parameter, ν is the outward normal to ∂ Ω , Σ , Σ are smooth (N − 1) -dimensional submanifolds of ∂ Ω such that Σ ∪ Σ = ∂ Ω , Σ ∩ Σ = ∅ and Σ ∩ Σ ¯ = Γ is a smooth (N − 2) -dimensional submanifold of ∂ Ω. [ABSTRACT FROM AUTHOR]

By using linking and [Formula: see text]-theorems in this paper we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet–Neumann boundary data, (−Δ)su = λu + f(x,u)in Ω,u = 0 on Σ𝒟,∂u ∂ν = 0 on Σ𝒩, where [Formula: see text], [Formula: see text], is the spectral fractional Laplacian operator, [Formula: see text], [Formula: see text], is a smooth bounded domain, [Formula: see text] is a real parameter, [Formula: see text] is the outward normal to [Formula: see text], [Formula: see text], [Formula: see text] are smooth [Formula: see text]-dimensional submanifolds of [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] is a smooth [Formula: see text]-dimensional submanifold of [Formula: see text].