Subcritical nonlocal problems with mixed boundary conditions

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].