Abstract
In this work we study the existence of solutions to the critical Brezis–Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet–Neumann boundary conditions, i.e.,{(−Δ)su=λu+u2s⁎−1,u>0inΩ,u=0onΣD,∂u∂ν=0onΣN, where Ω⊂RN is a regular bounded domain, 12<s<1, 2s⁎ is the critical fractional Sobolev exponent, 0≤λ∈R, ν is the outwards normal to ∂Ω, ΣD, ΣN are smooth (N−1)-dimensional submanifolds of ∂Ω such that ΣD∪ΣN=∂Ω, ΣD∩ΣN=∅, and ΣD∩Σ‾N=Γ is a smooth (N−2)-dimensional submanifold of ∂Ω.
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