Solvability of a critical semilinear problem with the spectral Neumann fractional Laplacian

Abstract

Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in Ω ∈ C 2 : \Omega \in C^2: ( − Δ ) S p s u ( x ) + u ( x ) = u 2 s ∗ − 1 ( x ) (-\Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x) . Here ( − Δ ) S p s (-\Delta )_{Sp}^s stands for the s s th power of the conventional Neumann Laplacian in Ω ⋐ R n \Omega \Subset \mathbb {R}^n , n ≥ 3 n \geq 3 , s ∈ ( 0 , 1 ) s \in (0, 1) , 2 s ∗ = 2 n / ( n − 2 s ) 2^*_s = 2n/(n-2s) . For the local case where s = 1 s = 1 , corresponding results were obtained earlier for the Neumann Laplacian and Neumann p p -Laplacian operators.