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 $\Omega \in C^2:$ $(-\Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x)$. Here $(-\Delta )_{Sp}^s$ stands for the $s$th power of the conventional Neumann Laplacian in $\Omega \Subset \mathbb {R}^n$, $n \geq 3$, $s \in (0, 1)$, $2^*_s = 2n/(n-2s)$. For the local case where $s = 1$, corresponding results were obtained earlier for the Neumann Laplacian and Neumann $p$-Laplacian operators.