The Neumann problem for a class of semilinear fractional equations with critical exponent

TL;DR

This paper proves the existence of solutions for a fractional Laplacian Neumann problem with critical exponent, overcoming variational challenges using Sobolev inequalities and establishing solution uniqueness in small domains.

Contribution

It introduces new methods to handle the lack of Palais-Smale condition for critical exponent problems involving fractional Laplacian with Neumann boundary conditions.

Findings

Existence of solutions for the fractional Neumann problem with critical exponent.

Development of bounds for Rayleigh quotient in nonlocal setting.

Proof of solution uniqueness in small domains.

Abstract

We establish the existence of solutions to the following semilinear Neumann problem for fractional Laplacian and critical exponent: \begin{align*}\left\{\begin{array}{l l} { (-\Delta)^{s}u+ \lambda u= \abs{u}^{p-1}u } & \text{in $ \Omega,$ } \\ \hspace{0.8cm} { \mathcal{N}_{s}u(x)=0 } & \text{in $ \mathbb{R}^{n}\setminus \overline{\Omega},$} \\ \hspace{1.6cm} {u \geq 0}& \text{in $\Omega,$} \end{array} \right.\end{align*} where $\lambda > 0$ is a constant and $\Omega \subset \mathbb{R}^{n}$ is a bounded domain with smooth boundary. Here, $p=\frac{n+2s}{n-2s}$ is a critical exponent, $n > \max\left\{4s, \frac{8s+2}{3}\right\},$ $s\in(0, 1).$ Due to the critical exponent in the problem, the corresponding functional $J_{\lambda}$ does not satisfy the Palais-Smale (PS)-condition and therefore one cannot use standard variational methods to find the critical points of $J_{\lambda}.$ We overcome such difficulties by establishing a bound for Rayleigh quotient and with the aid of nonlocal version of the Cherrier's optimal Sobolev inequality in bounded domains. We also show the uniqueness of these solutions in small domains.