Turkish Journal of Mathematics
Abstract
In this paper, we consider the following critical fractional semilinear Neumann problem \begin{equation*} \begin{cases} (-\Delta)^{1/2}u+\lambda u=u^{\frac{n+1}{n-1}},~u>0\quad&\, \mathrm{in}\ \Omega,\\ \partial_\nu{u}=0 &\mathrm{on}\ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^n~(n\geq5)$ is a smooth bounded domain, $\lambda>0$ and $\nu$ is the outward unit normal to $\partial\Omega$. We prove that there exists a constant $\lambda_0>0$ such that the above problem admits a minimal energy solution for $\lambda<\lambda_0$. Moreover, if $\Omega$ is convex, we show that this solution is constant for sufficiently small $\lambda$.
DOI
10.55730/1300-0098.3458
Keywords
Fractional Laplacian operator, Neumann boundary condition, critical exponent
First Page
1715
Last Page
1732
Recommended Citation
JIN, Z, & SUN, H (2023). Fractional semilinear Neumann problem with critical nonlinearity. Turkish Journal of Mathematics 47 (6): 1715-1732. https://doi.org/10.55730/1300-0098.3458
