ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS TO FRACTIONAL NEUMANN PROBLEMS
Somnath Gandal, Jagmohan Tyagi · Journal of the Australian Mathematical Society · 2024
Abstract We study the asymptotic behaviour of the least energy solutions to the following class of nonlocal Neumann problems: $$ \begin{align*} \begin{cases} { d(-\Delta)^{s}u+ u= \vert u\vert^{p-1}u } & \text{in } \Omega, \\ {u>0} & \text{in } \Omega, \\ { \mathcal{N}_{s}u=0 } & \text{in } \mathbb{R}^{n}\setminus \overline{\Omega}, \end{cases} \end{align*} $$ where $\Omega \subset \mathbb {R}^{n}$ is a bounded domain of class $C^{1,1}$ , $1 \max \{1, 2s \}, 00$ and $\mathcal {N}_{s}u$ is the nonlocal Neumann derivative. We show that for small $d,$ the least energy solutions $u_d$ of the above problem achieve an $L^{\infty }$ -bound independent of $d.$ Using this together with suitable $L^{r}$ -estimates on $u_d,$ we show that the least energy solution $u_d$ achieves a maximum on the boundary of $\Omega $ for d sufficiently small.