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ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS TO FRACTIONAL NEUMANN PROBLEMS

Part of: Elliptic equations and systems Miscellaneous topics - Partial differential equations Partial differential equations Representations of solutions

Published online by Cambridge University Press:  13 September 2024

SOMNATH GANDAL  and
JAGMOHAN TYAGI
SOMNATH GANDAL
Affiliation:
Indian Institute of Technology Gandhinagar Palaj, Gnadhinagar, Gujarat 382055, India e-mail: gandal_somnath@iitgn.ac.in
JAGMOHAN TYAGI*
Affiliation:
Indian Institute of Technology Gandhinagar Palaj, Gandhinagar, Gujarat 382055, India e-mail: jtyagi1@gmail.com
*
e-mail: jtyagi@iitgn.ac.in
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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 $C^{1,1}$, $1<p<({n+s})/({n-s}),\,n>\max \{1, 2s \}, 0<s<1, d>0$ 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.

Keywords

semilinear Neumann problemfractional Laplacianpositive solutionsasymptotic behaviour

MSC classification

Primary: 35J60: Nonlinear elliptic equations 35B09: Positive solutions 35B40: Asymptotic behavior of solutions
Secondary: 35J61: Semilinear elliptic equations 35R11: Fractional partial differential equations 35D30: Weak solutions
Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 32
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Florica C. Cîrstea

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