Differential and Integral Equations

Limiting behavior of solutions to an equation with the fractional Laplacian

Xiaoli Chen and Jianfu Yang

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In this paper, we investigate the limiting behavior of solutions $u_p$ of the following subcritical problem \begin{equation}\label{eq:0.1} (-\Delta)^{\frac \alpha 2} u = |x|^\beta u ^ {p-1},\quad u > 0, \quad x\in\Omega ;\quad u = 0,\quad x \in \partial \Omega \end{equation} as $p$ tending to the critical exponent $2^{*}_{\alpha}$, where $\Omega$ is the unit ball in $\mathbb{R}^n$ centered at the origin, and $\beta>0,0 <\alpha <2,\ 2 < p <2^{*}_{\alpha}=\frac{2N}{N-\alpha}$. We show that $u_p$ concentrates on a point on the boundary of the domain as $p\to 2^{*}_{\alpha}$.

Article information

Differential Integral Equations, Volume 27, Number 1/2 (2014), 157-179.

First available in Project Euclid: 12 November 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J25: Boundary value problems for second-order elliptic equations 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx] 35B45: A priori estimates 35J70: Degenerate elliptic equations


Chen, Xiaoli; Yang, Jianfu. Limiting behavior of solutions to an equation with the fractional Laplacian. Differential Integral Equations 27 (2014), no. 1/2, 157--179. https://projecteuclid.org/euclid.die/1384282858

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