## Differential and Integral Equations

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

### Limiting behavior of solutions to an equation with the fractional Laplacian

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 November 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1384282858

**Mathematical Reviews number (MathSciNet)**

MR3161600

**Zentralblatt MATH identifier**

1313.35084

**Subjects**

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

#### Citation

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