## Differential and Integral Equations

### 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 $$\label{eq:0.1} (-\Delta)^{\frac \alpha 2} u = |x|^\beta u ^ {p-1},\quad u > 0, \quad x\in\Omega ;\quad u = 0,\quad x \in \partial \Omega$$ 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