Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent

Xudong Shang; Jihui Zhang; Yang Yang

doi:10.3934/cpaa.2014.13.567

Article Contents

2014, Volume 13, Issue 2: 567-584. Doi: 10.3934/cpaa.2014.13.567

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Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent

1.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046
2.
Jiangsu Key Laboratory for NSLSCS, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu
3.
School of Science, Jiangnan University, Wuxi, 214122

Received: October 2012

Revised: June 2013

Published: March 2014

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Abstract

In this paper, we study the following problem \begin{eqnarray} (-\Delta)^{\frac{\alpha}{2}}u = K(x)|u|^{2_{\alpha}^{*}-2}u + f(x) \quad in \ \Omega,\\ u=0 \quad on \ \partial \Omega, \end{eqnarray} where $\Omega\subset R^N$ is a smooth bounded domain, $0< \alpha < 2$, $N>\alpha$, $ 2_{\alpha}^{*}= \frac{2N}{N-\alpha}$, $f\in H^{-\frac{\alpha}{2}}(\Omega)$ and $K(x)\in L^\infty(\Omega)$. Under appropriate assumptions on $K$ and $f$, we prove that this problem has at least two positive solutions. When $\alpha = 1$, we also establish a nonexistence result for a positive solution in a class of linear positive-type domains are more general than star-shaped ones.

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Mathematics Subject Classification: 35J60, 35R11, 35J20, 35J05.

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