Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent

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_{\alpha}^{*}= \frac{2N}{N-\alpha}$, $f\inH^{-\frac{\alpha}{2}}(\Omega)$ and $K(x)\in L^\infty(\Omega)$.Under appropriate assumptions on $K$ and $f$, we prove that thisproblem has at least two positive solutions. When $\alpha = 1$, wealso establish a nonexistence result for a positive solution in aclass of linear positive-type domains are more general thanstar-shaped ones.