Equations involving fractional Laplacian operator: Compactness and application

Abstract

In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-Δ)^α u= |u|^{2^*_α-2-\varepsilon}u + λu\,\, {\rm in}\,\, Ω,\quad u=0 \,\, {\rm on}\, \, \partialΩ, \end{equation} where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$, $\epsilon \in [0,2_{\alpha }^{\ast }-2)$, $0<\alpha <1,2_{\alpha }^{\ast }=\frac{2N}{N-2\alpha }$. We show that for any sequence of solutions $u_n$ of \eqref{eq:0.1} corresponding to ${\epsilon }_n\in [0,2_{\alpha }^{\ast }-2)$, satisfying $\Vert u_n{\Vert }_H\le C$ in the Sobolev space $H$ defined in \eqref{eq:1.1a}, $u_n$ converges strongly in $H$ provided that $N>6\alpha$ and $\lambda >0$. An application of this compactness result is that problem \eqref{eq:0.1} possesses infinitely many solutions under the same assumptions.

34 pages