Equations involving fractional Laplacian operator: Compactness and application

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 $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $\varepsilon\in [0, 2^*_α-2)$, $0<α<1,\, 2^*_α= \frac {2N}{N-2α}$. We show that for any sequence of solutions $u_n$ of \eqref{eq:0.1} corresponding to $\varepsilon_n\in [0, 2^*_α-2)$, satisfying $\|u_n\|_{H}\le C$ in the Sobolev space $H$ defined in \eqref{eq:1.1a}, $u_n$ converges strongly in $H$ provided that $N>6α$ and $λ>0$. An application of this compactness result is that problem \eqref{eq:0.1} possesses infinitely many solutions under the same assumptions.

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