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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} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \, \partial\Omega, \end{equation} where Ω\Omega is a smooth bounded domain in RN\mathbb{R}^N, ε∈[0,2α∗−2)\varepsilon\in [0, 2^*_\alpha-2), 0<α<1, 2α∗=2NN−2α0<\alpha<1,\, 2^*_\alpha = \frac {2N}{N-2\alpha}. We show that for any sequence of solutions unu_n of \eqref{eq:0.1} corresponding to εn∈[0,2α∗−2)\varepsilon_n\in [0, 2^*_\alpha-2), satisfying ∥un∥H≤C\|u_n\|_{H}\le C in the Sobolev space HH defined in \eqref{eq:1.1a}, unu_n converges strongly in HH provided that N>6αN>6\alpha and λ>0\lambda>0. An application of this compactness result is that problem \eqref{eq:0.1} possesses infinitely many solutions under the same assumptions.Comment: 34 page

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