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A multiplicity result for a non-local critical problem. (English) Zbl 1427.35027

Summary: In this paper, we are interested in the multiple solutions of the following fractional critical problem \[\begin{cases} (-\Delta)^s u = |u|^{2_s^*-2} u + \lambda u &\text{in} \; \Omega, \\ u = 0 & \text{on} \; \mathbb{R}^N \setminus \Omega, \end{cases}\] where \(s \in (0,1), N > 4s, 2^*_s = 2N/(N-2s), \Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) and \((-\Delta)^s\) is the fractional Laplace operator. Because the nonlocal property of fractional Laplacian makes the variational functional of the fractional critical problem different from the one of local operator \(-\Delta \). To the best of our knowledge, it is still unknown whether multiple solutions of the fractional critical problem exist for all \(\lambda > 0\). In this paper, we give a partial answer. Precisely, by introducing some new ideas and careful estimates, we prove that for any \(s \in (0,1)\), the fractional critical problem has at least \([(N+1)/2]\) pairs of nontrivial solutions if \(0 < \lambda \neq \lambda_n\), and has \([(N+1-l)/2]\) pairs if \(\lambda = \lambda_n\) with multiplicity number \(0 < l < \min \{n,N+2\} \), via constraint method and Krasnoselskii genus. Here \(\lambda_n\) denotes the \(n\)-th eigenvalue of \((-\Delta)^s\) with zero Dirichlet boundary data in \(\Omega\) and \([a]\) denotes the least positive integer \(k\) such that \(k \geq a\).

MSC:

35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35R11 Fractional partial differential equations
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