A Multiplicity Result for a Non-local Critical Problem

In this paper, we are interested in the multiple solutions of the following fractional critical problem {(-Delta)(s) u = vertical bar u vertical bar(2)s*(-2)u + lambda u in Omega, u = 0 on R-N\Omega, where s is an element of (0, 1), N > 4s, 2(s)* = 2N/(N -2s), Omega is a smooth bounded domain in 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 is an element of (0, 1), the fractional critical problem has at least [(N + 1)/2] pairs of nontrivial solutions if 0 < A and has [(N +1 - l)/2] pairs if 0 < lambda not equal 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 >= a.