Sign-changing solutions of fractional Laplacian system with critical exponent

作者：Li, Qi; Wen, Shixin^*

来源：MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 2024.

DOI：10.1002/mma.9952

• fractional critical Sobolev exponent
• fractional Laplacian system
• sign-changing solutions

This paper deals with the following fractional Laplacian system with critical exponent: @@@ {(-Delta)(s)u = lambda(1)u + mu(1)vertical bar u vertical bar(2s)*(-2)u + beta vertical bar u vertical bar(2s)*(/2-2)u vertical bar v vertical bar(2s)*(/2), in Omega, @@@ (-Delta)(s)v = lambda(2)v + mu 2 vertical bar v vertical bar(2s)*(-2)v + beta vertical bar v vertical bar(2s)*(/2-2)v vertical bar u vertical bar(2s)*(/2), in Omega, @@@ u = v = 0, in R-N\Omega, @@@ where Omega is a bounded smooth open connected set in R-N, 2(s)* = 2N/N-2s is the fractional critical Sobolev exponent, s is an element of(0, 1), mu(1), mu(2) > 0, 0 < lambda(1), lambda(2) < lambda(1)(Omega) and lambda(1)(Omega) is the first eigenvalue of fractional Laplacian (-Delta)(s) under the condition u = 0 in R-N\Omega. We prove that, for each fixed beta is an element of(-root mu(1)mu(2), 0) and lambda(1), lambda(2) slightly smaller than lambda(1)(Omega), the above system with N >= 5s admits a sign-changing solution in the following sense: one component changes sign, while the other one is positive. Our result includes the lower dimensional case N is an element of[5s, 6s). Compared with the classical Laplacian case, our problem is nonlocal and the first component of the solution is sign-changing, some new difficulties arise and new arguments and estimates should be introduced.