Normalized ground states for the critical fractional NLS equation with a perturbation

作者：Zhen, Maoding; Zhang, Binlin^*

来源：Revista Matematica Complutense, 2022, 35(1): 89-132.

DOI：10.1007/s13163-021-00388-w

• Fractional Laplacian
• Critical exponent
• Normalized ground state solution.

In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: @@@ {(-Delta)(s)u =lambda u + mu vertical bar u vertical bar(q-2)u + vertical bar u vertical bar(2)*(s) (-2)u, x is an element of R-N, @@@ integral(N)(R) u(2)dx = a(2), @@@ where (-Delta)(s) is the fractional Laplacian, 0 < s < 1, N > 2s, 2 < q < 2* s = 2N/(N-2s) is a fractional critical Sobolev exponent, a > 0, mu is an element of R. By using Jeanjean's trick in Jeanjean (Nonlinear Anal 28:1633-1659, 1997), and the standard method which can be found in Brezis and Nirenberg (Commun PureAppl Math 36:437-477, 1983) to overcome the lack of compactness, we first prove several existence and nonexistence results for a L-2-subcritical (or L-2-critical or L-2-supercritical) perturbation mu vertical bar u vertical bar(q-2)u, then we give some results about the behavior of the ground state obtained above as mu -> 0(+). Our results extend and improve the existing ones in several directions.