A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass

In this article, we study the following fractional Kirchhoff-type problems with critical and sublinear nonlinearities:<br /> {{a+b (RNxRN integral integral)u(x) = u(y)(2 )/ Ix - yN+2s dxdy] -Delta(s)u(q-1 + )u(s-1)(2 )u > 0 , in Omega<br /> -dxdy (-4)3u = lambda u9-1+ u2-1, u> 0, in Q,<br /> ju = 0, in R<^>\Q,<br /> Su2dx = c2,<br /> where (-4) is the fractional Laplacian, QC RN is a bounded domain with Lipschitz boundary, 0 < s < 1,2s <N < 4s, 1 <q <2, lambda > 0, a > 0, b > 0, c> 0. First, we prove that the bounded Palais-Smale sequence has a profile decomposition in the fractional Laplacian setting. Then, by utilizing decomposition techniques and variational methods, we acquire that there are two positive normalized solutions for the aforementioned problems.