Existence of positive solutions for fractional Kirchhoff equation.

We study the following Kirchhoff equation involving fractional Laplacian in R N where N ≥ 2 , a ≥ 0 , b , μ > 0 , 0 < s < 1 , and (- Δ) s is the fractional Laplacian with order s. By reducing (K) to an equivalent system, we obtain the existence of a positive solution of (K) with general nonlinearities. The positive solution is unique if g (u) = | u | p - 1 u , 1 < p < N + 2 s N - 2 s . Moreover, if the function g is odd, the existence of infinitely many (sign-changing) solutions is concluded. As we shall see, for the case where 0 < s ≤ N 4 , a necessary condition of existence of nontrivial solutions of (K) is that b is small. Our method works well for the so-called degenerate case a = 0 . [ABSTRACT FROM AUTHOR]