Existence of positive solutions for fractional Kirchhoff equation

We study the following Kirchhoff equation involving fractional Laplacian in R-N. (a + b integral R-N x R-N vertical bar u(x) - u(y)vertical bar(2/vertical bar)x - y vertical bar(N + 2s)dxdy) (-Delta)(s)u + uc = g(u), (K) where N >= 2, a >= 0, b, mu > 0, 0 < s < 1, and (-Delta)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) = vertical bar u vertical bar(p-1)u, 1 < p < N+2s/N-2s. 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.

Automated summary

This paper studies the problem of the Kirchhoff equation involving fractional Laplacian in R-N. By reducing the equation to an equivalent system, the existence and uniqueness of a positive solution with general nonlinearities are obtained. It is concluded that there exist infinitely many sign-changing solutions if the function g is odd. It is also found that a small value of b is a necessary condition for the existence of nontrivial solutions of (K) in the case where 0 < s <= N/4. The method used in this paper works well for the degenerate case a = 0.