A priori bounds and existence of positive solutions for fractional Kirchhoff equations - Li, Pengfei; Xie, Junhui; Mu, Dan (Topological Methods in Nonlinear Analysis, 2022)
In this paper, we are concerned with the following Kirchhoff equations involving the fractional Laplacian, equation0001 cases (a+b[u]^2_s)(-Δ)^su=u^p+h(x,u, abla u), &xΩ, u> 0, &xΩ, u=0, &x otinΩ, cases equation where Ω is a smooth bounded domain in R ^N(N≥3), 0< s< 1, a,b> 0 and 0< p< (N+2s)/(N-2s) are constants. Under suitable conditions on h(x,u, abla u), using the defining integral, we carry on a blowing-up and rescaling argument directly on the nonlocal equations and thus obtain a priori estimates on the positive solutions. Moreover, existence results for positive solutions of problem (0001) are proved by Leray-Schauder degree theory and the above estimates.