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,\begin{equation}\label{0001}\begin{cases}\left(a+b[u]^2_s\right)(-\Delta)^su=u^p+h(x,u, abla u), &x\in\Omega,\\u> 0, &x\in\Omega,\\u=0, &x otin\Omega,\\\end{cases}\end{equation}where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N(N\geq3)$, $0< s< 1$, $a,b> 0$ 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 (\ref{0001}) are proved by Leray-Schauder degree theory and the above estimates.