A priori bounds and existence of positive solutions for fractional Kirchhoff equations
DOI:
https://doi.org/10.12775/TMNA.2021.053Keywords
Fractional Kirchhoff equations, existence of solution, a priori bounds, Leray-Schauder degree theoryAbstract
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,\nabla u), &x\in\Omega,\\ u> 0, &x\in\Omega,\\ u=0, &x\notin\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 $h(x,u,\nabla u)$, 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.References
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