Abstract
In this paper we deal with the following fractional Kirchhoff problem
$$\begin{array}{}
\displaystyle
\left\{
{\begin{array}{l}
\left[M\left(\displaystyle \iint_{\mathbb R^n\times \mathbb R^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} dx dy\right)\right]^{p-1}(-\Delta)^{s}_{p}u
= f(x, u)+\lambda |u|^{r-2}u \\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\mbox{ in } \, \Omega, \\
\\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
u=0 \, \, ~\mbox{ in } \, \mathbb R^n\setminus \Omega.
\end{array}}
\right.
\end{array}$$
Here Ω ⊂ ℝn is a smooth bounded open set with continuous boundary ∂Ω, p ∈ (1, +∞), s ∈ (0, 1), n > sp,
$\begin{array}{}
(-\Delta)^{s}_{p}
\end{array}$ is the fractional p-Laplacian, M is a Kirchhoff function, f is a continuous function with subcritical growth, λ is a nonnegative parameter and r >
$\begin{array}{}
p^*_s
\end{array}$, where
$\begin{array}{}
p^*_s=\frac{np}{n-sp}
\end{array}$ is the fractional critical Sobolev exponent. By combining variational techniques and a truncation argument, we prove two existence results for this problem, provided that the parameter λ is sufficiently small.
Asymptotic behavior and existence results for a biharmonic
equation involving the critical Sobolev exponent in a five-dimensional domain
