Supercritical fractional Kirchhoff type problems

Abstract

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.