Nonhomogeneous fractional $p$-Kirchhoff problems involving a critical nonlinearity

Abstract

This paper is concerned with the existence of solutions for a kind of nonhomogeneous critical $p$-Kirchhoff type problem driven by an integro-differential operator $\mathcal{L}^{p}_{K}$. In particular, we investigate the equation: \begin{align*} \mathcal{M}\left(\iint_{\mathbb{R}^{2n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+ps}}dxdy\right) \mathcal{L}^{p}_{K}v(x)=\mu g(x)|v|^{q-2}v+|v|^{p_{s}^{*}-2}v+\mu f(x) \quad\mbox{in}~\mathbb{R}^{n}, \end{align*} where $g(x)>0$, and $f(x)$ may change sign, $\mu>0$ is a real parameter, $0ps$, $1<q<p<p_{s}^{*}$, $p_{s}^{*}=\frac{np}{n-ps}$ is the critical exponent of the fractional Sobolev space $W^{s,p}_{K}(\mathbb{R}^{n}).$ By exploiting Ekeland's variational principle, we show the existence of non-trivial solutions. The main feature and difficulty of this paper is the fact that $\mathcal{M}$ may be zero and lack of compactness at critical level $L^{p_{s}^{*}}(\mathbb{R}^{n})$. Our conclusions improve the related results on this topic.

Keywords

• fractional $p$-kirchhoff problems
• non-homogeneous
• critical nonlinearity
• ekeland's variational principle