Fractional weighted $ p $-Kirchhoff equations with general nonlinearity

The aim of this paper is to study the existence of solutions for a class of weighted fractional $ p $-Kirchhoff problems with general nonlinearities$ \begin{align*} \begin{cases} M\left( \displaystyle {\iint }_{\mathbb{R}^{2N}}\frac{|x|^{\alpha p}|y|^{\alpha p}|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p,|x|^{\alpha}}^s u\\ = |x|^\beta(\lambda f(x,u)+|u|^{q-2}u),&\ \mathrm{in} \ \Omega,\\ u = 0,&\ \mathrm{in}\ \mathbb{R}^N\setminus\Omega, \end{cases} \end{align*} $where $ s\in(0,1) $, $ 1<p<N/s $, $ \alpha>-\frac{N-sp}{p} $, $ \beta>\alpha Np/(N-sp) $, $ \lambda>0 $, $ M\in C[0,\infty) $, $ (-\Delta)_{p,|x|^{\alpha}}^s $ is the weighted fractional $ p $-Laplacian, $ p<q<Np/(N-sp) $ and $ f\in C(\Omega\times\mathbb{R}) $. Under suitable assumptions, the existence of solutions is obtained by using the mountain pass theorem combined with a truncation argument. Moreover, we investigate the behavior of solutions as $ \lambda\rightarrow0 $. The main feature of our paper is that we only assume that the nonlinear term $ f $ satisfies $ p $-superlinear at zero and infinity respectively. Moreover, the nonlinear term $ f $ may satisfy critical or supercritical growth. Our result generalizes and improves the existence results of fractional $ p $-Laplacian problems in the literature.