Existence of at least $k$ solutions to a fractional $p$-Kirchhoff problem involving singularity and critical exponent

Abstract

We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity \begin{align} \mathfrak{M}\left(\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s} u&=\frac{\lambda}{|u|^{\gamma-1}u}+|u|^{p_s^*-2}u~\text{in}~\Omega,\nonumber u&>0~\text{in}~\Omega,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminus\Omega,\nonumber \end{align} where $\Omega\subset\mathbb{R}^N$, is a bounded domain with Lipschitz boundary, $\lambda>0$, $N>ps$, $0<s,\gamma<1$, $(-\Delta)_{p}^{s}$ is the fractional $p$-Laplacian operator for $1<p<\infty$ and $p_s^*=\frac{Np}{N-ps}$ is the critical Sobolev exponent. We employ a {\it cut-off} argument to obtain the existence of $k$ (being an arbitrarily large integer) solutions. Furthermore, by using the Moser iteration technique, we prove a uniform $L^{\infty}({\Omega})$ bound for the solutions. The novelty of this work lies in proving the existence of small energy solutions by using the symmetric mountain pass theorem in spite of the presence of a critical nonlinear term which, of course, is super-linear.