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

TL;DR

This paper proves the existence of multiple solutions for a nonlocal fractional p-Kirchhoff problem with singularity and critical exponent, using variational methods and a cut-off technique.

Contribution

It introduces a novel approach to establish multiple solutions for a fractional p-Kirchhoff problem with critical nonlinearities.

Findings

Existence of at least k solutions for the problem.

Uniform boundedness of solutions in L-infinity norm.

Application of symmetric mountain pass theorem in a critical setting.

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.