Ground State Solutions of $p$-Laplacian singular Kirchhoff Problem involving a Riemann-Liouville fractional derivative

Abstract

The purpose of this paper is to study the existence and multiplicity of solutions to the following Kirchhoff equation with singular nonlinearity and Riemann-Liouville FractionalDerivative:\begin{eqnarray*}{\rm (P_\lambda)}\left\{\begin{array}{ll}& \left(a+b\int_{0}^{T} |_{0}D_{t}^{\alpha}(u(t))|^{p}dt\right)^{p-1}\,_{t}D_{T}^{\alpha}\left(\Phi_{p}(_{0}D_{t}^{\alpha}u(t))\right)\\ &=\frac{\lambda g(t)}{u^{\gamma}(t)}+ f(t,u(t)),\;t\in (0,T);\\ \\ & u(0)=u(T)=0,\end{array}\right. \end{eqnarray*}where $a\geq1,\;b,\;\lambda>0$, $p>1$ are constants, $\frac{1}{p}<\alpha\leq 1,$ $0<\gamma<1,$ $g\in C([0,1])$ and $f\in C^{1}([0,T]\times \mathbb{R},\mathbb{R})$. Under appropriate assumptionson the function $f,$ we employ variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $\lambda$.