High energy solutions for $p$-Kirchhoff elliptic problems with Hardy-Littlewood-Sobolev nonlinearity

This article deals with the study of the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left(, \int\limits_{$\mathbb{R}$^N}|\nabla u|^p\right) (-Δ_p) u + V(x)|u|^{p-2}u = \left(, \int\limits_{$\mathbb{R}$^N}\frac{F(u)(y)}{|x-y|^μ},dy \right) f(u), ;;$\text{in}$ ; $\mathbb{R}$^N, u > 0, ;; $\text{in}$ ; $\mathbb{R}$^N, \end{array} \end{equation*} where $M$ models Kirchhoff-type nonlinear term of the form $M(t)=a+b{t}^{\theta -1}$, where $a,b>0$ are given constants; $1<p<N$, ${\mathrm{\Delta }}_p=\text{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)$ is the $p$-Laplacian operator; potential $V\in C^2({\mathbb{R}}^N)$; $f$ is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for $\theta \in [1,\frac{2N-\mu }{N-p})$ via the Pohožaev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem.