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