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

Abstract

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) (-\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{equation*} where $M$ models Kirchhoff-type nonlinear term of the form $M(t) = a + bt^{\theta-1}$, where $a, b > 0$ are given constants; $1<p<N$, $\Delta_p = \text{div}(|\nabla u|^{p-2}\nabla u)$ 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 \left[1, \frac{2N-\mu}{N-p}\right) $ via the Poho\v{z}aev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem.