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: $$\begin{aligned} \begin{array}{cc} \displaystyle M\left( \, \int \limits _{{\mathbb {R}}^N}|\nabla u|^p\right) (-\Delta _p) u + V(x)|u|^{p-2}u = \left( \, \int \limits _{{\mathbb {R}}^N}\frac{F(u)(y)}{|x-y|^{\mu }}\,dy \right) f(u), \;\;\text {in} \; {\mathbb {R}}^N,\\ u > 0, \;\; \text {in} \; {\mathbb {R}}^N, \end{array} \end{aligned}$$ M ∫ R N | ∇ u | p ( - Δ p ) u + V ( x ) | u | p - 2 u = ∫ R N F ( u ) ( y ) | x - y | μ d y f ( u ) , in R N , u > 0 , in R N , where M models Kirchhoff-type nonlinear term of the form $$M(t) = a + bt^{\theta -1}$$ M ( t ) = a + b t θ - 1 , where $$a, b > 0$$ a , b > 0 are given constants; $$1<p<N$$ 1 < p < N , $$\Delta _p = \text {div}(|\nabla u|^{p-2}\nabla u)$$ Δ p = div ( | ∇ u | p - 2 ∇ u ) is the p-Laplacian operator; potential $$V \in C^2({\mathbb {R}}^N)$$ V ∈ C 2 ( R N ) ; f is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for $$\theta \in \left[ 1, \frac{2N-\mu }{N-p}\right) $$ θ ∈ 1 , 2 N - μ 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.