In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, − a + b ∫ R N | ∇ u | 2 d x Δ u = α k ( x ) | u | q − 2 u + β ∫ R N | u ( y ) | 2 μ ∗ | x − y | μ d y | u | 2 μ ∗ − 2 u , x ∈ R N , $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N , N ≥ 3, α and β are positive real parameters, 2 μ ∗ = ( 2 N − μ ) / ( N − 2 ) $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ L ^r (ℝ ^N ), with r = 2 ^∗ /(2 ^∗ − q ) if 1 < q < 2 ^* and r = ∞ if q ≥ 2 ^∗ . According to the different range of q , we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

In this paper, we deal with a class of Kirchhoff-type critical elliptic equations involving the $p$ -sub-Laplacians operators on the Heisenberg group of the form $$\left\{\begin{array}{c}M(\parallel D_{H}u{\parallel }_p^p+\parallel u{\parallel }_{p,V}^p)[-{\mathrm{\Delta }}_{H,p}u+V(\xi )|u{|}^{p-2}u]=\lambda f(\xi ,u)+|u{|}^{p^{\ast }-2}u,\xi \in {ℍ}^n,\hfill \\ u\in {\mathrm{\text{HW}}}_V^{1,p}({ℍ}^n),\hfill \end{array}\right.$$ where ${\mathrm{\Delta }}_{H,p}u:={\text{div}}_H(|D_{H}u{|}_H^{p-2}{D}_{H}u)$ is the $p$ -sub-Laplacian, $1<p<Q$ , ${\mathrm{\text{HW}}}_V^{1,p}({ℍ}^n)$ is the horizontal Sobolev space on ${ℍ}^n$ . And $Q=2n+2$ is the homogeneous dimension of ${ℍ}^n$ , $\lambda$ is a real parameter, $p^{\ast }=Qp/(Q-p)$ is the critical Sobolev exponent on the Heisenberg group. Under some proper assumptions on the Kirchhoff function $M$ , the potential function $V$ and $f$ , together with the mountain pass theorem and the concentration-compactness principles for classical Sobolev spaces on the Heisenberg group, we prove the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases on the Heisenberg group. The results of this paper extend or complete recent papers and are new in several directions for the non-degenerate and degenerate critical Kirchhoff equations involving the $p$ -Laplacian type operators on the Heisenberg group.

In this paper, we are concerned with the existence and multiplicity of solutions for the fractional Choquard‐type Schrödinger‐Kirchhoff equations with electromagnetic fields and critical nonlinearity: where is the fractional magnetic operator with 0<s0 is a positive parameter. The electric potential satisfies V(x)=0 in some region of , which means that this is the critical frequency case. We first prove the (PS)_c condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term M can vanish at zero.

In this paper, we deal with the following critical Choquard-Kirchhoff problem on the Heisenberg group of the form: M ( ‖ u ‖ 2 ) ( − Δ H u + V ( ξ ) u ) = ∫ H N ∣ u ( η ) ∣ Q λ ∗ ∣ η − 1 ξ ∣ λ d η ∣ u ∣ Q λ ∗ − 2 u + μ f ( ξ , u ) , where M is the Kirchhoff function, Δ H is the Kohn Laplacian on the Heisenberg group H N , f is a Carathéodory function, μ > 0 is a parameter and Q λ ∗ = 2 Q − λ Q − 2 is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We first establish a new version of the concentration-compactness principle for the Choquard equation on the Heisenberg group. Then, combining with the mountain pass theorem, we obtain the existence of nontrivial solutions to the aforementioned problem in the case of nondegenerate and degenerate cases.