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.

This paper is concerned with the existence of multiple normalized solutions for a class of Choquard equation involving the biharmonic operator and competing potentials in [ ]: [ ] where [ ] is the biharmonic operator, [ ], [ ], [ ] is an unknown parameter that appears as the Lagrange multiplier, the absorption potential V, reaction potential G and nonlinear term f are continuous functions and satisfy some assumptions. With the help of the minimization techniques and Lusternik–Schnirelmann category, we show that the number of normalized solutions is not less than the number of global minimum points of V when the parameter [ ] is sufficiently small. Furthermore, and more interestingly, we can prove the existence of multiple solutions for this problem by using the Morse theory. As far as we know, this study seems to be the first contribution regarding the concentration behavior for Choquard equation involving the biharmonic operator.

In this paper, we consider the following critical Schrödinger equation involving \((2,q)\)-Laplacian: \[\begin{cases} -\Delta u-\Delta_{q} u=\lambda u+\mu |u|^{\gamma-2}u+|u|^{2^*-2}u \quad\text{in }\mathbb{R}^N, \\ \int_{\mathbb{R}^N} |u|^{2}dx=a^2,\end{cases}\] where \(\Delta_q u =\operatorname{div} (|\nabla u|^{q-2}\nabla u)\) is the \(q\)-Laplacian operator, \(\mu, a\gt 0,\) \(\lambda\in\mathbb{R}\), \(\gamma\in(2,2^*)\), \(q\in(\frac{2N}{N+2},2)\) and \(N\geq3\). The meaningful and interesting phenomenon is the simultaneous occurrence of \((2,q)\)-Laplacian and critical nonlinearity in the above equation. In order to obtain existence of multiple normalized solutions for such equation, we need to make a detailed estimate. More precisely, for the \(L^2\)-subcritical case, we use the truncation technique, concentration-compactness principle and the genus theory to get the existence of multiple normalized solutions. For the \(L^2\)-supercritical case, we obtain a couple of normalized solution for the above equation by a fiber map and concentration-compactness principle.

In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: − a − b ∫ Ω | ∇ H u | 2 d ξ Δ H u + μ ϕ u = λ | u | q − 2 u + | u | 2 u , in Ω , − Δ H ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega,\\ u=\phi=0,\quad &\mbox{on}\, \partial\Omega, \end{cases} \end{equation*}$$ where Δ _H is the Kohn-Laplacian on the first Heisenberg group H 1 $ \mathbb{H}^1 $ , and Ω ⊂ H 1 $ \Omega\subset \mathbb{H}^1 $ is a smooth bounded domain, a , b > 0, 1 < q < 2 or 2 < q < 4, λ > 0 and μ ∈ R $ \mu\in \mathbb{R} $ are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient ( a − b ∫ _Ω ∣∇ _H u ∣ ² dx ) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.