Existence And Multiplicity Of Solutions For Two Classes Of Critical Equations On The Heisenberg Group

In recent years,more and more attention has been paid to the study of Heisenberg group.In this paper,we mainly study the existence and multiplicity of solutions for two classes of critical equations on the Heisenberg group.The paper is structured as follows:In Chapter 1,we briefly summarize the research background,research status,research problems and preparatory knowledge.In Chapter 2,we are interested in Q-Laplacian problem with exponential nonlinear terms on the Heisenberg group:(?)where ΔQ(·):=divHn(| ▽Hn(·)|HnQ-2▽Hn(·))is the Q-Laplacian operator,Ω is an open,smooth bounded domain on the Heisenberg group Hn.By using constrained variational method and quantitative deformation lemma,we obtain the existence and multiplicity of sign-changing solutions for the above problem in the subcritical and critical cases.The results extend the results for the signchanging solutions of the N-Laplacian problem[72]from Euclidean space to the Heisenberg group.In Chapter 3,we are devoted to the critical Choquard-Kichhoff problem on the Heisenberg group:M(‖u‖2)(-ΔHu+V(ξ)u)=(∫Hn|η-1ξ|λ/|u(η)|Qλ*dη)|u|Qλ*-2u+μf(ξ,u),where ΔH is the Kohn Laplacian on the Heisenberg group Hn,Qλ*=2Q-λ/Q-2is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.We overcome the difficulty of lack of compactness due to the appearance of critical terms by establishing the concentrated compactness principle for the Choquard equation on the Heisenberg group.Then,together with the Mountain Pass Theorem,the existence and multiplicity of nontrivial solutions to this problem are proved in degenerate and non-degenerate cases.The results are generalization of the concentrated compactness principle for the Choquard equation[28]from Euclidean spaces to the Heisenberg group.And the results are new in Euclidean spaces.