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Some existence results for critical nonlocal Choquard equation on the Heisenberg group. (English) Zbl 1526.35165

Summary: This paper deals with the following critical nonlocal Choquard equation on the Heisenberg group: \[ \begin{cases} -\left(a-b\int_\Omega|\nabla_Hu|^2d\xi\right)\Delta_Hu=\mu|u|^{q-2}u+\int_\Omega\frac{|u(\eta)|^{Q_\lambda^\ast}}{|\eta^{-1}\xi|^\lambda}d\eta|u|^{Q_\lambda^\ast-2}u & \text{ in }\Omega, \\ u=0 & \text{ on }\partial\Omega, \end{cases} \] where \(\Omega\subset\mathbb{H}^N\) is a smooth bounded domain, \(\Delta_H\) is the Kohn-Laplacian on the Heisenberg group \(\mathbb{H}^N\), \(1<q<2\) or \(2<q<Q_\lambda^\ast\), \(a,b>0\), \(\mu>0\), \(0<\lambda<4\), and \(Q_\lambda^\ast=\frac{2Q-\lambda}{Q-2}\) is the critical exponent. Existence results are obtained by using the Ekeland variational principle, Clark critical point theorem, mountain pass theorem, and Krasnoselskii genus theorem, respectively. Due to critical nonlinearities as well as the presence of the double non-local teams, there are some difficulties on the Heisenberg group’s framework. Our results are new even in the Euclidean case.

MSC:

35J61 Semilinear elliptic equations
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
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