Existence of solution of sub-elliptic equations on the Heisenberg group with critical growth and double singularities (Q2871591)

This paper deals with a class of sub-elliptic equations on the Heisenberg group \(H^N\) having a nonlinearity with critical nonlinear growth \(p_*(s)\) and Hardy type singularities \({|z|^p\over\rho^{2p}}\), \({|z|^s\over\rho^{2s}}\), where \(1<p< 2N+2\), \(0\leq s<p\), \(\xi= (x,y,t)\in H^N\), \(z=(x,y)\in \mathbb{R}^{2N}\) and \(\rho\) is the distance of \(\xi\in H^N\) to the origin.NEWLINENEWLINE In the main result, Theorem 1.2, the authors prove the existence of least energy solution by developing new techniques based on Nehari constraint. In fact, the embeddings NEWLINE\[NEWLINED^{1,p}_0(\Omega)\hookrightarrow L^p\Biggl(\Omega, {|z|^p\over \rho^{2p}}\,d\xi\Biggr),\;D^{1,p}_0(\Omega)\hookrightarrow L^{p_*(s)}\Biggl(\Omega, {|z|^s\over \rho^{2s}}\,d\xi\Biggr)NEWLINE\]NEWLINE are not compact, and standard variational arguments cannot be used directly. Theorem 1.2 generalizes some previous results of \textit{Y. Han} and \textit{P. Niu} [Manuscr. Math. 118, No. 2, 235--252 (2005; Zbl 1078.22004)].