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PAPER: Existence of solutions of sub-elliptic equations on the Heisenberg group with critical growth and singularity

REF. DATE:12 February 2013 CREATED/MODIFIED: 15 January 2016 246 views No Comment

Existence of solutions of sub-elliptic equations on the Heisenberg group with critical growth and singularity (with J. Chen), Opuscula Math. 33:2(2013), 237-254.


ABSTRACT: We consider a class of sub-elliptic equations on the Heisenberg group \mathbb{H}^N with a Hardy type singularity and a critical nonlinear growth

    \[-\Delta_{H,p} u -\lambda\frac{|z|^p}{\rho^{2p}}|u|^{p-2}u = \frac{|z|^s}{\rho^{2s}}|u|^{p_\ast(s)-2}u\quad\mbox{ in }\mathbb{H}^N\backslash\{0\},\]

where \Delta_{H,p} is the p-sub-Laplacian with respect to a fixed bases for the corresponding Lie algebra of the Heisenberg group, (z,t)=(x,y,t)\in\mathbb{H}^N, \rho(\xi) is the distance of \xi\in\mathbb{H}^N to the origin given by the homogeneous distance

    \[d(\xi,\xi')=\left((|x-x'|^2+|y-y'|^2)^2+(t-t'-2(x,y')_{\mathbb{R}}-2(x',y)_{\mathbb{R}})^2\right)^{\frac{1}{4}},\]

Q=2N+2 is the homogeneous dimension, \mathcal{D}^{1,p}_0\left(\mathbb{H}^N\right) denotes the closure of C^\infty_0\left(\mathbb{H}^N\right) under the norm \|u\|=(\int_{\mathbb{H}^N}|\nabla_H u|^p\,d\xi)^{\frac{1}{p}} and p_\ast(s)=p(Q-s)(Q-p)^{-1} the critical exponent of embedding

    \[\mathcal{D}^{1,p}_0\left(\mathbb{H}^N\right) \hookrightarrow L^{p_\ast(s)}\left(\mathbb{H}^N\right, \frac{|z|^s}{\rho^{2s}}d\xi).\]

We prove the existence of energy solutions by developing new techniques based on the Nehari constraint. This result extends previous works, e.g., by Han et al. [Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math. 118 (2005), 235–252].


Download: Preprint version (RIA |  download) / Publisher version (doi:10.7494/OpMath.2013.33.2.237)
Extra info: (P080/2013) (free/open access)

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