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Sign changing solutions of a semilinear equation on Heisenberg group, (with J. Chen) accepted in J. Non. and Convex Anal. (2015).


ABSTRACT: This paper is concerned with the existence of multiple solutions to the semilinear equation

    \[\Delta_H\, u + |u|^{\frac{4}{Q-2}}u+\mu|\xi|^\alpha_H u=0\]

in a bounded domain of the Heisenberg group \mathbb{H}^N with Dirichlet boundary condition, where \alpha > 0 and |\xi|_H is a distance in \mathbb{H}^N. By using variational methods, we prove that this problem possesses at least one positive solution and one sign changing solution for some values of \alpha and \mu.


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Extra info: (P090/2015-06)

REF. DATE: 18 June 2015 CREATED/MODIFIED: 18 January 2016 VIEW POST 24 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].


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Extra info: (P080/2013) (free/open access)

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

A class of sub-elliptic equations on the Heisenberg group and related interpolation inequalities (with J. Chen) in Advances in Harmonic Analysis and Operator Theory, Operator Theory: Advances and Applications 229, 123–137, (2013).


ABSTRACT: We firstly prove the existence of least energy solutions to a class of sub-elliptic equations on the Heisenberg group \mathbb{H}^N of the form

    \[-\Delta_H u + u = |u|^{p-2}u\quad u\in S^{1,2}_0(\mathbb{H}^N),\]

where S^{1,2}_0(\mathbb{H}^N) is the closure of C^{\infty}_0(\mathbb{H}^N) under the norm \|u\|=(\int_{\mathbb{H}^N}|\nabla_H u|^2+|u|^2)^{\frac{1}{2}}. Then we use this least energy solution to give a sharp estimate to the smallest positive constant in the Gagliardo-Nirenberg inequality on the Heisenberg group, 2 < p < 2+\frac{2}{N},

    \[\int_{\mathbb{H}^N}|u|^p\,d\xi \leq C\left(\int_{\mathbb{H}^N}|\nabla_H u|^2\,d\xi\right)^{\frac{Q(p-2)}{4}}\left(\int_{\mathbb{H}^N}|u|^2\,d\xi\right)^{\frac{2p-Q(p-2)}{4}}.\]

We also point out some extensions to the quasilinear sub-elliptic case.


Download: Preprint version (RIA |  download) / Publisher version (doi:10.1007/978-3-0348-0516-2)
Extra info: (P085/2013-02-05)

REF. DATE: 5 February 2013 CREATED/MODIFIED: 15 January 2016 VIEW POST 199 views No Comment