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Positive solutions of Yamabe-type equations on the Heisenberg group. (English) Zbl 0948.35027
Let \({\mathbb H}^n\) be the Heisenberg group of real dimension \(2n+1.\) As it is well-known, \({\mathbb H}^n={\mathbb C}^n\times{\mathbb R}\) is a manifold, and the group structure is given by \((z,t)\cdot(z',t')=(z+z',t+t'+2\operatorname {Im}\langle z,z'\rangle_{{\mathbb C}^n})\), where \(\langle\cdot,\cdot\rangle_{{\mathbb C}^n}\) denotes the usual Hermitian structure on \({\mathbb C}^n.\) Let us consider the Kohn-Spencer Laplacian defined by \(\Delta_{{\mathbb H}^n}:=\sum_{j=1}^n(X_j^2+Y_j^2),\) where \[ X_j=2 \operatorname {Re} {{\partial}\over{\partial z_j}}+2 \operatorname {Im}z_j{{\partial}\over{\partial t}},\quad Y_j=2 \operatorname {Im} {{\partial}\over{\partial z_j}}-2 \operatorname {Re}z_j{{\partial}\over{\partial t}}, j=1,\ldots,n, \] that, along with \(\partial/\partial t,\) span the Lie algebra of left-invariant vector fields on \({\mathbb H}^n.\) In this paper, the authors consider the equation \[ \Delta_{{\mathbb H}^n}u+a(z,t)u-b(z,t)|u|^{\sigma -1}u=0, \] where \(\sigma >1.\) Their aim is to give conditions on the coefficients \(a(z,t)\) and \(b(z,t)\) in order for the above equation to have, or not to have, a positive solution on \({\mathbb H}^n.\) They recall how the above equation is related to the classical one relative to the CR-Yamabe problem, pointing out that one of the difficulties in the case they treat is due to the non-compactness of the manifold they work on. They prove a number of results about existence and non-existence of a positive solution \(u\) on \({\mathbb H}^n\) based on growth assumptions of the coefficients \(a\) and \(b\) of the kind \[ \psi(z,t)a_1(d(z,t))\leq a(z,t)\qquad \text{(and/or)}\qquad a(z,t)\leq\psi(z,t)a_2(d(z,t)), \] \[ \psi(z,t)b_1(d(z,t))\leq b(z,t)\qquad \text{(and/or)}\qquad b(z,t)\leq\psi(z,t)b_2(d(z,t)), \] where \(d(z,t)=(|z|^4+t^2)^{1/4},\) \(\psi(z,t):=|z|^2/d(z,t)^2\) (\((z,t)\not=(0,0)\)), and where \(a_i,b_i\) are suitable nonnegative continuous functions on \([0,+\infty).\)

35H10 Hypoelliptic equations
22E25 Nilpotent and solvable Lie groups
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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[1] P. Aviles and R. McOwen, Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds , J. Differential Geom. 27 (1988), no. 2, 225-239. · Zbl 0648.53021
[2] B. Bianchini and M. Rigoli, Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifolds , Trans. Amer. Math. Soc., to appear. JSTOR: · Zbl 0892.53019 · doi:10.1090/S0002-9947-97-01810-2 · links.jstor.org
[3] Jean-Michel Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés , Ann. Inst. Fourier (Grenoble) 19 (1969), no. fasc. 1, 277-304 xii. · Zbl 0176.09703 · doi:10.5802/aif.319 · numdam:AIF_1969__19_1_277_0 · eudml:73982
[4] K. S. Cheng and J. T. Lin, On the elliptic equations \(\Delta u=K(x)u^ \sigma\) and \(\Delta u=K(x)e^ 2u\) , Trans. Amer. Math. Soc. 304 (1987), no. 2, 639-668. JSTOR: · Zbl 0635.35027 · doi:10.2307/2000734 · links.jstor.org
[5] K. S. Cheng and W. M. Ni, On the structure of the conformal scalar curvature equation on \(\mathbf R^ n\) , Indiana Univ. Math. J. 41 (1992), no. 1, 261-278. · Zbl 0764.35037 · doi:10.1512/iumj.1992.41.41015
[6] D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in \(3\)-manifolds of nonnegative scalar curvature , Comm. Pure Appl. Math. 33 (1980), no. 2, 199-211. · Zbl 0439.53060 · doi:10.1002/cpa.3160330206
[7] G. B. Folland and E. M. Stein, Estimates for the \(\bar \partial \sbb\) complex and analysis on the Heisenberg group , Comm. Pure Appl. Math. 27 (1974), 429-522. · Zbl 0293.35012 · doi:10.1002/cpa.3160270403
[8] N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation , Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 313-356. · Zbl 0694.22003 · doi:10.5802/aif.1215 · numdam:AIF_1990__40_2_313_0 · eudml:74879
[9] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents , Acta Math. 139 (1977), no. 1-2, 95-153. · Zbl 0366.22010 · doi:10.1007/BF02392235
[10] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[11] L. Hörmander, Hypoelliptic second order differential equations , Acta Math. 119 (1967), 147-171. · Zbl 0156.10701 · doi:10.1007/BF02392081
[12] D. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. II , J. Funct. Anal. 43 (1981), no. 2, 224-257. · Zbl 0493.58022 · doi:10.1016/0022-1236(81)90031-8
[13] D. Jerison and J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds , Microlocal analysis (Boulder, Colo., 1983), Contemp. Math., vol. 27, Amer. Math. Soc., Providence, RI, 1984, pp. 57-63. · Zbl 0577.53035
[14] D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds , J. Differential Geom. 25 (1987), no. 2, 167-197. · Zbl 0661.32026
[15] D. Jerison and J. M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem , J. Amer. Math. Soc. 1 (1988), no. 1, 1-13. · Zbl 0634.32016 · doi:10.2307/1990964 · numdam:SEDP_1983-1984____A18_0 · eudml:111849
[16] M. Naito, A note on bounded positive entire solutions of semilinear elliptic equations , Hiroshima Math. J. 14 (1984), no. 1, 211-214. · Zbl 0555.35044
[17] W. M. Ni, On the elliptic equation \(\Delta u+K(x)u^(n+2)/(n-2)=0\), its generalizations, and applications in geometry , Indiana Univ. Math. J. 31 (1982), no. 4, 493-529. · Zbl 0496.35036 · doi:10.1512/iumj.1982.31.31040
[18] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations , Prentice-Hall Inc., Englewood Cliffs, N.J., 1967. · Zbl 0153.13602
[19] A. Ratto, M. Rigoli, and L. Veron, Courbure scalaire et déformations conformes des variétés riemanniennes non compactes , C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 7, 665-670. · Zbl 0798.53042
[20] D. Sattinger, Topics in stability and bifurcation theory , Lecture Notes in Math., Springer-Verlag, Berlin, 1973. · Zbl 0248.35003
[21] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus , Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189-258. · Zbl 0151.15401 · doi:10.5802/aif.204 · numdam:AIF_1965__15_1_189_0 · eudml:73861
[22] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals , Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. · Zbl 0821.42001
[23] C. A. Swanson, Comparison and oscillation theory of linear differential equations , Mathematics in Science and Engineering, vol. 48, Academic Press, New York, 1968. · Zbl 0191.09904
[24] G. N. Watson, A Treatise on the Theory of Bessel Functions , Cambridge University Press, Cambridge, England, 1944. · Zbl 0063.08184
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