Asymptotic behavior and non-existence theorems for semilinear Dirichlet problems involving critical exponent on unbounded domains of the Heisenberg group

Lanconelli, E.; Uguzzoni, F.

Lanconelli, E. ; Uguzzoni, F.

Bollettino dell'Unione Matematica Italiana, Tome 1-A (1998), p. 139-168 / Harvested from Biblioteca Digitale Italiana di Matematica

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Résumé

In questa nota dimostriamo stime asintotiche ottimali per le soluzioni deboli non negative del problema al contorno $- Δ_{H^{n}} u = u^{(Q + 2) / (Q - 2)} i n Ω, u = 0 in \partial Ω .$ $- Δ_{H^{n}}$ è il Laplaciano di Kohn sul gruppo di Heisenberg $H^{n}$ , $Ω$ è un aperto non limitato e $Q = 2 n + 2$ è la dimensione omogenea di $H^{n}$ . Utilizziamo successivamente le stime ottenute per dimostrare un teorema di non esistenza per (*) nel caso in cui $Ω$ sia un semispazio di $H^{n}$ con bordo parallelo al centro del gruppo.

Publié le : 1998-02-01

MR 1618972 | Zbl 0902.22006

@article{BUMI_1998_8_1B_1_139_0,
     author = {E. Lanconelli and F. Uguzzoni},
     title = {Asymptotic behavior and non-existence theorems for semilinear Dirichlet problems involving critical exponent on unbounded domains of the Heisenberg group},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {1-A},
     year = {1998},
     pages = {139-168},
     zbl = {0902.22006},
     mrnumber = {1618972},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_1998_8_1B_1_139_0}
}

Lanconelli, E.; Uguzzoni, F. Asymptotic behavior and non-existence theorems for semilinear Dirichlet problems involving critical exponent on unbounded domains of the Heisenberg group. Bollettino dell'Unione Matematica Italiana, Tome 1-A (1998) pp. 139-168. http://gdmltest.u-ga.fr/item/BUMI_1998_8_1B_1_139_0/

Bibliographie

[BC] Bahri, A.-Coron, J. M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. | MR 929280 | Zbl 0649.35033

[BeC] Benci, V.-Cerami, G., Existence of positive solutions of the equation $- Δ u + a (x) u = u^{(N + 2) / (N - 2)}$ in $R^{N}$ , J. Funct. Anal., 88 (1990), 90-117. | MR 1033915 | Zbl 0705.35042

[BCC] Birindelli, I.-Capuzzo Dolcetta, I.-Cutrì, A., Liouville theorems for semilinear equations on the Heisenberg group, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear. | MR 1450950 | Zbl 0876.35033

[BK] Brezis, H.-Kato, T., Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151. | MR 539217 | Zbl 0408.35025

[BN] Brezis, H.-Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. | MR 709644 | Zbl 0541.35029

[B] Burger, N., Espace des fonctions à variation moyenne bornée sur un espace de nature homogène, C. R. Acad. Sci. Paris, Serie A, 286 (1978), 139-142. | MR 467176 | Zbl 0368.46037

[C] Citti, G., Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian, Ann. Mat. Pura Appl., 169 (1995), 375-392. | MR 1378482 | Zbl 0848.35040

[CGL] Citti, G.-Garofalo, N.-Lanconelli, E., Harnack's inequality for sum of squares of vector fields plus a potential, Amer. J. Math., 115 (1993), 699-734. | MR 1221840 | Zbl 0795.35018

[Cy] Cygan, J., Wiener's test for the Brownian motion on the Heisenberg group, Colloquium Math., 39 (1978), 367-373. | MR 522380 | Zbl 0409.60075

[EL] Esteban, M. J.-Lions, P.-L., Existence and non-existence results for semilinear elliptic problems in unbounded domains, Proc. Royal Soc. Edinburgh, 93A (1982), 1-14. | MR 688279 | Zbl 0506.35035

[Fe] Federer, H., Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, 153, Springer, New York (1969). | MR 257325 | Zbl 0176.00801

[F] Folland, G. B., A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc., 79 (1973), 373-376. | MR 315267 | Zbl 0256.35020

[FS] Folland, G. B.-Stein, E. M., Estimates for the ${\bar{\partial}}_{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27 (1974), 429-522. | MR 367477 | Zbl 0293.35012

[GL1] Garofalo, N.-Lanconelli, E., Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier Grenoble, 40 (1990), 313-356. | MR 1070830 | Zbl 0694.22003

[GL2] Garofalo, N.-Lanconelli, E., Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J., 41 (1992), 71-98. | MR 1160903 | Zbl 0793.35037

[GT] Gilbarg, D.-Trudinger, N. S., Elliptic partial differential equations of second order, Die Grundlehren der mathematischen Wissenschaften, 224, Springer-Verlag, New York (1977). | MR 473443 | Zbl 0361.35003

[J] Jerison, D. S., The Dirichlet problem for the Kohn Laplacian on the Heisenberg group I, J. Funct. Anal., 43 (1981), 97-142. | MR 639800 | Zbl 0493.58021

[JL1] Jerison, D.-Lee, J. M., Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom., 29 (1989), 303-343. | MR 982177 | Zbl 0671.32016

[JL2] Jerison, D.-Lee, J. M., Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc., 1 (1988), 1-13. | MR 924699 | Zbl 0634.32016

[KN] Kohn, J. J.-Nirenberg, L., Non-coercive boundary value problems, Comm. Pure Appl. Math., 18 (1965), 443-492. | MR 181815 | Zbl 0125.33302

[L] Lions, P-L., The concentration-compactness principle in the calculus of variations. The limit case, Rev. Mat. Iberoamericana, 1 (1985), n. 1, 145-201, n. 2, 45-121. | MR 850686 | Zbl 0704.49005

[T] Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., Serie IV, 110 (1976), 353-372. | MR 463908 | Zbl 0353.46018

[U] Uguzzoni, F., A Liouville-type theorem on halfspaces for the Kohn Laplacian, Proc. Amer. Math. Soc., to appear. | MR 1458268 | Zbl 0907.31006