On the elliptic equation $Lu - k + K \exp [2u] = 0$

Kenig, Carlos E.; Ni, Wei-Ming

On the elliptic equation

L u - k + K exp [2 u] = 0

Kenig, Carlos E.; Ni, Wei-Ming

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 12 (1985) no. 2, p. 191-224

MR 829052 | Zbl 0593.35044

@article{ASNSP_1985_4_12_2_191_0,
     author = {Kenig, Carlos and Ni, Wei-Ming},
     title = {On the elliptic equation $Lu - k + K \exp [2u] = 0$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 12},
     number = {2},
     year = {1985},
     pages = {191-224},
     zbl = {0593.35044},
     mrnumber = {829052},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_1985_4_12_2_191_0}
}

Kenig, Carlos E.; Ni, Wei-Ming. On the elliptic equation $Lu - k + K \exp [2u] = 0$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 12 (1985) no. 2, pp. 191-224. http://www.numdam.org/item/ASNSP_1985_4_12_2_191_0/

References

[A] L.V. Ahlfors, An extension of Schwarz's Lemma, Trans. Amer. Math. Soc., 43 (1938), pp. 359-364. | JFM 64.0315.04 | MR 1501949 | Zbl 0018.41002

[AB] L.V. Ahlfors - L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math., 72 (1960), pp. 385-404. | MR 115006 | Zbl 0104.29902

[Av] P. Aviles, Prescribing conformal complete metrics with given positive Gaussian curvature in R2, preprint.

[FJK] E.B. Fabes - D.S. Jerison - C.E. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble), 32 (1982), pp. 151-182. | Numdam | MR 688024 | Zbl 0488.35034

[FKS] E.B. Fabes - C.E. Kenig - R.P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Communications in Partial Differential Equations, 7 (1982), pp. 77-116. | MR 643158 | Zbl 0498.35042

[F] A. Friedman, Bounded entire solutions of elliptic equations, Pacific J. Math., 44, (1973), pp. 497-507. | MR 320506 | Zbl 0256.35024

[Kp] L. Karp, The exterior Dirichlet problem for nonuniformly elliptic equations and applications, Workshop on Nonlinear Elliptic P.D.E., Jan. 1983 MSRI, Berkeley.

[K] J. Kazdan, Gaussian and scalar curvature, an update, Ann. of Math. Studies, 102 (S.-T. Yau, Ed.), pp. 185-192. | Zbl 0481.53035

[KW] J. Kazdan - F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math., 99 (1974), pp. 14-47. | MR 343205 | Zbl 0273.53034

[Kr] J.B. Keller, On solutions of Δu = f(u), Comm. Pure Appl. Math., 10 (1957), pp. 503-510. | Zbl 0090.31801

[KN1] C.E. Kenig - W.-M. NI, An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math., 106 (1984), pp. 689-702. | MR 745147 | Zbl 0559.35025

[KN2] C.E. Kenig - W.-M. NI, On the elliptic equation Lu - k + K exp [2u] = 0 and conformal metrics with prescribed curvatures, Abstracts Amer. Math. Soc., 3, no. 5 (Aug. 1982), 796-35-81, p. 343.

[KN3] C.E. Kenig - W.-M. Ni, Conformal metrics with prescribed curvatures, Abstracts Amer. Math. Soc., 4, no. 3 (April 1983), 803-35-84, p. 254.

[LSW] W. Littman - G. Stampacchia - H. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa, Ser. III, 17 (1963), pp. 45-79. | Numdam | MR 161019 | Zbl 0116.30302

[M1] R. Mcowen, On the elliptic equation Δu + K exp [2u] = f and prescribed negative curvature in R2, preprint.

[M 2] R. Mcowen, Conformal metrics in R2 with prescribed Gaussian and positive total curvature, preprint. | Zbl 0546.53032

[N1] W.-M. Ni, On the elliptic equation Δu + K(x) exp [2u] = 0 and conformal metrics with prescribed Gaussian curvatures, Invent. Math., 66 (1982), pp. 343-352. | Zbl 0487.35042

[N2] W.-M. Ni, On the elliptic equation Δu + K(x)u (n+2)/(n-2) = 0, its generalization, and applications in geometry, Indiana Univ. Math. J., 31 (1982), pp. 493-529. | Zbl 0496.35036

[O] O.A. Oleinik, On the equation Δu + k(x) exp [u] = 0, Russian Math. Surveys, 33 (1978), pp. 243-244. | Zbl 0401.35051

[Os] R. Osserman, On the inequality Δu ≽ f(u), Pacific J. Math., 7 (1957), pp. 1641-1647. | Zbl 0083.09402

[S] D.H. Sattinger, Conformal metrics in R2 with prescribed curvature, Indiana Univ. Math. J., 22 (1972), pp. 1-4. | MR 305307 | Zbl 0236.53009

[T] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup., Ser. IV, 3 (1976), pp. 697-718. | Numdam | MR 601601 | Zbl 0341.35031

[W] H. Wittich, Ganze Lösungen der Differentialgleichung Δu = exp [u], Math. Z., 49 (1944), pp. 579-582. | Zbl 0028.41001