Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domains


Authors: Melvyn S. Berger and Martin Schechter
Journal: Trans. Amer. Math. Soc. 172 (1972), 261-278
MSC: Primary 46E35; Secondary 35J65, 35L60, 49F99
DOI: https://doi.org/10.1090/S0002-9947-1972-0312241-X
MathSciNet review: 0312241
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Sobolev-Kondrachov embedding and compactness theorems are extended to cover general unbounded domains, by introducing appropriate weighted $ {L_p}$ norms. These results are then applied to the Dirichlet problem for quasi-linear elliptic partial differential equations and isoperimetric variational problems defined on general unbounded domains in $ {{\mathbf{R}}^N}$.


References [Enhancements On Off] (What's this?)

  • 1. M. S. Berger and M. Schechter, $ {L_p}$ embedding and nonlinear eigenvalue problems for unbounded domains, Bull. Amer. Math. Soc. 76 (1970), 1299-1302. MR 42 #2294. MR 0267392 (42:2294)
  • [1] A. M. Molcanov, On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order, Trudy Moskov. Mat. Obšč. 2 (1953), 169-199. (Russian) MR 15, 224. MR 0057422 (15:224g)
  • [2] M. S. Birman, On the spectrum of singular boundary-value problems, Mat. Sb. 55 (97) (1961), 125-174; English transl., Amer. Math. Soc. Transl. (2) 53 (1966), 23-80. MR 26 #463. MR 0142896 (26:463)
  • [3] M. Schechter, On the essential spectrum of an elliptic operator perturbed by a potential, J. Analyse Math. 22 (1969), 87-115. MR 40 #579. MR 0247311 (40:579)
  • [4] V. P. Gluško and S. G. Kreĭn, Inequalities for the norms of derivatives in weighted $ {L_p}$ spaces, Sibirsk. Mat. Ž. 1 (1960), 343-382; English transl., Amer. Math. Soc. Transl. (2) 85 (1969), 1-50. MR #A3507. MR 0133681 (24:A3507)
  • [5] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod; Gauthier-Villars, Paris, 1969. MR 41 #4326. MR 0259693 (41:4326)
  • [6] B. R. Vaĭnberg, On elliptic problems in unbounded domains, Mat. Sb. 75 (117) (1968), 454-480 = Math. USSR Sb. 4 (1968), 419-444. MR 37 #6601. MR 0231044 (37:6601)
  • [7] D. M. Èĭdus, The principle of limiting amplitude, Uspehi Mat. Nauk 24 (1969), no. 3 (147), 91-156 = Russian Math. Surveys 24 (1969), no. 3, 97-168. MR 0601072 (58:29156)
  • [8] L. Bers, F. John and M. Schechter, Partial differential equations, Lectures in Appl. Math., vol. 3, Interscience, New York, 1964. MR 29 #346. MR 598466 (82c:35001)
  • [9] M. Vaĭnberg, Variational methods for the study of non-linear operators, GITTL, Moscow, 1956; English transl., Holden-Day, San Francisco, Calif., 1964. MR 19, 567; MR 31 #638.
  • [10] M. S. Berger and M. Berger, Perspectives in nonlinearity. An introduction to nonlinear analysis, Benjamin, New York, 1968. MR 40 #4971. MR 0251744 (40:4971)
  • [11] J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach, New York, 1969. MR 0433481 (55:6457)
  • [12] R. Palais, Ljusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115-132. MR 41 #4584. MR 0259955 (41:4584)
  • [13] M. S. Berger, An eigenvalue problem for nonlinear elliptic partial differential equations, Trans. Amer. Math. Soc. 120 (1965), 145-184. MR 31 #6047. MR 0181821 (31:6047)
  • [14] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 31 #426. MR 0176151 (31:426)
  • [15] N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, Ann. Inst. Fourier (Grenoble) 11 (1961), 385-475. MR 26 #1485. MR 0143935 (26:1485)
  • [16] M. Schechter, Spectra of partial differential operators, North-Holland, Amsterdam, 1971. MR 869254 (88h:35085)
  • [17] J. L. Lions and E. Magenes, Problemi ai limiti non omogenei. III, Ann. Scuola Norm. Sup. Pisa (3) 15 (1961), 41-103. MR 26 #4048. MR 0146526 (26:4048)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46E35, 35J65, 35L60, 49F99

Retrieve articles in all journals with MSC: 46E35, 35J65, 35L60, 49F99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0312241-X
Keywords: $ {L_p}$ embedding, nonlinear eigenvalue problems, isoperimetric problems, critical point theory, quasi-linear elliptic boundary value problems
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society