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Transactions of the American Mathematical Society

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An eigenvalue problem for nonlinear elliptic partial differential equations


Author: Melvyn S. Berger
Journal: Trans. Amer. Math. Soc. 120 (1965), 145-184
MSC: Primary 35.47
MathSciNet review: 0181821
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  • [1] Shmuel Agmon, The 𝐿_{𝑝} approach to the Dirichlet problem. I. Regularity theorems, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 405–448. MR 0125306
  • [2] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 0125307
  • [3] S. Banach, Über homogene polynome in $ {L^2}$, Studia Math. 7 (1938), 36-44.
  • [4] G. Bratu, Sur les équations intégrales non linéaires, Bull. Soc. Math. France 42 (1914), 113–142 (French). MR 1504727
  • [5] Felix E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1960/1961), 22–130. MR 0209909
  • [6] Felix E. Browder, Functional analysis and partial differential equations. II, Math. Ann. 145 (1961/1962), 81–226. MR 0136857
  • [7] Felix E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc. 69 (1963), 862–874. MR 0156116, 10.1090/S0002-9904-1963-11068-X
  • [8] -, Variational boundary value problems for quasilinear elliptic equations. I, II, III, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 31-37, 592-598, 794-798.
  • [9] R. Courant and D. Hilbert, Methods of mathematical physics, Vols. 1, 2, New York, 1953, 1962.
  • [10] Ju. A. Dubinskiĭ, Some imbedding theorems in Orlicz classes, Dokl. Akad. Nauk SSSR 152 (1963), 529–532 (Russian). MR 0160109
  • [11] G. F. D. Duff, Modified boundary value problems for a quasi-linear elliptic equation, Canad. J. Math. 8 (1956), 203–219. MR 0078560
  • [12] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745
  • [13] L. Gårding, Dirichlet's problem for linear elliptic partial differential equations, Math. Scand. 1 (1953), 55-72.
  • [14] David Gilbarg, Some local properties of elliptic equations, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 127–141. MR 0133578
  • [15] M. Golomb, Zur theorie der nichtlinearen Integralgleichungen, Math. Z. 39 (1934), 45-75.
  • [16] A. Granas, Introduction to the toplogy of function spaces, Lecture notes, Univ. of Chicago, Chicago, Ill., 1961.
  • [17] A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math. 54 (1930), no. 1, 117–176 (German). MR 1555304, 10.1007/BF02547519
  • [18] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, London, 1934.
  • [19] A. I. Koshelev, A priori estimates in $ {L_p}$ and generalized solutions of elliptic eqations and systems, Uspehi Mat. Nauk 13 (1958), 29-88; Amer. Math. Soc. Transl. (2) 20 (1962), 105-171.
  • [20] M. A. Krasnoselskii and Y. Rutitskii, Convex functions, Noordhoff, Groningen, 1961.
  • [21] M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Matem. Nauk (N. S.) 3 (1948), no. 1(23), 3–95 (Russian). MR 0027128
  • [22] O. A. Ladyženskaya, Investigation of the Navier-Stokes equation for stationary motion of an incompressible fluid, Uspehi Mat. Nauk 14 (1959), no. 3, 75–97 (Russian). MR 0119676
  • [23] Norman Levinson, Positive eigenfunctions for Δ𝑢+𝜆𝑓(𝑢)=0, Arch. Rational Mech. Anal. 11 (1962), 258–272. MR 0145216
  • [24] J.-L. Lions, Équations différentielles opérationnelles et problèmes aux limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961 (French). MR 0153974
  • [25] L. Ljusternik, Sur une class d'équations différentielles non-lineaires, Mat.Sb. 44 (1937), 1143-1168.
  • [26] -, On a class of non-linear operators in Hilbert space, Izv. Akad. Nauk. SSSR (1939), 257-264.
  • [27] Zeev Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math. 105 (1961), 141–175. MR 0123775
  • [28] Louis Nirenberg, Estimates and existence of solutions of elliptic equations, Comm. Pure Appl. Math. 9 (1956), 509–529. MR 0091402
  • [29] Louis Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8 (1955), 649–675. MR 0075415
  • [30] George H. Pimbley Jr., A superlinear Sturm-Liouville problem, Trans. Amer. Math. Soc. 103 (1962), 229–248. MR 0138821, 10.1090/S0002-9947-1962-0138821-6
  • [31] Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
  • [32] H. H. Schaeffer, Some non-linear eigenvalue problems, Non-Linear Problems Conference Univ. of Wisconsin, Madison, Wis., 1963.
  • [33] Laurent Schwartz, Théorie des distributions. Tome II, Actualités Sci. Ind., no. 1122 = Publ. Inst. Math. Univ. Strasbourg 10, Hermann & Cie., Paris, 1951 (French). MR 0041345
  • [34] M. M. Vaĭnberg, On the continuity of some operators of special type, Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 253–255 (Russian). MR 0036451
  • [35] M. M. Vaĭnberg, The existence of characteristic functions for nonlinear integral equations with nonpositive kernels, Doklady Akad. Nauk SSSR (N.S.) 78 (1951), 1077–1080 (Russian). MR 0043354
  • [36] -, Variational methods for the investigation of non-linear operators, GITTL, Moscow, 1956.
  • [37] M. I. Višik, Boundary-value problems for quasilinear strongly elliptic systems of equations having divergence form, Dokl. Akad. Nauk SSSR 138 (1961), 518–521 (Russian). MR 0150454
  • [38] Aurel Wintner, On the Hölder restrictions in the theory of partial differential equations, Amer. J. Math. 72 (1950), 731–738. MR 0040503
  • [39] I. M. Gel′fand, Some problems in the theory of quasi-linear equations, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 87–158 (Russian). MR 0110868

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DOI: https://doi.org/10.1090/S0002-9947-1965-0181821-3
Article copyright: © Copyright 1965 American Mathematical Society