Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients

Gossez, Jean-Pierre

doi:10.1090/S0002-9947-1974-0342854-2

Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients
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Trans. Amer. Math. Soc. 190 (1974), 163-205 Request permission

Abstract:

Variational boundary value problems for quasilinear elliptic systems in divergence form are studied in the case where the nonlinearities are nonpolynomial. Monotonicity methods are used to derive several existence theorems which generalize the basic results of Browder and Leray-Lions. Some features of the mappings of monotone type which arise here are that they act in nonreflexive Banach spaces, that they are unbounded and not everywhere defined, and that their inverse is also unbounded and not everywhere defined.

References

Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
Haïm Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 1, 115–175 (French). MR 270222, DOI 10.5802/aif.280
Felix E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc. 69 (1963), 862–874. MR 156116, DOI 10.1090/S0002-9904-1963-11068-X
Felix E. Browder, Nonlinear elliptic functional equations in nonreflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 89–95. MR 185287, DOI 10.1090/S0002-9904-1966-11432-5
Felix E. Browder, Nonlinear maximal monotone operators in Banach space, Math. Ann. 175 (1968), 89–113. MR 223942, DOI 10.1007/BF01418765
Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
Felix E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 1–60. MR 0269962
Felix E. Browder, Nonlinear elliptic boundary value problems and the generalized topological degree, Bull. Amer. Math. Soc. 76 (1970), 999–1005. MR 264222, DOI 10.1090/S0002-9904-1970-12530-7
Felix E. Browder, Existence theory for boundary value problems for quasilinear elliptic systems with strongly nonlinear lower order terms, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 269–286. MR 0340815

Nonlinear elliptic eigenvalue problems with strong lower order nonlinearities

Felix E. Browder and Peter Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis 11 (1972), 251–294. MR 0365242, DOI 10.1016/0022-1236(72)90070-5
Thomas Donaldson, Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces, J. Differential Equations 10 (1971), 507–528. MR 298472, DOI 10.1016/0022-0396(71)90009-X
Thomas K. Donaldson and Neil S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Functional Analysis 8 (1971), 52–75. MR 0301500, DOI 10.1016/0022-1236(71)90018-8
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523

Opérateurs elliptiques du calcul des variations à coefficients très fortement non linéaires

274

Jean-Pierre Gossez, Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs, J. Math. Anal. Appl. 34 (1971), 371–395 (French). MR 313890, DOI 10.1016/0022-247X(71)90119-3
Jean-Pierre Gossez, On the range of a coercive maximal monotone operator in a nonreflexive Banach space, Proc. Amer. Math. Soc. 35 (1972), 88–92. MR 298492, DOI 10.1090/S0002-9939-1972-0298492-7
Jean-Pierre Gossez, On the subdifferential of a saddle function, J. Functional Analysis 11 (1972), 220–230. MR 0350416, DOI 10.1016/0022-1236(72)90092-4
Jean-Pierre Gossez, Boundary value problems for quasilinear elliptic equations with rapidly increasing coefficients, Bull. Amer. Math. Soc. 78 (1972), 753–758. MR 318669, DOI 10.1090/S0002-9904-1972-13019-2
A. Granas, Sur la notion du degré topologique pour une certaine classe de transformations multivalentes dans les espaces de Banach, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 7 (1959), 191–194 (unbound insert) (French, with Russian summary). MR 0108743
A. Granas, Theorem on antipodes and theorems on fixed points for a certain class of multi-valued mappings in Banach spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 7 (1959), 271–275 (unbound insert) (English, with Russian summary). MR 0117588
Peter Hess, On nonlinear mappings of monotone type with respect to two Banach spaces, J. Math. Pures Appl. (9) 52 (1973), 13–26. MR 636418
Peter Hess, A strongly nonlinear elliptic boundary value problem, J. Math. Anal. Appl. 43 (1973), 241–249. MR 318670, DOI 10.1016/0022-247X(73)90272-2
M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron. MR 0126722
Jean Leray and Jacques-Louis Lions, Quelques résulatats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France 93 (1965), 97–107 (French). MR 194733, DOI 10.24033/bsmf.1617
Joram Lindenstrauss, On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 967–970. MR 205040, DOI 10.1090/S0002-9904-1966-11606-3
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
Wilhelmus Anthonius Josephus Luxemburg, Banach function spaces, Technische Hogeschool te Delft, Delft, 1955. Thesis. MR 0072440
W. A. J. Luxemburg and A. C. Zaanen, Conjugate spaces of Orlicz spaces, Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 217–228. MR 0077899, DOI 10.1016/S1385-7258(56)50029-7
George J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346. MR 169064
R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216. MR 262827, DOI 10.2140/pjm.1970.33.209
R. T. Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969), 397–407. MR 253014, DOI 10.1307/mmj/1029000324
R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75–88. MR 282272, DOI 10.1090/S0002-9947-1970-0282272-5
M. I. Višik, Quasi-linear strongly elliptic systems of differential equations of divergence form, Trudy Moskov. Mat. Obšč. 12 (1963), 125–184 (Russian). MR 0156085
M. I. Višik, On the solvability of the first boundary-value problem for quasi-linear equations with coefficients of rapid growth in Orlicz classes, Dokl. Akad. Nauk SSSR 151 (1963), 758–761 (Russian). MR 0155090