Nonlinear second order elliptic partial differential equations at resonance

Iannacci, R.; Nkashama, M. N.; Ward, J. R.

doi:10.1090/S0002-9947-1989-0951886-3

Nonlinear second order elliptic partial differential equations at resonance
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by R. Iannacci, M. N. Nkashama and J. R. Ward

Trans. Amer. Math. Soc. 311 (1989), 711-726

DOI: https://doi.org/10.1090/S0002-9947-1989-0951886-3

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Abstract:

In this paper we study the solvability of boundary value problems for semilinear second order elliptic partial differential equations of resonance type in which the nonlinear perturbation is not (necessarily) required to satisfy the Landesman-Lazer condition or the monotonicity assumption. The nonlinearity may be unbounded and some crossing of eigenvalues is allowed. Selfadjoint and nonselfadjoint resonance problems are considered.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Shair Ahmad, Nonselfadjoint resonance problems with unbounded perturbations, Nonlinear Anal. 10 (1986), no. 2, 147–156. MR 825213, DOI 10.1016/0362-546X(86)90042-8
Herbert Amann and Michael G. Crandall, On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J. 27 (1978), no. 5, 779–790. MR 503713, DOI 10.1512/iumj.1978.27.27050
Henri Berestycki and Djairo Guedes de Figueiredo, Double resonance in semilinear elliptic problems, Comm. Partial Differential Equations 6 (1981), no. 1, 91–120. MR 597753, DOI 10.1080/03605308108820172
Jean-Michel Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A333–A336 (French). MR 223711
Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
H. Brézis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 225–326. MR 513090
L. Cesari and R. Kannan, Qualitative study of a class of nonlinear boundary value problems at resonance, J. Differential Equations 56 (1985), no. 1, 63–81. MR 772121, DOI 10.1016/0022-0396(85)90100-7
Lamberto Cesari and Patrizia Pucci, Existence theorems for nonselfadjoint semilinear elliptic boundary value problems, Nonlinear Anal. 9 (1985), no. 11, 1227–1241. MR 813655, DOI 10.1016/0362-546X(85)90032-X
E. N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 4, 283–300. MR 499709, DOI 10.1017/S0308210500019648
Djairo G. de Figueiredo and Jean-Pierre Gossez, Conditions de non-résonance pour certains problèmes elliptiques semi-linéaires, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 15, 543–545 (French, with English summary). MR 845644
Djairo Guedes de Figueiredo and Wei Ming Ni, Perturbations of second order linear elliptic problems by nonlinearities without Landesman-Lazer condition, Nonlinear Anal. 3 (1979), no. 5, 629–634. MR 541873, DOI 10.1016/0362-546X(79)90091-9
Pavel Drábek, On the resonance problem with nonlinearity which has arbitrary linear growth, J. Math. Anal. Appl. 127 (1987), no. 2, 435–442. MR 915069, DOI 10.1016/0022-247X(87)90121-1
Svatopluk Fučík, Surjectivity of operators involving linear noninvertible part and nonlinear compact perturbation, Funkcial. Ekvac. 17 (1974), 73–83. MR 365255
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
José Valdo A. Gonçalves, On bounded nonlinear perturbations of an elliptic equation at resonance, Nonlinear Anal. 5 (1981), no. 1, 57–60. MR 597281, DOI 10.1016/0362-546X(81)90070-5
Chaitan P. Gupta, Perturbations of second order linear elliptic problems by unbounded nonlinearities, Nonlinear Anal. 6 (1982), no. 9, 919–933. MR 677617, DOI 10.1016/0362-546X(82)90011-6
Chaitan P. Gupta, Solvability of a boundary value problem with the nonlinearity satisfying a sign condition, J. Math. Anal. Appl. 129 (1988), no. 2, 482–492. MR 924305, DOI 10.1016/0022-247X(88)90266-1
R. Iannacci and M. N. Nkashama, Unbounded perturbations of forced second order ordinary differential equations at resonance, J. Differential Equations 69 (1987), no. 3, 289–309. MR 903389, DOI 10.1016/0022-0396(87)90121-5
R. Iannacci and M. N. Nkashama, Nonlinear boundary value problems at resonance, Nonlinear Anal. 11 (1987), no. 4, 455–473. MR 887655, DOI 10.1016/0362-546X(87)90064-2
R. Iannacci and M. N. Nkashama, Nonlinear two-point boundary value problems at resonance without Landesman-Lazer condition, Proc. Amer. Math. Soc. 106 (1989), no. 4, 943–952. MR 1004633, DOI 10.1090/S0002-9939-1989-1004633-9
R. Kannan, J. J. Nieto, and M. B. Ray, A class of nonlinear boundary value problems without Landesman-Lazer condition, J. Math. Anal. Appl. 105 (1985), no. 1, 1–11. MR 773569, DOI 10.1016/0022-247X(85)90093-9
M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Translation 1950 (1950), no. 26, 128. MR 0038008
J. Mawhin, Topological degree methods in nonlinear boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 40, American Mathematical Society, Providence, R.I., 1979. Expository lectures from the CBMS Regional Conference held at Harvey Mudd College, Claremont, Calif., June 9–15, 1977. MR 525202
Jean Mawhin, Nonresonance conditions of nonuniform type in nonlinear boundary value problems, Dynamical systems, II (Gainesville, Fla., 1981) Academic Press, New York, 1982, pp. 255–276. MR 703699
Jean Mawhin, A Neumann boundary value problem with jumping monotone nonlinearity, Delft Progr. Rep. 10 (1985), no. 1, 44–52. MR 787670
J. Mawhin, J. R. Ward, and M. Willem, Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem, Proc. Amer. Math. Soc. 93 (1985), no. 4, 667–674. MR 776200, DOI 10.1090/S0002-9939-1985-0776200-X
J. Mawhin, J. R. Ward Jr., and M. Willem, Variational methods and semilinear elliptic equations, Arch. Rational Mech. Anal. 95 (1986), no. 3, 269–277. MR 853968, DOI 10.1007/BF00251362
J. Mawhin and M. Willem, Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 6, 431–453 (English, with French summary). MR 870864
Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825, DOI 10.1007/978-1-4612-5282-5
Renate Schaaf and Klaus Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc. 306 (1988), no. 2, 853–859. MR 933322, DOI 10.1090/S0002-9947-1988-0933322-5
Martin Schechter, Jack Shapiro, and Morris Snow, Solution of the nonlinear problem $Au=N(u)$ in a Banach space, Trans. Amer. Math. Soc. 241 (1978), 69–78. MR 492290, DOI 10.1090/S0002-9947-1978-0492290-7

A note on the Dirichlet problem for some semi-linear elliptic equations