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Nonlinear second order elliptic partial differential equations at resonance. (English) Zbl 0686.35045
The authors consider nonlinear second order elliptic partial differential equations at resonance. More precisely, they study the solvability of selfadjoint boundary value problems of the form $(1)\quad Lu+\lambda_ 1u+g(x,u)=h\quad in\quad \Omega,\quad u/\partial \Omega =0,$ and the corresponding nonselfadjoint problems $(2)\quad Au+\lambda_ 1u+g(x,u)=h\quad in\quad \Omega,\quad u/\partial \Omega =0,$ where h is a given function on $$\Omega$$ and $$\lambda_ 1$$ is the first (resp. principal) eigenvalue of a uniformly elliptic operator -L (resp. -A) on a bounded smooth domain $$\Omega \subset {\mathbb{R}}^ N:$$ $Lu=\sum_{i,j}\partial /\partial x_ i(a_{ij}(x)\partial u/\partial x_ j)-a_ 0(u)u,\quad Au=Lu+\sum_{i}b_ i(x)\partial u/\partial x_ i,$ with the coefficients satisfying suitable regularity conditions. The objective is to show solvability of (1) or (2) for any h orthogonal to the first eigenfunction, in situations where the nonlinearity satisfies neither a monotonicity condition nor a Landesman- Lazer type condition. Instead, the nonlinearity is assumed to satisfy a sign condition g(x,u), $$u\geq 0$$, and a linear growth allowing “interaction” with the first and second eigenvalues. Moreover, some crossing of eigenvalues is allowed in certain cases.
Reviewer: D.Costa

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J15 Second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations
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##### References:
  Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030  Shair Ahmad, Nonselfadjoint resonance problems with unbounded perturbations, Nonlinear Anal. 10 (1986), no. 2, 147 – 156. · Zbl 0599.35069 · doi:10.1016/0362-546X(86)90042-8 · doi.org  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. · Zbl 0391.35030 · doi:10.1512/iumj.1978.27.27050 · doi.org  Henri Berestycki and Djairo Guedes de Figueiredo, Double resonance in semilinear elliptic problems, Comm. Partial Differential Equations 6 (1981), no. 1, 91 – 120. · Zbl 0468.35043 · doi:10.1080/03605308108820172 · doi.org  Jean-Michel Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A333 – A336 (French). · Zbl 0164.16803  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].  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. · Zbl 0386.47035  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. · Zbl 0554.34009 · doi:10.1016/0022-0396(85)90100-7 · doi.org  Lamberto Cesari and Patrizia Pucci, Existence theorems for nonselfadjoint semilinear elliptic boundary value problems, Nonlinear Anal. 9 (1985), no. 11, 1227 – 1241. · Zbl 0535.35026 · doi:10.1016/0362-546X(85)90032-X · doi.org  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. · Zbl 0351.35037 · doi:10.1017/S0308210500019648 · doi.org  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). · Zbl 0596.35049  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. · Zbl 0429.35035 · doi:10.1016/0362-546X(79)90091-9 · doi.org  Pavel Drábek, On the resonance problem with nonlinearity which has arbitrary linear growth, J. Math. Anal. Appl. 127 (1987), no. 2, 435 – 442. · Zbl 0642.34009 · doi:10.1016/0022-247X(87)90121-1 · doi.org  Svatopluk Fučík, Surjectivity of operators involving linear noninvertible part and nonlinear compact perturbation, Funkcial. Ekvac. 17 (1974), 73 – 83. · Zbl 0294.47041  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. · Zbl 0562.35001  José Valdo A. Gonçalves, On bounded nonlinear perturbations of an elliptic equation at resonance, Nonlinear Anal. 5 (1981), no. 1, 57 – 60. · Zbl 0473.35040 · doi:10.1016/0362-546X(81)90070-5 · doi.org  Chaitan P. Gupta, Perturbations of second order linear elliptic problems by unbounded nonlinearities, Nonlinear Anal. 6 (1982), no. 9, 919 – 933. · Zbl 0509.35035 · doi:10.1016/0362-546X(82)90011-6 · doi.org  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. · Zbl 0638.34015 · doi:10.1016/0022-247X(88)90266-1 · doi.org  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. · Zbl 0627.34008 · doi:10.1016/0022-0396(87)90121-5 · doi.org  R. Iannacci and M. N. Nkashama, Nonlinear boundary value problems at resonance, Nonlinear Anal. 11 (1987), no. 4, 455 – 473. · Zbl 0676.35023 · doi:10.1016/0362-546X(87)90064-2 · doi.org  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. · Zbl 0684.34025  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. · Zbl 0589.34013 · doi:10.1016/0022-247X(85)90093-9 · doi.org  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. · Zbl 0030.12902  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. · Zbl 0414.34025  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.  Jean Mawhin, A Neumann boundary value problem with jumping monotone nonlinearity, Delft Progr. Rep. 10 (1985), no. 1, 44 – 52. · Zbl 0595.35050  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. · Zbl 0559.34014  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. · Zbl 0656.35044 · doi:10.1007/BF00251362 · doi.org  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). · Zbl 0678.35091  Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. · Zbl 0549.35002  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. · Zbl 0657.34021  Martin Schechter, Jack Shapiro, and Morris Snow, Solution of the nonlinear problem \?\?=\?(\?) in a Banach space, Trans. Amer. Math. Soc. 241 (1978), 69 – 78. · Zbl 0403.47030  J. R. Ward. Jr., A note on the Dirichlet problem for some semi-linear elliptic equations, (preprint 1986).
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