Characterizations of the ranges of some nonlinear operators and applications to boundary value problems

Brézis, H.; Nirenberg, L.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 5 (1978) no. 2, pp. 225-326.

[A-L-P] S. Ahmad - A.C. Lazer - J.L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J., 25 (1976), pp. 933-944. | MR | Zbl

[A-M] A. Ambrosetti - G. Mancini, Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance. - I: The case of simple eigenvalue, to appear. | MR | Zbl

[Ba-Ha] J.B. Baillon - G. Haddad, Quelques propriétés des opérateurs anglebornés et n-cycliquement monotones, Israel J. Math., 26 (1977), pp. 137-150. | MR | Zbl

[B-H-V] P. Baras - J.C. Hassan - L. Veron, Oompacité de l'opérateur définissant la solution d'une équation d'évolution non homogène, C. R. Acad. Sc., 284 (1977), pp. 799-802. | MR | Zbl

[B-S] M. Berger - M. Schechter, On the solvability of semilinear operator equations and elliptic boundary value problems, Bull. A.M.S., 78 (1972), pp. 741-745. | MR | Zbl

[Bré-1] H. Brézis, On some degenerate nonlinear parabolic equations, Nonlinear Functional Analysis, Proc. Symp. Pure Math., Vol. 18 (part 1), F. Browder ed. AMS (1970), pp. 28-38. | MR | Zbl

[Bré-2] H. Brézis, Opérateurs maximaux monotones, Lecture Notes, Vol. 5, North-Holland, 1973.

[B-B] H. Brézis - F. Browder, Nonlinear integral equations and systems of Hammerstein type, Advances in Math., 18 (1975), pp. 115-147. | MR | Zbl

[Br-Ha] H. Brézis - A. Haraux, Image d'une somme d'opérateurs monotones et applications, Israel J. Math., 23 (1976), pp. 165-186. | MR | Zbl

[Br-N] H. Brézis - L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math., 31 (1978). | MR | Zbl

[Bro-1] F. Browder, Nonlinear Operators in Banach Spaces, Proc. Symp. Pure Math., Vol. 18 (part 2) AMS (1976). | MR

[Bro-2] F. Browder, On a principte of H. Brézis and its applications, J. Funct. Anal., 25 (1977), pp. 356-365. | MR | Zbl

[C-G] B. Calvert - C. Gupta, Nonlinear elliptic boundary value problems in Lp spaces and sums of ranges of accretive operators, Nonlinear Analysis, TMA2 (1978), pp. 1-26. | MR | Zbl

[C] L. Cesari, Alternative methods in nonlinear analysis, International Conference on Differential Equations, Antosiewicz ed. Acad. Press. (1975). | MR | Zbl

[Da-1] E.N. Dancer, On a nonlinear elliptic boundary value problem, Bull. Aust. Math. Soc., 12 (1975), pp. 399-405. | MR | Zbl

[Da-2] E.N. Dancer, On the Dirichlet problem for weakly nonlinear elliptic partial differential equations, to appear. | Zbl

[DeF-1] D.G. De Figueiredo, The Dirichlet problem for nonlinear elliptic equations : A Hilbert space approach, Partial differential equations and related topics, Springer Verlag Lecture Notes 446 (1975), pp. 144-165. | MR | Zbl

[DeF-2] D.G. De Figueiredo, Semilinear elliptic problems with nonlinearities near the first eigenvalue, Proc. Symp. Dynamical Systems, Univ. of Florida (1976), to appear. | Zbl

[DeF-G] D.G. De Figueiredo - J.P. Gossez, Nonlinear perturbations of a linear elliptic problem near its first eigenvalues, to appear. | Zbl

[DeS-T] L. De Simon - G. Torelli, Soluzione periodiche di equazioni a derivate parziali di tipo iperbolico nonlineari, Rend. Sem. Math. Univ. Padova, 40 (1968), pp. 380-401. | Numdam | MR | Zbl

[Du] J.C. Dunn, Convexity, monotonicity and gradient processes in Hilbert space, J. Math. Anal. Applic., 53 (1976), pp. 145-158. | MR | Zbl

[F-1] S Fučik, Surjectivity of operators involving linear noninvertible part and nonlinear compact perturbation, Funk. Ekv., 17 (1974), pp. 73-83. | MR | Zbl

[F-2] S Fučik, Open problems in the solvability of nonlinear equations, to appear.

[F-K-N] S Fučik - M. Ku - J. Ne, Ranges of nonlinear asymptotically linear operators, J. Diff. Eq., 17 (1975), pp. 375-394. | MR | Zbl

[G-H] C. Gupta - P. Hess, Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems, J. Diff. Eq., 22 (1976), pp. 305-313. | MR | Zbl

[Ha] J. Hale, Periodic solutions of a class of hyperbolic equations containing a small parameter, Arch. Rat. Mech. Anal., 23 (1967), pp. 380-398. | MR | Zbl

[He-1] P. Hess, On semicoercive nonlinear problems, Indiana Univ. Math. J., 23 (1974), pp. 645-654. | MR | Zbl

[He-2] P. Hess, On a theorem by Landesman and Lazer, Indiana Univ. Math. J. 23 (1974), pp. 827-830. | MR | Zbl

[K-W] J. Kazdan - F. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), pp. 567-597. | MR | Zbl

[L-S-U] O. Ladyzenskaja - V. Solonnikov - N. Ural'Ceva, Linear and quasilinear equations of parabolic type, Translations of Math. Monographs, Vol. 23 AMS (1968). | MR | Zbl

[La-La] E. Landesman - A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), pp. 609-623. | MR | Zbl

[Le-Li] J. Leray - J.L. Lions, Quelques résultats de Visik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. Fr., 93 (1965), pp. 97-107. | Numdam | MR | Zbl

[L-S] J.L. Lions - G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), pp. 493-519. | MR | Zbl

[Li] W. Littman, A strong maximum principle for weakly L-subharmonic functions, J. Math. Mech., 8 (1959), pp. 761-770. | MR | Zbl

[Lo] H. Lovicarova, Periodic solutions of weakly nonlinear wave equation in one dimension, Czech. Math. J., 19 (1969), pp. 324-342. | MR | Zbl

[McK-R] P Mckenna - J. Rauch, Strongly nonlinear perturbations of nonnegative boundary value problems with kernel, Jour. Diff. Equa., to appear. | MR | Zbl

[M-1] J. Mawhin, Nonlinear perturbations of Fredholm mappings in normed spaces and applications to differential equations, Univ. de Brasilia, Trabalho de Mat., No. 61 (1974).

[M-2] J. Mawhin, Periodic solutions of nonlinear telegraph equations, Proc. Int. Symp. Dyn. Syst. Gainesville, Florida (1976). | MR | Zbl

[M-3] J. Mawhin, Solutions periodiques d'équations aux dérivées partielles hyperboliques nonlinéaires (to appear in a volume dedicated to TH. VOGEL).

[M-4] J. Mawhin, Lecture at Oberwolfach, January 1976.

[N-1] L. Nirenberg, An application of generalized degree to a class of nonlinear problems, Coll. Analyse Fonctionnelle, Liège, 1970 (Math. Vander., pp. 57-74). | MR | Zbl

[N-2] L. Nirenberg, Generalized degree and nonlinear problems, Contribution to Nonlinear Functional Analysis, Zarantonello ed. (1971), Acad. Press, pp. 1-9. | MR

[N-3] L. Nirenberg, Topics in Nonlinear Functional Analysis, New York Univ. Lecture Notes (1974). | MR | Zbl

[O-S] J.E. Osborn - D. Sather, Alternative problems and monotonicity, J. Diff. Eq'ns., 18 (1975), pp. 393-410. | MR | Zbl

[P] G. Prodi, Soluzioni periodiche delle equazioni a derivate parziali di tipo iperbolico nonlineari, Annali Mat. Pura Appl., 42 (1956), pp. 25-49. | MR | Zbl

[Ra-1] P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 (1967), pp. 145-205. | MR | Zbl

[Ra-2] P. Rabinowitz, Time periodic solutions of nonlinear wave equations, Manuscripta Math., 5 (1971), pp. 165-194. | MR | Zbl

[Ra-3] P. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, to appear. | MR | Zbl

[Ro] R.T. Rockafellar, Convex analysis, Princeton Univ. Press, 1970. | MR | Zbl

[Scha] M. Schatzman, Problèmes aux limites, nonlinéaires, noncoercifs, Ann. Scuola Norm. Sup. Pisa, 27 (1973), pp. 641-686. | Numdam | MR | Zbl

[Sche] M. Schechter, A nonlinear elliptic boundary value problem, Ann. Scuola Norm. Sup. Pisa, 27 (1973), pp. 707-716. | Numdam | MR | Zbl

[Sta] G. Stampacchia, Equations elliptiques du second ordre à coefficients discontinus, Presses Univ. Montreal, (1966). | MR | Zbl

[Str] W. Strauss, On weak solutions of semilinear hyperbolic equations, Anais Acad. Brasileira Ciencias, 42 (1970), pp. 645-651. | MR | Zbl

[V] O. Vejvoda, Periodic solutions of nonlinear partial differential equations of evolution, Proc. Symp. Diff. Eq. and Applications, Bratislava, (1969), pp. 293-300. | MR | Zbl

[W] S. Williams, A sharp sufficient condition for solution of nonlinear elliptic boundary value problem, J. Diff. Eq., 8 (1970), pp. 580-588. | MR | Zbl