Multiple solutions for a class of elliptic equations with jumping nonlinearities

Molle, Riccardo ; Passaseo, Donato

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 529-553 / Harvested from Numdam

Résumé

Nous considérons un problème de Dirichlet semi-linéaire avec le terme non linéaire qui interfère avec les valeurs propres de l'opérateur linéaire. Avec des méthodes variationnelles, nous montrons que le nombre de solutions est arbitrairement grand pourvu que le nombre de valeurs propres qui interfèrent avec le terme non linéaire soit suffisamment grand. Pour la démonstration nous prouvons que pour tout $ k\in \mathbb{N}$ le problème a une solution qui présente k pics quand un paramètre est suffisamment grand. Nous décrivons aussi le comportement asymptotique et la forme de cette solution quand ce paramètre tend à l'infini.

Abstract

We consider a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using variational methods, we show that the number of solutions tends to infinity as the number of jumped eigenvalues tends to infinity. In order to prove this fact, for every positive integer k we prove that, when a parameter is large enough, there exists a solution which presents k interior peaks. We also describe the asymptotic behaviour and the profile of this solution as the parameter tends to infinity.

Détail

Publié le : 2010-01-01

MR 2595191 | Zbl 1185.35099 | 1 citation dans Numdam

DOI : https://doi.org/10.1016/j.anihpc.2009.09.005
Classification:  35J20,  35J60,  35J65

Bibliographie

[1] H. Amann, P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 84 no. 1–2 (1979), 145-151 | MR 549877 | Zbl 0416.35029

[2] H. Amann, E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 7 no. 4 (1980), 539-603 | Numdam | MR 600524 | Zbl 0452.47077

[3] A. Ambrosetti, Elliptic equations with jumping nonlinearities, J. Math. Phys. Sci. 18 no. 1 (1984), 1-12 | MR 755462 | Zbl 0589.35043

[4] A. Ambrosetti, G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. (4) 93 (1972), 231-246 | MR 320844 | Zbl 0288.35020

[5] A. Bahri, Résolution générique d'une équation semi-linéaire, C. R. Acad. Sci. Paris Sér. A–B 291 no. 4 (1980), A251-A254

[6] H. Berestycki, Le nombre de solutions de certains problémes semi-linéaires elliptiques, J. Funct. Anal. 40 no. 1 (1981), 1-29 | MR 607588 | Zbl 0452.35038

[7] M.S. Berger, E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1974/1975), 837-846 | MR 377274 | Zbl 0329.35026

[8] B. Breuer, P.J. Mckenna, M. Plum, Multiple solutions for a semilinear boundary value problem: A computational multiplicity proof, J. Differential Equations 195 no. 1 (2003), 243-269 | MR 2019251 | Zbl 1156.35359

[9] R. Caccioppoli, Un principio di inversione per le corrispondenze funzionali e sue applicazioni alle equazioni alle derivate parziali, Atti Acc. Naz. Lincei 16 (1932), 392-400 | JFM 58.1117.01

[10] N.P. Các, On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue, J. Differential Equations 80 no. 2 (1989), 379-404 | MR 1011156 | Zbl 0713.35036

[11] N.P. Các, On a boundary value problem with nonsmooth jumping nonlinearity, J. Differential Equations 93 no. 2 (1991), 238-259 | MR 1125219 | Zbl 0768.35031

[12] D.G. Costa, D.G. De Figueiredo, P.N. Srikanth, The exact number of solutions for a class of ordinary differential equations through Morse index computation, J. Differential Equations 96 no. 1 (1992), 185-199 | MR 1153315 | Zbl 0764.34013

[13] E.N. Dancer, Multiple solutions of asymptotically homogeneous problems, Ann. Mat. Pura Appl. (4) 152 (1988), 63-78 | MR 980972 | Zbl 0850.35043

[14] E.N. Dancer, A counterexample to the Lazer–McKenna conjecture, Nonlinear Anal. 13 no. 1 (1989), 19-21 | MR 973364 | Zbl 0691.35039

[15] E.N. Dancer, On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math. 25 no. 3 (1995), 957-975 | MR 1357103 | Zbl 0846.35046

[16] E.N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 76 no. 4 (1976/1977), 283-300 | MR 499709 | Zbl 0351.35037

[17] E.N. Dancer, Generic domain dependence for nonsmooth equations and the open set problem for jumping nonlinearities, Topol. Methods Nonlinear Anal. 1 no. 1 (1993), 139-150 | MR 1215262 | Zbl 0817.35026

[18] E.N. Dancer, S. Yan, On the superlinear Lazer–McKenna conjecture, J. Differential Equations 210 no. 2 (2005), 317-351 | MR 2119987 | Zbl 1190.35082

[19] E.N. Dancer, S. Yan, The Lazer–McKenna conjecture and a free boundary problem in two dimensions, J. Lond. Math. Soc. (2) 78 no. 3 (2008), 639-662 | MR 2456896 | Zbl 1202.35088

[20] D.G. De Figueiredo, On the superlinear Ambrosetti–Prodi problem, Nonlinear Anal. 8 no. 6 (1984), 655-665 | MR 746723 | Zbl 0554.35045

[21] D.G. De Figueiredo, S. Solimini, A variational approach to superlinear elliptic problems, Comm. Partial Differential Equations 9 no. 7 (1984), 699-717 | MR 745022 | Zbl 0552.35030

[22] O. Druet, The critical Lazer–McKenna conjecture in low dimensions, J. Differential Equations 245 no. 8 (2008), 2199-2242 | MR 2446190 | Zbl 1155.35030

[23] S. Fučík, Nonlinear equations with noninvertible linear part, Czechoslovak Math. J. 24 no. 99 (1974), 467-495 | MR 348568 | Zbl 0315.47038

[24] S. Fučík, Boundary value problems with jumping nonlinearities, Časopis Pěst. Mat. 101 no. 1 (1976), 69-87 | MR 447688 | Zbl 0332.34016

[25] T. Gallouët, O. Kavian, Résultats d'existence et de non-existence pour certains problèmes demi-linéaires à l'infini, Ann. Fac. Sci. Toulouse Math. (5) 3 no. 3–4 (1981), 201-246 | Numdam | MR 658734 | Zbl 0495.35001

[26] J.A. Hempel, Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J. 20 (1970/1971), 983-996 | MR 279423 | Zbl 0225.35045

[27] H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 no. 4 (1982), 493-514 | MR 682663 | Zbl 0488.47034

[28] A.C. Lazer, P.J. Mckenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 no. 1 (1981), 282-294 | MR 639539 | Zbl 0496.35039

[29] A.C. Lazer, P.J. Mckenna, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Roy. Soc. Edinburgh Sect. A 95 no. 3–4 (1983), 275-283 | MR 726879 | Zbl 0533.35037

[30] A.C. Lazer, P.J. Mckenna, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues, Comm. Partial Differential Equations 10 no. 2 (1985), 107-150 | MR 777047 | Zbl 0572.35036

[31] A.C. Lazer, P.J. Mckenna, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues, II, Comm. Partial Differential Equations 11 no. 15 (1986), 1653-1676 | MR 871108 | Zbl 0654.35082

[32] A. Marino, A.M. Micheletti, A. Pistoia, A nonsymmetric asymptotically linear elliptic problem, Topol. Methods Nonlinear Anal. 4 no. 2 (1994), 289-339 | MR 1350975 | Zbl 0844.35035

[33] R. Molle, D. Passaseo, in preparation

[34] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. vol. 65, American Mathematical Society, Providence, RI (1986) | MR 845785

[35] B. Ruf, On nonlinear elliptic problems with jumping nonlinearities, Ann. Mat. Pura Appl. (4) 128 (1981), 133-151 | MR 640779 | Zbl 0475.35046

[36] B. Ruf, Remarks and generalizations related to a recent multiplicity result of A. Lazer and P. McKenna, Nonlinear Anal. 9 no. 12 (1985), 1325-1330 | MR 820643 | Zbl 0626.34017

[37] S. Solimini, Existence of a third solution for a class of BVP with jumping nonlinearities, Nonlinear Anal. 7 no. 8 (1983), 917-927 | MR 709044 | Zbl 0522.35045

[38] S. Solimini, Some remarks on the number of solutions of some nonlinear elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 no. 2 (1985), 143-156 | Numdam | MR 794004 | Zbl 0583.35044

[39] G. Sweers, On the maximum of solutions for a semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A 108 no. 3–4 (1988), 357-370 | MR 943809 | Zbl 0681.35013

[40] J. Wei, S. Yan, Lazer–McKenna conjecture: The critical case, J. Funct. Anal. 244 no. 2 (2007), 639-667 | MR 2297039 | Zbl 1231.35072