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) no. 2, pp. 529-553.

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.

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.

[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 | Zbl

[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 | EuDML | Numdam | MR | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | EuDML | Numdam | MR | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

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

[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 | Zbl

[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 | Zbl

[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 | EuDML | Numdam | MR | Zbl

[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 | Zbl

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