Infinitely many new curves of the Fučík spectrum
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, pp. 1145-1171.

In this paper we present some results on the Fučík spectrum for the Laplace operator, that give new information on its structure. In particular, these results show that, if Ω is a bounded domain of N with N>1, then the Fučík spectrum has infinitely many curves asymptotic to the lines {λ 1 }× and ×{λ 1 }, where λ 1 denotes the first eigenvalue of the operator −Δ in H 0 1 (Ω). Notice that the situation is quite different in the case N=1; in fact, in this case the Fučík spectrum may be obtained by direct computation and one can verify that it includes only two curves asymptotic to these lines.

Nous présentons des résultats qui donnent de nouvelles informations sur la structure du spectre de Fučík pour l'opérateur de Laplace. En particulier, ces résultats montrent que, si Ω est un domaine borné de N avec N>1, alors le spectre de Fučík a un nombre infini de courbes qui ont comme asymptotes les droites {λ 1 }× et ×{λ 1 }, où λ 1 est la première valeur propre de l'operateur −Δ in H 0 1 (Ω). La situation est bien différente dans le cas N=1 ; en effect, dans ce cas on peut vérifier qu'il y a seulement deux courbes dans le spectre de Fučík, qui ont ces droites comme asymptotes.

DOI: 10.1016/j.anihpc.2014.05.007
Classification: 35J20,  35J60,  35J66
Keywords: Elliptic operators, Fučík spectrum, Variational methods, Multiplicity results, Asymptotic behaviours
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     title = {Infinitely many new curves of the {Fu\v{c}{\'\i}k} spectrum},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     publisher = {Elsevier},
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Molle, Riccardo; Passaseo, Donato. Infinitely many new curves of the Fučík spectrum. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, pp. 1145-1171. doi : 10.1016/j.anihpc.2014.05.007. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.05.007/

[1] 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

[2] M. Arias, J. Campos, Radial Fučík spectrum of the Laplace operator, J. Math. Anal. Appl. 190 no. 3 (1995), 654 -666 | MR | Zbl

[3] A.K. Ben-Naoum, C. Fabry, D. Smets, Structure of the Fučík spectrum and existence of solutions for equations with asymmetric nonlinearities, Proc. R. Soc. Edinb. A 131 no. 2 (2001), 241 -265 | MR | Zbl

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

[5] L.E.J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 no. 1 (1911), 97 -115 | EuDML | JFM | MR

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

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

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

[9] G. Cerami, D. Passaseo, S. Solimini, Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients, Commun. Pure Appl. Math. 66 no. 3 (2013), 372 -413 | MR | Zbl

[10] G. Cerami, D. Passaseo, S. Solimini, Nonlinear scalar field equations: existence of a solution with infinitely many bumps, Ann. Inst. Henri Poincaré, Anal. Non Linéaire (2014), http://dx.doi.org/10.1016/j.anihpc.2013.08.008 | Numdam | MR | Zbl

[11] M. Cuesta, J.-P. Gossez, A variational approach to nonresonance with respect to the Fučík spectrum, Nonlinear Anal. 19 no. 5 (1992), 487 -500 | MR | Zbl

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

[13] E.N. Dancer, On the existence of solutions of certain asymptotically homogeneous problems, Math. Z. 177 no. 1 (1981), 33 -48 | EuDML | MR | Zbl

[14] 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

[15] D.G. De Figueiredo, J.-P. Gossez, On the first curve of the Fučík spectrum of an elliptic operator, Differ. Integral Equ. 7 no. 5–6 (1994), 1285 -1302 | MR | Zbl

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

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

[18] S. Fučík, A. Kufner, Nonlinear Differential Equations, Stud. Appl. Mech. vol. 2 , Elsevier Scientific Publishing Co., Amsterdam, New York (1980) | MR | Zbl

[19] 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

[20] T. Gallouët, O. Kavian, Resonance for jumping nonlinearities, Commun. Partial Differ. Equ. 7 no. 3 (1982), 325 -342 | MR | Zbl

[21] J.V.A. Gonçalves, C.A. Magalhães, Semilinear Elliptic Problems with Crossing of the Singular Set, Trabalhos Mat. vol. 263 , Univ. de Brasilia (1992)

[22] J. Horák, W. Reichel, Analytical and numerical results for the Fučík spectrum of the Laplacian, J. Comput. Appl. Math. 161 no. 2 (2003), 313 -338 | MR | Zbl

[23] C. Li, S. Li, Z. Liu, J. Pan, On the Fučík spectrum, J. Differ. Equ. 244 no. 10 (2008), 2498 -2528 | MR | Zbl

[24] C.A. Magalhães, Semilinear elliptic problem with crossing of multiple eigenvalues, Commun. Partial Differ. Equ. 15 no. 9 (1990), 1265 -1292 | MR | Zbl

[25] C.A. Margulies, W. Margulies, An example of the Fučík spectrum, Nonlinear Anal. 29 no. 12 (1997), 1373 -1378 | MR | Zbl

[26] C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital. (2) 3 (1940), 5 -7 | JFM | MR

[27] R. Molle, D. Passaseo, Multiple solutions for a class of elliptic equations with jumping nonlinearities, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27 no. 2 (2010), 529 -553 | Numdam | MR | Zbl

[28] R. Molle, D. Passaseo, Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities, J. Funct. Anal. 259 no. 9 (2010), 2253 -2295 | MR | Zbl

[29] R. Molle, D. Passaseo, Elliptic equations with jumping nonlinearities involving high eigenvalues, Calc. Var. Partial Differ. Equ. 49 no. 1–2 (2014), 861 -907 | MR | Zbl

[30] R. Molle, D. Passaseo, New properties of the Fučík spectrum, C. R. Math. Acad. Sci. Paris 351 no. 17–18 (2013), 681 -685 | MR | Zbl

[31] R. Molle, D. Passaseo, On the first curve of the Fučík spectrum for elliptic operators, Rend. Lincei Mat. Appl. 25 no. 2 (2014), 141 -146 | MR | Zbl

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

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

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

[35] M. Schechter, The Fučík spectrum, Indiana Univ. Math. J. 43 no. 4 (1994), 1139 -1157 | MR | Zbl

[36] M. Schechter, Type (II) regions between curves of the Fucik spectrum, Nonlinear Differ. Equ. Appl. 4 no. 4 (1997), 459 -476 | MR | Zbl

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

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