Resonant nonlinear Neumann problems with indefinite weight
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, p. 729-788

We consider nonlinear Neumann problems driven by the p-Laplacian plus an indefinite potential. First we develop the spectral properties of such differential operators. Subsequently, using these spectral properties and variational methods based on critical point theory, truncation techniques and Morse theory, we prove existence and multiplicity theorems for resonant problems.

Published online : 2018-06-21
Classification:  35J20,  35J65,  58E05
@article{ASNSP_2012_5_11_4_729_0,
     author = {Mugnai, Dimitri and Papageorgiou, Nikolaos S.},
     title = {Resonant nonlinear Neumann problems with indefinite weight},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {4},
     year = {2012},
     pages = {729-788},
     zbl = {1270.35215},
     mrnumber = {3060699},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2012_5_11_4_729_0}
}
Mugnai, Dimitri; Papageorgiou, Nikolaos S. Resonant nonlinear Neumann problems with indefinite weight. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 4, pp. 729-788. http://www.numdam.org/item/ASNSP_2012_5_11_4_729_0/

[1] S. Aizicovici, N. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neumann problems, Ann. Mat. Pura Appl. (4) 188 (2009), 679–719. | MR 2533962 | Zbl 1188.35057

[2] W. Allegretto and Y.X. Huang, A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal. 32 (1998), 819–830. | MR 1618334 | Zbl 0930.35053

[3] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. 7 (1983), 981–1012. | MR 713209 | Zbl 0522.58012

[4] T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001), 117–152. | MR 1863294 | Zbl 1211.58003

[5] T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 28 (1997), 419–441. | MR 1420790 | Zbl 0872.58018

[6] P. A. Binding and B. P. Rynne, Variational and non-variational eigenvalues of the p-Laplacian, J. Differential Equations 244 (2008), 24–39. | MR 2373652 | Zbl 1136.35061

[7] I. Birindelli and F. Demengel, Existence of solutions for semi-linear equations involving the p-Laplacian: the non coercive case, Calc. Var. Partial Differential Equations 20 (2004), 343–366. | MR 2071925 | Zbl 1154.35344

[8] H. Brezis and L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 465–472. | MR 1239032 | Zbl 0803.35029

[9] K.-C. Chang, “Infinite-dimensional Morse Theory and Multiple Solution Problems”, In: Progress in Nonlinear Differential Equations and their Applications 6, Birkhäuser Boston, MA, 1993. | MR 1196690 | Zbl 0779.58005

[10] K.-C. Chang, “Methods in Nonlinear Analysis”, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. | MR 2170995 | Zbl 1081.47001

[11] M. Cuesta, Eigenvalue problems for the p-Laplacian with indefinite weights, Electron. J. Differential Equations 2001, No. 33, 1–9. | MR 1836801 | Zbl 0964.35110

[12] M. Cuesta and H. Ramos Quoirin, A weighted eigenvalue problem for the p-Laplacian plus a potential, NoDEA Nonlinear Differential Equations Appl. 16 (2009), 469–491. | MR 2525513 | Zbl 1174.35089

[13] M. Cuesta, D. de Figueiredo and J. P. Gossez, The beginning of the Fučik spectrum for the p-Laplacian, J. Differential Equations 159 (1999), 212–238. | MR 1726923 | Zbl 0947.35068

[14] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 493–516. | Numdam | MR 1632933 | Zbl 0911.35009

[15] L. M Del Pezzo, J. Fernández Bonder and J. D. Rossi, An optimization problem for the first weighted eigenvalue problem plus a potential, Proc. Amer. Math. Soc. 138 (2010), 3551–3567. | MR 2661555 | Zbl 1198.49047

[16] J. Fernández Bonder and L. M. Del Pezzo, An optimization problem for the first eigenvalue of the p-Laplacian plus a potential, Commun. Pure Appl. Anal. 5 (2006), 675–690. | MR 2246002 | Zbl 1175.35092

[17] J. P. García Azorero, I. Peral Alonso and J. J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2 (2000), 385–404. | MR 1776988 | Zbl 0965.35067

[18] L. Gasinski and N. S. Papageorgiou, “Nonlinear analysis”, Series in Mathematical Analysis and Applications 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. | MR 2168068 | Zbl 1086.47001

[19] N. Ghoussoub, “Duality and Perturbation Methods in Critical Point Theory”, Cambridge Tracts in Mathematics 107, Cambridge University Press, Cambridge, 1993. | MR 1251958 | Zbl 0790.58002

[20] S. Goldberg, “Unbounded Linear Operators. Theory and Applications”, McGraw-Hill Book Co., New York, 1966. | MR 200692 | Zbl 0925.47001

[21] A. Granas and J. Dugundji, “Fixed Point Theory”, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. | MR 1987179 | Zbl 1025.47002

[22] Z. Guo and Z. Zhang, W 1,p versus C 1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl. 286 (2003), 32–50. | MR 2009616 | Zbl 1160.35382

[23] S. Hu and N. S. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Commun. Pure Appl. Anal. in press. | MR 2787434 | Zbl 1242.58008

[24] Q. Jiu and J. Su, Existence and multiplicity results for perturbations of the p-Laplacian, J. Math. Anal. Appl. 281 (2003), 587–601. | MR 1982676 | Zbl 1146.35358

[25] A. Lê, Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64 (2006), 1057–1099. | MR 2196811 | Zbl 1208.35015

[26] L. Leadi and A. Yechoui, Principal eigenvalue in an unbounded domain with indefinite potential, NoDEA Nonlinear Differential Equations Appl. 17 (2010), 391–409. | MR 2677824 | Zbl 1198.35168

[27] C. Li, The existence of infinitely many solutions of a class of nonlinear elliptic equations with Neumann boundary condition for both resonance and oscillation problems, Nonlinear Anal. 54 (2003), 431–443. | MR 1978420 | Zbl 1126.35320

[28] Z. Liang and J. Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl. 354 (2009), 147–158. | MR 2510425 | Zbl 1166.35023

[29] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219. | MR 969499 | Zbl 0675.35042

[30] S. Liu and S. Li, Existence of solutions for asymptotically ‘linear’ p-Laplacian equations, Bull. London Math. Soc. 36 (2004), 81–87. | MR 2011981 | Zbl 1088.35025

[31] J. Liu and S. Wu, Calculating critical groups of solutions for elliptic problem with jumping nonlinearity, Nonlinear Anal. 49 (2002), 779–797. | MR 1894784 | Zbl 1012.35022

[32] J. López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations 127 (1996), 263–294. | MR 1387266 | Zbl 0853.35078

[33] E. Medeiros and K. Perera, Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index, Nonlinear Anal. 71 (2009), 3654–3660. | MR 2536276 | Zbl 1173.35480

[34] D. Motreanu, V. Motreanu and N.S. Papageorgiou, Nonlinear Neumann problems near resonance, Indiana Univ. Math. J. 58 (2009), 1257–1279. | MR 2541367 | Zbl 1168.35018

[35] D. Mugnai, Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA Nonlinear Differential Equations Appl. 11 (2004), 379–391, and a comment on the generalized Ambrosetti-Rabinowitz condition, NoDEA Nonlinear Differential Equations Appl. 19 (2004), 299–301. | MR 2090280 | Zbl 1247.35023

[36] N. S. Papageorgiou and S. T. Kyritsi, “Handbook of Applied Analysis”, Advances in Mechanics and Mathematics 19, Springer, New York, 2009. | MR 2527754 | Zbl 1189.49003

[37] P. Pucci and J. Serrin, “The Maximum Principle”, Progress in Nonlinear Differential Equations and their Applications 73, Birkhäuser Verlag, Basel, 2007. | MR 2356201 | Zbl 1134.35001

[38] A. Qian, Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem, Bound. Value Probl. 2005, 329–335. | MR 2202221 | Zbl 1220.35052

[39] R. E. Showalter, “ Hilbert Space Methods for Partial Differential Equations”, Monographs and Studies in Mathematics 1, Pitman, London, 1977. | MR 477394 | Zbl 0364.35001

[40] M. Struwe, “Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems”, Fourth edition, Springer-Verlag, Berlin, 2008. | MR 2431434 | Zbl 0746.49010

[41] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. | MR 226198 | Zbl 0153.42703

[42] J. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202. | MR 768629 | Zbl 0561.35003

[43] M. Zhang, The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials, J. London Math. Soc. (2) 64 (2001), 125–143. | MR 1840775 | Zbl 1109.35372