Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential

Papageorgiou, Nikolaos; Rădulescu, Vicenţiu

doi:10.1090/S0002-9947-2014-06518-5

Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential
HTML articles powered by AMS MathViewer

by Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu PDF

Trans. Amer. Math. Soc. 367 (2015), 8723-8756 Request permission

Abstract:

We examine semilinear Neumann problems driven by the Laplacian plus an unbounded and indefinite potential. The reaction is a Carathéodory function which exhibits linear growth near $\pm \infty$. We allow for resonance to occur with respect to a nonprincipal nonnegative eigenvalue, and we prove several multiplicity results. Our approach uses critical point theory, Morse theory and the reduction method (the Lyapunov-Schmidt method).

References

Sergiu Aizicovici, Nikolaos S. Papageorgiou, and Vasile Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc. 196 (2008), no. 915, vi+70. MR 2459421, DOI 10.1090/memo/0915
Herbert Amann, Saddle points and multiple solutions of differential equations, Math. Z. 169 (1979), no. 2, 127–166. MR 550724, DOI 10.1007/BF01215273
Thomas Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001), no. 1, 117–152. MR 1863294, DOI 10.1006/jfan.2001.3789
Thomas Bartsch and Zhi-Qiang Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 115–131. MR 1422008, DOI 10.12775/TMNA.1996.005
Haïm Brezis and Louis Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 939–963. MR 1127041, DOI 10.1002/cpa.3160440808
Alfonso Castro, Jorge Cossio, and Carlos Vélez, Existence of seven solutions for an asymptotically linear Dirichlet problem without symmetries, Ann. Mat. Pura Appl. (4) 192 (2013), no. 4, 607–619. MR 3081637, DOI 10.1007/s10231-011-0239-5
Alfonso Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl. (4) 120 (1979), 113–137. MR 551063, DOI 10.1007/BF02411940
Kung-ching Chang, Infinite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, vol. 6, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1196690, DOI 10.1007/978-1-4612-0385-8
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
Michael Filippakis and Nikolaos S. Papageorgiou, Multiple nontrivial solutions for resonant Neumann problems, Math. Nachr. 283 (2010), no. 7, 1000–1014. MR 2677284, DOI 10.1002/mana.200710045
Leszek Gasiński and Nikolaos S. Papageorgiou, Nonlinear analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2168068
Leszek Gasiński and Nikolaos S. Papageorgiou, Dirichlet problems with double resonance and an indefinite potential, Nonlinear Anal. 75 (2012), no. 12, 4560–4595. MR 2927120, DOI 10.1016/j.na.2011.09.014
Leszek Gasiński and Nikolaos S. Papageorgiou, Pairs of nontrivial solutions for resonant Neumann problems, J. Math. Anal. Appl. 398 (2013), no. 2, 649–663. MR 2990090, DOI 10.1016/j.jmaa.2012.09.034
Andrzej Granas and James Dugundji, Fixed point theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1987179, DOI 10.1007/978-0-387-21593-8
Helmut Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), no. 4, 493–514. MR 682663, DOI 10.1007/BF01457453
Shouchuan Hu and Nikolas S. Papageorgiou, Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities, Commun. Pure Appl. Anal. 11 (2012), no. 5, 2005–2021. MR 2911122, DOI 10.3934/cpaa.2012.11.2005
Sophia Th. Kyritsi and Nikolaos S. Papageorgiou, Multiple solutions for superlinear Dirichlet problems with an indefinite potential, Ann. Mat. Pura Appl. (4) 192 (2013), no. 2, 297–315. MR 3035141, DOI 10.1007/s10231-011-0224-z
Shibo Liu, Remarks on multiple solutions for elliptic resonant problems, J. Math. Anal. Appl. 336 (2007), no. 1, 498–505. MR 2348521, DOI 10.1016/j.jmaa.2007.01.051
Jia Quan Liu and Shu Jie Li, An existence theorem for multiple critical points and its application, Kexue Tongbao (Chinese) 29 (1984), no. 17, 1025–1027 (Chinese). MR 802575
Shibo Liu and Shujie Li, Critical groups at infinity, saddle point reduction and elliptic resonant problems, Commun. Contemp. Math. 5 (2003), no. 5, 761–773. MR 2017717, DOI 10.1142/S0219199703001129
C. R. F. Maunder, Algebraic topology, Dover Publications, Inc., Mineola, NY, 1996. Reprint of the 1980 edition. MR 1402473
D. Motreanu, V. V. Motreanu, and N. S. Papageorgiou, On resonant Neumann problems, Math. Ann. 354 (2012), no. 3, 1117–1145. MR 2983082, DOI 10.1007/s00208-011-0763-z
D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3527–3535. MR 2813384, DOI 10.1090/S0002-9939-2011-10884-0
Donal O’Regan, Nikolaos S. Papageorgiou, and George Smyrlis, Neumann problems with double resonance, Topol. Methods Nonlinear Anal. 39 (2012), no. 1, 151–173. MR 2952308
Nikolaos S. Papageorgiou and Sophia Th. Kyritsi-Yiallourou, Handbook of applied analysis, Advances in Mechanics and Mathematics, vol. 19, Springer, New York, 2009. MR 2527754, DOI 10.1007/b120946
Nikolaos S. Papageorgiou and Francesca Papalini, Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries, Israel J. Math., in press.
Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu, Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity, Recent trends in nonlinear partial differential equations. II. Stationary problems, Contemp. Math., vol. 595, Amer. Math. Soc., Providence, RI, 2013, pp. 293–315. MR 3156380, DOI 10.1090/conm/595/11801
Nikolaos S. Papageorgiou and George Smyrlis, On a class of parametric Neumann problems with indefinite and unbounded potential, Forum Math. DOI: 10.1515/forum-2012-0042.
Patrizia Pucci and James Serrin, The maximum principle, Progress in Nonlinear Differential Equations and their Applications, vol. 73, Birkhäuser Verlag, Basel, 2007. MR 2356201, DOI 10.1007/978-3-7643-8145-5
Chun-Lei Tang, Multiple solutions of Neumann problem for elliptic equations, Nonlinear Anal. 54 (2003), no. 4, 637–650. MR 1983440, DOI 10.1016/S0362-546X(03)00091-9
Chun-Lei Tang and Xing-Ping Wu, Existence and multiplicity for solutions of Neumann problem for semilinear elliptic equations, J. Math. Anal. Appl. 288 (2003), no. 2, 660–670. MR 2020187, DOI 10.1016/j.jmaa.2003.09.034
Klaus Thews, Nontrivial solutions of elliptic equations at resonance, Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), no. 1-2, 119–129. MR 566069, DOI 10.1017/S0308210500011732
J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191–202. MR 768629, DOI 10.1007/BF01449041
Xu Jia Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations 93 (1991), no. 2, 283–310. MR 1125221, DOI 10.1016/0022-0396(91)90014-Z