Abstract

Through variational methods, sub-supersolution and truncation techniques we prove the existence of three nontrivial solutions for a quasilinear elliptic equation with Neumann boundary condition. We provide sign information for each of these solutions: two of them are of opposite constant sign and the third one is sign changing.

Author information

Contact details are reproduced from the original publication and may be historical.

Giuseppina Barletta

MECMAT, Università di Reggio Calabria, Via Graziella, Località Feo di Vito, 89100 Reggio Calabria, Italy

giuseppina.barletta@unirc.it

Pasquale Candito

MECMAT, Università di Reggio Calabria, Via Graziella, Località Feo di Vito, 89100 Reggio Calabria, Italy

pasquale.candito@unirc.it

Dumitru Motreanu

Dép. de Mathématiques, Université de Perpignan, Avenue Paul Alduy 52, 66860 Perpignan, France

motreanu@univ-perp.fr

Keywords

  • Quasilinear elliptic equations; sub-supersolution; critical point; Neumann problem; constant sign solutions; sign-changing solutions

Mathematics Subject Classification

35J62, 49J45

References

  1. S. Aizicovici, N. S. Papageorgiou, V. Staicu: The spectrum and an index formula for the Neumann p-Laplacian and multiple solutions for problems with a crossing nonlinearity, Discrete Contin. Dyn. Syst. 25 (2009) 431–456.https://doi.org/10.3934/dcds.2009.25.431Google Scholar
  2. G. Bonanno, P. Candito: Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. 80 (2003) 424–429.https://doi.org/10.1007/s00013-003-0479-8Google Scholar
  3. G. Barletta, N. S. Papageorgiou: A multiplicity theorem for the Neumann p-Laplacian with an asymmetric nonsmooth potential, J. Glob. Optim. 39 (2007) 365–392.https://doi.org/10.1007/s10898-007-9142-4Google Scholar
  4. S. Carl, D. Motreanu: Constant-sign and sign-changing solutions for nonlinear eigenvalue problems, Nonlinear Anal. 68 (2008) 2668–2676.https://doi.org/10.1016/j.na.2007.02.013Google Scholar
  5. S. Carl, D. Motreanu: Multiple and sign-changing solutions for the multivalued p-Laplacian equation, Math. Nachr. 283 (2010) 965–981.https://doi.org/10.1002/mana.200710049Google Scholar
  6. S. Carl, V. K. Le, D. Motreanu: Nonsmooth Variational Problems and their Inequalities. Comparison Principles and Applications, Springer Monogr. Math., Springer, New York (2007).https://doi.org/10.1007/978-0-387-46252-3Google Scholar
  7. S. Carl, K. Perera: Sign-changing and multiple solutions for the p-Laplacian, Abstr. Appl. Anal. 7 (2002) 613–625.https://doi.org/10.1155/S1085337502207010Google Scholar
  8. E. DiBenedetto: C 1,α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983) 827–850.https://doi.org/10.1016/0362-546X(83)90061-5Google Scholar
  9. L. Gasiński, N. S. Papageorgiou: Nonlinear Analysis, Chapman & Hall / CRC Press, Boca Raton (2006).Google Scholar
  10. A. Lê: Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64 (2006) 1057–1099.https://doi.org/10.1016/j.na.2005.05.056Google Scholar
  11. S. Miyajima, D. Motreanu, M. Tanaka: Multiple existence results of solutions for the Neumann problems via super- and sub-solutions, J. Funct. Anal. 262 (2012) 1921–1953.https://doi.org/10.1016/j.jfa.2011.11.028Google Scholar
  12. D. Motreanu, V. V. Motreanu, N. S. Papageorgiou: Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 10 (2011) 729–755.https://doi.org/10.2422/2036-2145.2011.3.09Google Scholar
  13. D. Motreanu, M. Tanaka: Sign-changing and constant-sign solutions for p−Laplacian problems with jumping nonlinearities, J. Differ. Equ. 249 (2010) 3352–3376.https://doi.org/10.1016/j.jde.2010.08.017Google Scholar
  14. D. Motreanu, Z. Zhang: Constant and sign changing solutions for systems of quasilinear elliptic equations, Set-Valued Var. Anal. 19 (2011) 255–269.https://doi.org/10.1007/s11228-010-0142-zGoogle Scholar
  15. J. L. Vázquez: A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization 12 (1984) 191–202.https://doi.org/10.1007/BF01449041Google Scholar
  16. P. Winkert: Sign-changing and extremal constant-sign solutions of nonlinear elliptic Neumann boundary value problems, Bound. Value Probl. 2010 (2010), Art. ID 139126.https://doi.org/10.1155/2010/139126Google Scholar
  17. P. Winkert: Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values, Adv. Differ. Equ. 15(5–6) (2010) 561–599.https://doi.org/10.57262/ade/1355854681Google Scholar
  18. P. Winkert: Multiple solution results for elliptic Neumann problems involving set-valued nonlinearities, J. Math. Anal. Appl. 377(1) (2011) 121–134.https://doi.org/10.1016/j.jmaa.2010.10.029Google Scholar
  19. J. Zhang, S. Li, Y. Wang, X. Xue: Multiple solutions for semilinear elliptic equations with Neumann boundary condition and jumping nonlinearities, J. Math. Anal. Appl. 371(2) (2010) 682–690.https://doi.org/10.1016/j.jmaa.2010.05.045Google Scholar

Suggested citation

G. Barletta, P. Candito, D. Motreanu. “Constant Sign and Sign-Changing Solutions for Quasilinear Elliptic Equations with Neumann Boundary Condition.” Journal of Convex Analysis 21 (2014), No. 1, 53–66. https://doi.org/10.68381/jca21003

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.