Osaka Journal of Mathematics

Multiple solutions for superlinear $p$-Laplacian Neumann problems

Sergiu Aizicovici, Nikolaos S. Papageorgiou, and Vasile Staicu

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Abstract

Our main goal is to prove the existence of multiple solutions with precise sign information for a Neumann problem driven by the $p$-Laplacian differential operator with a ($p-1$)-superlinear term which does not satisfy the Ambrosetti--Rabinowitz condition. Using minimax methods we show that the problem has five nontrivial smooth solutions, two positive, two negative and the fifth nodal. In the semilinear case ($p = 2$), using Morse theory, we produce a second nodal solution (for a total of six nontrivial smooth solutions).

Article information

Source
Osaka J. Math., Volume 49, Number 3 (2012), 699-740.

Dates
First available in Project Euclid: 15 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1350306594

Mathematical Reviews number (MathSciNet)
MR2993064

Zentralblatt MATH identifier
1260.35037

Subjects
Primary: 35J25: Boundary value problems for second-order elliptic equations 35J70: Degenerate elliptic equations 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

Citation

Aizicovici, Sergiu; Papageorgiou, Nikolaos S.; Staicu, Vasile. Multiple solutions for superlinear $p$-Laplacian Neumann problems. Osaka J. Math. 49 (2012), no. 3, 699--740. https://projecteuclid.org/euclid.ojm/1350306594


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References

  • S. Aizicovici, N.S. Papageorgiou and V. Staicu: Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Mem. Amer. Math. Soc. 196, 2008.
  • S. Aizicovici, N.S. 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.
  • S. Aizicovici, N.S. Papageorgiou and V. Staicu: On a $p$-superlinear Neumann $p$-Laplacian equation, Topol. Methods Nonlinear Anal. 34 (2009), 111–130.
  • S. Aizicovici, N.S. Papageorgiou and 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.
  • G. Anello: Existence of infinitely many weak solutions for a Neumann problem, Nonlinear Anal. 57 (2004), 199–209.
  • G. Barletta and N.S. Papageorgiou: A multiplicity theorem for the Neumann $p$-Laplacian with an asymmetric nonsmooth potential, J. Global Optim. 39 (2007), 365–392.
  • 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.
  • T. Bartsch and Z. Liu: On a superlinear elliptic $p$-Laplacian equation, J. Differential Equations 198 (2004), 149–175.
  • P.A. Binding, P. Drábek and Y.X. Huang: Existence of multiple solutions of critical quasilinear elliptic Neumann problems, Nonlinear Anal. 42 (2000), Ser. A: Theory Methods, 613–629.
  • G. Bonanno and P. Candito: Three solutions to a Neumann problem for elliptic equations involving the $p$-Laplacian, Arch. Math. (Basel) 80 (2003), 424–429.
  • F. Cammaroto, A. Chinn\`\i and B. Di Bella: Some multiplicity results for quasilinear Neumann problems, Arch. Math. (Basel) 86 (2006), 154–162.
  • D.G. Costa and C.A. Magalhães: Existence results for perturbations of the $p$-Laplacian, Nonlinear Anal. 24 (1995), 409–418.
  • E.N. Dancer and Y. Du: A note on multiple solutions of some semilinear elliptic problems, J. Math. Anal. Appl. 211 (1997), 626–640.
  • N. Dunford and J.T. Schwartz: Linear Operators, I, Interscience Publishers, Inc., New York, 1958.
  • G. Fei: On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differential Equations (2002), 1–12.
  • M. Filippakis, L. Gasiński and N.S. Papageorgiou: Multiplicity results for nonlinear Neumann problems, Canad. J. Math. 58 (2006), 64–92.
  • J.P. Garcí a 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.
  • L. Gasiński and N.S. Papageorgiou: Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, FL, 2006.
  • T. Godoy, J.-P. Gossez and S. Paczka: On the antimaximum principle for the $p$-Laplacian with indefinite weight, Nonlinear Anal. 51 (2002), 449–467.
  • 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.
  • S. Heikkilä and V. Lakshmikantham: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Dekker, New York, 1994.
  • A. Iannizzotto and N.S. Papageorgiou: Existence of three nontrivial solutions for nonlinear Neumann hemivariational inequalities, Nonlinear Anal. 70 (2009), 3285–3297.
  • C. Li, S. Li and J. Liu: Splitting theorem, Poincaré-Hopf theorem and jumping nonlinear problems, J. Funct. Anal. 221 (2005), 439–455.
  • G.M. Lieberman: Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.
  • \begingroup J. Mawhin and M. Willem: Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. \endgroup
  • D. Motreanu, V.V. Motreanu and N.S. Papageorgiou: Multiple nontrivial solutions for nonlinear eigenvalue problems, Proc. Amer. Math. Soc. 135 (2007), 3649–3658.
  • D. Motreanu and N.S. Papageorgiou: Existence and multiplicity of solutions for Neumann problems, J. Differential Equations 232 (2007), 1–35.
  • E.H. Papageorgiou and N.S. Papageorgiou: A multiplicity theorem for problems with the $p$-Laplacian, J. Funct. Anal. 244 (2007), 63–77.
  • N.S. Papageorgiou, E.M. Rocha and V. Staicu: Multiplicity theorems for superlinear elliptic problems, Calc. Var. Partial Differential Equations 33 (2008), 199–230.
  • B. Ricceri: Infinitely many solutions of the Neumann problem for elliptic equations involving the $p$-Laplacian, Bull. London Math. Soc. 33 (2001), 331–340.
  • J.L. Vázquez: A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.
  • X. Wu and K.-K. Tan: On existence and multiplicity of solutions of Neumann boundary value problems for quasi-linear elliptic equations, Nonlinear Anal. 65 (2006), 1334–1347.