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July  2014, 13(4): 1491-1512. doi: 10.3934/cpaa.2014.13.1491

Multiple solutions for a class of nonlinear Neumann eigenvalue problems

Leszek Gasiński 1, and  Nikolaos S. Papageorgiou 2,

1. 

Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków
2. 

Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received  June 2013 Revised  December 2014 Published  February 2014

We consider a parametric nonlinear equation driven by the Neumann $p$-Laplacian. Using variational methods we show that when the parameter $\lambda > \widehat{\lambda}_1$ (where $\widehat{\lambda}_1$ is the first nonzero eigenvalue of the negative Neumann $p$-Laplacian), then the problem has at least three nontrivial smooth solutions, two of constant sign (one positive and one negative) and the third nodal. In the semilinear case (i.e., $p=2$), using Morse theory and flow invariance argument, we show that the problem has three nodal solutions.
Keywords: flow invariance., critical groups, Smooth solutions of constant sign, first nonzero eigenvalue, negative gradient flow, mountain pass theorem, nodal solutions.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60, 35J92, 58E0.
Citation: Leszek Gasiński, Nikolaos S. Papageorgiou. Multiple solutions for a class of nonlinear Neumann eigenvalue problems. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1491-1512. doi: 10.3934/cpaa.2014.13.1491
References:
[1]

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 crossing nonlinearity,, \emph{Discrete Contin. Dyn. Syst.}, 25 (2009), 431. doi: 10.3934/dcds.2009.25.431. Google Scholar

[2]

A. Ambrosetti and D. Lupo, On a class of nonlinear Dirichlet problems with multiple solutions,, \emph{Nonlinear Anal.}, 8 (1984), 1145. doi: 10.1016/0362-546X(84)90116-0. Google Scholar

[3]

A. Ambrosetti and G. Mancini, Sharp nonuniqueness results for some nonlinear problems,, \emph{Nonlinear Anal.}, 3 (1979), 635. doi: 10.1016/0362-546X(79)90092-0. Google Scholar

[4]

T. Bartsch, Critical point theory on partially ordered Hilbert spaces,, \emph{J. Funct. Anal.}, 186 (2001), 117. doi: 10.1006/jfan.2001.3789. Google Scholar

[5]

H. Brézis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers,, \emph{C. R. Acad. Sci. Paris S{\'e}r. I Math.}, 317 (1993), 465. Google Scholar

[6]

A. Castro, J. Cossio and C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 123. doi: 10.3934/dcds.2013.33.123. Google Scholar

[7]

K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems,, Birkh{\, (1993). Google Scholar

[8]

G. M. Coclite and M. M. Coclite, On a Dirichlet problem in bounded domains with singular nonlinearity,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 4923. doi: 10.3934/dcds.2013.33.4923. Google Scholar

[9]

J. García Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations,, \emph{Commun. Contemp. Math.}, 2 (2000), 385. doi: 10.1142/S0219199700000190. Google Scholar

[10]

L. Gasiński and N. S. Papageorgiou, Existence of solutions and of multiple solutions for eigenvalue problems of hemivariational inequalities,, \emph{Adv. Math. Sci. Appl.}, 11 (2001), 437. Google Scholar

[11]

L. Gasiński and N. S. Papageorgiou, Multiple solutions for semilinear hemivariational inequalities at resonance,, \emph{Publ. Math. Debrecen}, 59 (2001), 121. Google Scholar

[12]

L. Gasiński and N. S. Papageorgiou, Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance,, \emph{Proc. Royal Soc. Edinburgh Section A, 131A (2001), 1091. doi: 10.1017/S0308210500001281. Google Scholar

[13]

L. Gasiński and N. S. Papageorgiou, A multiplicity result for nonlinear second order periodic equations with nonsmooth potential,, \emph{Bull. Belg. Math. Soc. Simon Stevin}, 9 (2002), 245. Google Scholar

[14]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis,, Chapman and Hall/ CRC Press, (2006). Google Scholar

[15]

L. Gasiński and N. S. Papageorgiou, Nodal and multiple constant sign solutions for resonant $p$-Laplacian equations with a nonsmooth potential,, \emph{Nonlinear Anal.}, 71 (2009), 5747. doi: 10.1016/j.na.2009.04.063. Google Scholar

[16]

L. Gasiński and N. S. Papageorgiou, Dirichlet (p,q)-equations at resonance,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2037. doi: 10.3934/dcds.2014.34.2037. Google Scholar

[17]

L. Gasiński and N. S. Papageorgiou, A pair of positive solutions for (p,q)-equations with combined nonlinearities,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 203. doi: 10.3934/cpaa.2014.13.203. Google Scholar

[18]

T. Godoy, J.-P. Gossez and S. Paczka, On the antimaximum principle for the $p$-Laplacian with indefinite weight,, \emph{Nonlinear Anal.}, 51 (2002), 449. doi: 10.1016/S0362-546X(01)00839-2. Google Scholar

[19]

S. Th. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 2469. doi: 10.3934/dcds.2013.33.2469. Google Scholar

[20]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, \emph{Nonlinear Anal.}, 12 (1988), 1203. doi: 10.1016/0362-546X(88)90053-3. Google Scholar

[21]

S. Li and Z. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems,, \emph{J. Anal. Math.}, 81 (2000), 373. doi: 10.1007/BF02788997. Google Scholar

[22]

S. A. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 815. doi: 10.3934/cpaa.2013.12.815. Google Scholar

[23]

A. Mercaldo, J. D. Rossi and S. Segura de León, C. Trombetti, Behaviour of $p$-Laplacian problems with Neumann boundary conditions when $p$ goes to 1,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 253. doi: 10.3934/cpaa.2013.12.253. Google Scholar

[24]

D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems,, \emph{J. Differential Equations}, 232 (2007), 1. doi: 10.1016/j.jde.2006.09.008. Google Scholar

[25]

D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operators,, \emph{Proc. Amer. Math. Soc.}, 139 (2011), 3527. doi: 10.1090/S0002-9939-2011-10884-0. Google Scholar

[26]

R. S. Palais, Homotopy theory of infinite dimensional manifolds,, \emph{Topology}, 5 (1966), 1. Google Scholar

[27]

E. H. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian,, \emph{J. Funct. Anal.}, 244 (2007), 63. doi: 10.1016/j.jfa.2006.11.015. Google Scholar

[28]

M. Struwe, A note on a result of Ambrosetti and Mancini,, \emph{Ann. Mat. Pura Appl.}, 131 (1982), 107. doi: 10.1007/BF01765148. Google Scholar

[29]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, Springer-Verlag, (2008). Google Scholar

[30]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues,, \emph{Nonlinear Anal.}, 48 (2002), 881. doi: 10.1016/S0362-546X(00)00221-2. Google Scholar

[31]

J. Tyagi, Multiple solutions for singular $N$-Laplace equations with a sign changing nonlinearity,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2381. doi: 10.3934/cpaa.2013.12.2381. Google Scholar

[32]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, \emph{Appl. Math. Optim.}, 12 (1984), 191. doi: 10.1007/BF01449041. Google Scholar

[33]

P. Winkert, Multiplicity results for a class of elliptic problems with nonlinear boundary condition,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 785. doi: 10.3934/cpaa.2013.12.785. Google Scholar

show all references

References:
[1]

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 crossing nonlinearity,, \emph{Discrete Contin. Dyn. Syst.}, 25 (2009), 431. doi: 10.3934/dcds.2009.25.431. Google Scholar

[2]

A. Ambrosetti and D. Lupo, On a class of nonlinear Dirichlet problems with multiple solutions,, \emph{Nonlinear Anal.}, 8 (1984), 1145. doi: 10.1016/0362-546X(84)90116-0. Google Scholar

[3]

A. Ambrosetti and G. Mancini, Sharp nonuniqueness results for some nonlinear problems,, \emph{Nonlinear Anal.}, 3 (1979), 635. doi: 10.1016/0362-546X(79)90092-0. Google Scholar

[4]

T. Bartsch, Critical point theory on partially ordered Hilbert spaces,, \emph{J. Funct. Anal.}, 186 (2001), 117. doi: 10.1006/jfan.2001.3789. Google Scholar

[5]

H. Brézis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers,, \emph{C. R. Acad. Sci. Paris S{\'e}r. I Math.}, 317 (1993), 465. Google Scholar

[6]

A. Castro, J. Cossio and C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 123. doi: 10.3934/dcds.2013.33.123. Google Scholar

[7]

K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems,, Birkh{\, (1993). Google Scholar

[8]

G. M. Coclite and M. M. Coclite, On a Dirichlet problem in bounded domains with singular nonlinearity,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 4923. doi: 10.3934/dcds.2013.33.4923. Google Scholar

[9]

J. García Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations,, \emph{Commun. Contemp. Math.}, 2 (2000), 385. doi: 10.1142/S0219199700000190. Google Scholar

[10]

L. Gasiński and N. S. Papageorgiou, Existence of solutions and of multiple solutions for eigenvalue problems of hemivariational inequalities,, \emph{Adv. Math. Sci. Appl.}, 11 (2001), 437. Google Scholar

[11]

L. Gasiński and N. S. Papageorgiou, Multiple solutions for semilinear hemivariational inequalities at resonance,, \emph{Publ. Math. Debrecen}, 59 (2001), 121. Google Scholar

[12]

L. Gasiński and N. S. Papageorgiou, Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance,, \emph{Proc. Royal Soc. Edinburgh Section A, 131A (2001), 1091. doi: 10.1017/S0308210500001281. Google Scholar

[13]

L. Gasiński and N. S. Papageorgiou, A multiplicity result for nonlinear second order periodic equations with nonsmooth potential,, \emph{Bull. Belg. Math. Soc. Simon Stevin}, 9 (2002), 245. Google Scholar

[14]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis,, Chapman and Hall/ CRC Press, (2006). Google Scholar

[15]

L. Gasiński and N. S. Papageorgiou, Nodal and multiple constant sign solutions for resonant $p$-Laplacian equations with a nonsmooth potential,, \emph{Nonlinear Anal.}, 71 (2009), 5747. doi: 10.1016/j.na.2009.04.063. Google Scholar

[16]

L. Gasiński and N. S. Papageorgiou, Dirichlet (p,q)-equations at resonance,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2037. doi: 10.3934/dcds.2014.34.2037. Google Scholar

[17]

L. Gasiński and N. S. Papageorgiou, A pair of positive solutions for (p,q)-equations with combined nonlinearities,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 203. doi: 10.3934/cpaa.2014.13.203. Google Scholar

[18]

T. Godoy, J.-P. Gossez and S. Paczka, On the antimaximum principle for the $p$-Laplacian with indefinite weight,, \emph{Nonlinear Anal.}, 51 (2002), 449. doi: 10.1016/S0362-546X(01)00839-2. Google Scholar

[19]

S. Th. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 2469. doi: 10.3934/dcds.2013.33.2469. Google Scholar

[20]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, \emph{Nonlinear Anal.}, 12 (1988), 1203. doi: 10.1016/0362-546X(88)90053-3. Google Scholar

[21]

S. Li and Z. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems,, \emph{J. Anal. Math.}, 81 (2000), 373. doi: 10.1007/BF02788997. Google Scholar

[22]

S. A. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 815. doi: 10.3934/cpaa.2013.12.815. Google Scholar

[23]

A. Mercaldo, J. D. Rossi and S. Segura de León, C. Trombetti, Behaviour of $p$-Laplacian problems with Neumann boundary conditions when $p$ goes to 1,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 253. doi: 10.3934/cpaa.2013.12.253. Google Scholar

[24]

D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems,, \emph{J. Differential Equations}, 232 (2007), 1. doi: 10.1016/j.jde.2006.09.008. Google Scholar

[25]

D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operators,, \emph{Proc. Amer. Math. Soc.}, 139 (2011), 3527. doi: 10.1090/S0002-9939-2011-10884-0. Google Scholar

[26]

R. S. Palais, Homotopy theory of infinite dimensional manifolds,, \emph{Topology}, 5 (1966), 1. Google Scholar

[27]

E. H. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian,, \emph{J. Funct. Anal.}, 244 (2007), 63. doi: 10.1016/j.jfa.2006.11.015. Google Scholar

[28]

M. Struwe, A note on a result of Ambrosetti and Mancini,, \emph{Ann. Mat. Pura Appl.}, 131 (1982), 107. doi: 10.1007/BF01765148. Google Scholar

[29]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, Springer-Verlag, (2008). Google Scholar

[30]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues,, \emph{Nonlinear Anal.}, 48 (2002), 881. doi: 10.1016/S0362-546X(00)00221-2. Google Scholar

[31]

J. Tyagi, Multiple solutions for singular $N$-Laplace equations with a sign changing nonlinearity,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2381. doi: 10.3934/cpaa.2013.12.2381. Google Scholar

[32]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, \emph{Appl. Math. Optim.}, 12 (1984), 191. doi: 10.1007/BF01449041. Google Scholar

[33]

P. Winkert, Multiplicity results for a class of elliptic problems with nonlinear boundary condition,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 785. doi: 10.3934/cpaa.2013.12.785. Google Scholar

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