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

Multiple solutions for a class of nonlinear Neumann eigenvalue problems

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.
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

[1]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[2]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345

[3]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745

[4]

Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou. Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 405-440. doi: 10.3934/dcds.2009.24.405

[5]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003

[6]

Dumitru Motreanu, Viorica V. Motreanu, Abdelkrim Moussaoui. Location of Nodal solutions for quasilinear elliptic equations with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 293-307. doi: 10.3934/dcdss.2018016

[7]

Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737

[8]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[9]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004

[10]

Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211

[11]

François James, Nicolas Vauchelet. Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1355-1382. doi: 10.3934/dcds.2016.36.1355

[12]

Norimichi Hirano, A. M. Micheletti, A. Pistoia. Existence of sign changing solutions for some critical problems on $\mathbb R^N$. Communications on Pure & Applied Analysis, 2005, 4 (1) : 143-164. doi: 10.3934/cpaa.2005.4.143

[13]

Yuxin Ge, Monica Musso, A. Pistoia, Daniel Pollack. A refined result on sign changing solutions for a critical elliptic problem. Communications on Pure & Applied Analysis, 2013, 12 (1) : 125-155. doi: 10.3934/cpaa.2013.12.125

[14]

Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256

[15]

Francesca Faraci, Antonio Iannizzotto. Three nonzero periodic solutions for a differential inclusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 779-788. doi: 10.3934/dcdss.2012.5.779

[16]

David L. Finn. Noncompact manifolds with constant negative scalar curvature and singular solutions to semihnear elliptic equations. Conference Publications, 1998, 1998 (Special) : 262-275. doi: 10.3934/proc.1998.1998.262

[17]

Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095

[18]

Teodora-Liliana Dinu. Entire solutions of the nonlinear eigenvalue logistic problem with sign-changing potential and absorption. Communications on Pure & Applied Analysis, 2003, 2 (3) : 311-321. doi: 10.3934/cpaa.2003.2.311

[19]

Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487

[20]

Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]