American Institute of Mathematical Sciences

Advanced
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

[1]

Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020292
[2]

Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028
[3]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[4]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454
[5]

Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326
[6]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448
[7]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011
[8]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[9]

Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020034
[10]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390
[11]

Joan Carles Tatjer, Arturo Vieiro. Dynamics of the QR-flow for upper Hessenberg real matrices. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1359-1403. doi: 10.3934/dcdsb.2020166
[12]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389
[13]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[14]

Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85
[15]

Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347
[16]

Kohei Nakamura. An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1093-1102. doi: 10.3934/dcdss.2020385
[17]

Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390
[18]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234
[19]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374
[20]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences