American Institute of Mathematical Sciences

Advanced
August  2012, 5(4): 765-777. doi: 10.3934/dcdss.2012.5.765

Multiple solutions to a Neumann problem with equi-diffusive reaction term

Giuseppina D’Aguì 1, , Salvatore A. Marano 2, and  Nikolaos S. Papageorgiou 3,

1. 

Engineering Faculty, University of Messina, 98166 Messina, Italy
2. 

Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania
3. 

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

Received  January 2011 Revised  February 2011 Published  November 2011

The existence of four solutions, one negative, one positive, and two sign-changing (namely, nodal), for a Neumann boundary-value problem with right-hand side depending on a positive parameter is established. Proofs make use of sub- and super-solution techniques as well as Morse theory.
Keywords: constant-sign solutions, critical groups, Morse identity., sub- and super-solutions, Neumann problem, nodal solutions.
Mathematics Subject Classification: Primary: 35J25, 35J70; Secondary: 58E0.
Citation: Giuseppina D’Aguì, Salvatore A. Marano, Nikolaos S. Papageorgiou. Multiple solutions to a Neumann problem with equi-diffusive reaction term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 765-777. doi: 10.3934/dcdss.2012.5.765
References:
[1]

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.  doi: 10.1007/s10231-009-0096-7.  Google Scholar

[2]

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.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[3]

T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems,, Math. Z., 233 (2000), 655.  doi: 10.1007/s002090050492.  Google Scholar

[4]

T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,, Nonlinear Anal., 28 (1997), 419.  doi: 10.1016/0362-546X(95)00167-T.  Google Scholar

[5]

K.-C. Chang, "Methods in Nonlinear Analysis,", Springer Monographs in Mathematics, (2005).   Google Scholar

[6]

A. Kristály and N. S. Papageorgiou, Multiple nontrivial solutions for Neumann problems involving the $p$-Laplacian: A Morse theoretical approach,, Adv. Nonlinear Stud., 10 (2010), 83.   Google Scholar

[7]

S. T. Kyritsi and N. S. Papageorgiou, Three nontrivial solutions for Neumann problems resonant at any positive eigenvalue,, Matematiche (Catania), 65 (2010), 79.   Google Scholar

[8]

An Lê, Eigenvalue problems for the $p$-Laplacian,, Nonlinear Anal., 64 (2006), 1057.  doi: 10.1016/j.na.2005.06.024.  Google Scholar

[9]

S. A. Marano and N. S. Papageorgiou, On a Neumann problem with $p$-Laplacian and non-coercive resonant nonlinearity,, Pacific J. Math., ().   Google Scholar

[10]

S. A. Marano and N. S. Papageorgiou, Constant-sign and nodal solutions for a Neumann problem with $p$-Laplacian and equi-diffusive reaction term,, Topol. Methods Nonlinear Anal., 38 (2011).   Google Scholar

[11]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Appl. Math. Sci., 74 (1989).   Google Scholar

[12]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations,, Manuscripta Math., 124 (2007), 507.  doi: 10.1007/s00229-007-0127-x.  Google Scholar

[13]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance,, Indiana Univ. Math. J., 58 (2009), 1257.  doi: 10.1512/iumj.2009.58.3565.  Google Scholar

[14]

W. Zou and J. Q. Liu, Multiple solutions for resonant elliptic equations via local linking theory and Morse theory,, J. Differential Equations, 170 (2001), 68.  doi: 10.1006/jdeq.2000.3812.  Google Scholar

show all references

References:
[1]

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.  doi: 10.1007/s10231-009-0096-7.  Google Scholar

[2]

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.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[3]

T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems,, Math. Z., 233 (2000), 655.  doi: 10.1007/s002090050492.  Google Scholar

[4]

T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,, Nonlinear Anal., 28 (1997), 419.  doi: 10.1016/0362-546X(95)00167-T.  Google Scholar

[5]

K.-C. Chang, "Methods in Nonlinear Analysis,", Springer Monographs in Mathematics, (2005).   Google Scholar

[6]

A. Kristály and N. S. Papageorgiou, Multiple nontrivial solutions for Neumann problems involving the $p$-Laplacian: A Morse theoretical approach,, Adv. Nonlinear Stud., 10 (2010), 83.   Google Scholar

[7]

S. T. Kyritsi and N. S. Papageorgiou, Three nontrivial solutions for Neumann problems resonant at any positive eigenvalue,, Matematiche (Catania), 65 (2010), 79.   Google Scholar

[8]

An Lê, Eigenvalue problems for the $p$-Laplacian,, Nonlinear Anal., 64 (2006), 1057.  doi: 10.1016/j.na.2005.06.024.  Google Scholar

[9]

S. A. Marano and N. S. Papageorgiou, On a Neumann problem with $p$-Laplacian and non-coercive resonant nonlinearity,, Pacific J. Math., ().   Google Scholar

[10]

S. A. Marano and N. S. Papageorgiou, Constant-sign and nodal solutions for a Neumann problem with $p$-Laplacian and equi-diffusive reaction term,, Topol. Methods Nonlinear Anal., 38 (2011).   Google Scholar

[11]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Appl. Math. Sci., 74 (1989).   Google Scholar

[12]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations,, Manuscripta Math., 124 (2007), 507.  doi: 10.1007/s00229-007-0127-x.  Google Scholar

[13]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance,, Indiana Univ. Math. J., 58 (2009), 1257.  doi: 10.1512/iumj.2009.58.3565.  Google Scholar

[14]

W. Zou and J. Q. Liu, Multiple solutions for resonant elliptic equations via local linking theory and Morse theory,, J. Differential Equations, 170 (2001), 68.  doi: 10.1006/jdeq.2000.3812.  Google Scholar

[1]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
[2]

Vladimir Lubyshev. Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem. Communications on Pure & Applied Analysis, 2009, 8 (3) : 999-1018. doi: 10.3934/cpaa.2009.8.999
[3]

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
[4]

Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212
[5]

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
[6]

Monica Musso, Donato Passaseo. Multiple solutions of Neumann elliptic problems with critical nonlinearity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 301-320. doi: 10.3934/dcds.1999.5.301
[7]

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
[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]

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
[10]

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
[11]

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
[12]

Xiao-Jing Zhong, Chun-Lei Tang. The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem. Communications on Pure & Applied Analysis, 2017, 16 (2) : 611-628. doi: 10.3934/cpaa.2017030
[13]

Zhongyuan Liu. Nodal Bubble-Tower Solutions for a semilinear elliptic problem with competing powers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5299-5317. doi: 10.3934/dcds.2017230
[14]

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
[15]

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
[16]

M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057
[17]

Monica Musso, A. Pistoia. Sign changing solutions to a Bahri-Coron's problem in pierced domains. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 295-306. doi: 10.3934/dcds.2008.21.295
[18]

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
[19]

Lorena Bociu, Petronela Radu, Daniel Toundykov. Errata: Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping. Evolution Equations & Control Theory, 2014, 3 (2) : 349-354. doi: 10.3934/eect.2014.3.349
[20]

Lorena Bociu, Petronela Radu, Daniel Toundykov. Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping. Evolution Equations & Control Theory, 2013, 2 (2) : 255-279. doi: 10.3934/eect.2013.2.255

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences