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

Previous Article
Three nonzero periodic solutions for a differential inclusion
DCDS-S Home
This Issue
Next Article
Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian

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

Abstract
Related Papers

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 and 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), 679-719. doi: 10.1007/s10231-009-0096-7.
[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), 981-1012. doi: 10.1016/0362-546X(83)90115-3.
[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), 655-677. doi: 10.1007/s002090050492.
[4]	T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28 (1997), 419-441. doi: 10.1016/0362-546X(95)00167-T.
[5]	K.-C. Chang, "Methods in Nonlinear Analysis," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
[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), 83-107.
[7]	S. T. Kyritsi and N. S. Papageorgiou, Three nontrivial solutions for Neumann problems resonant at any positive eigenvalue, Matematiche (Catania), 65 (2010), 79-95.
[8]	An Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099. doi: 10.1016/j.na.2005.06.024.
[9]	S. A. Marano and N. S. Papageorgiou, On a Neumann problem with $p$-Laplacian and non-coercive resonant nonlinearity, Pacific J. Math., in press.
[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), in press.
[11]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Appl. Math. Sci., 74, Springer-Verlag, New York, 1989.
[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), 507-531. doi: 10.1007/s00229-007-0127-x.
[13]	D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance, Indiana Univ. Math. J., 58 (2009), 1257-1279. doi: 10.1512/iumj.2009.58.3565.
[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), 68-95. doi: 10.1006/jdeq.2000.3812.

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), 679-719. doi: 10.1007/s10231-009-0096-7.
[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), 981-1012. doi: 10.1016/0362-546X(83)90115-3.
[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), 655-677. doi: 10.1007/s002090050492.
[4]	T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28 (1997), 419-441. doi: 10.1016/0362-546X(95)00167-T.
[5]	K.-C. Chang, "Methods in Nonlinear Analysis," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
[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), 83-107.
[7]	S. T. Kyritsi and N. S. Papageorgiou, Three nontrivial solutions for Neumann problems resonant at any positive eigenvalue, Matematiche (Catania), 65 (2010), 79-95.
[8]	An Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099. doi: 10.1016/j.na.2005.06.024.
[9]	S. A. Marano and N. S. Papageorgiou, On a Neumann problem with $p$-Laplacian and non-coercive resonant nonlinearity, Pacific J. Math., in press.
[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), in press.
[11]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Appl. Math. Sci., 74, Springer-Verlag, New York, 1989.
[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), 507-531. doi: 10.1007/s00229-007-0127-x.
[13]	D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance, Indiana Univ. Math. J., 58 (2009), 1257-1279. doi: 10.1512/iumj.2009.58.3565.
[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), 68-95. doi: 10.1006/jdeq.2000.3812.

[1]	Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 1999, 5 (2) : 301-320. doi: 10.3934/dcds.1999.5.301
[7]	Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure and Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[8]	Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
[9]	Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 2017, 37 (10) : 5299-5317. doi: 10.3934/dcds.2017230
[14]	Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[15]	M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057
[16]	Monica Musso, A. Pistoia. Sign changing solutions to a Bahri-Coron's problem in pierced domains. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 295-306. doi: 10.3934/dcds.2008.21.295
[17]	Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737
[18]	Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
[19]	Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292
[20]	Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
Giuseppina D’Aguì Salvatore A. Marano Nikolaos S. Papageorgiou
on Google Scholar
Giuseppina D’Aguì Salvatore A. Marano Nikolaos S. Papageorgiou

[Back to Top]