American Institute of Mathematical Sciences

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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), 679-719. 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), 981-1012. 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), 655-677. 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), 419-441. doi: 10.1016/0362-546X(95)00167-T.  Google Scholar

[5]

K.-C. Chang, "Methods in Nonlinear Analysis," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 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), 83-107.  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), 79-95. Google Scholar

[8]

An Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099. 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), in press. Google Scholar

[11]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Appl. Math. Sci., 74, Springer-Verlag, New York, 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), 507-531. 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), 1257-1279. 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), 68-95. 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), 679-719. 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), 981-1012. 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), 655-677. 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), 419-441. doi: 10.1016/0362-546X(95)00167-T.  Google Scholar

[5]

K.-C. Chang, "Methods in Nonlinear Analysis," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 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), 83-107.  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), 79-95. Google Scholar

[8]

An Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099. 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), in press. Google Scholar

[11]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Appl. Math. Sci., 74, Springer-Verlag, New York, 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), 507-531. 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), 1257-1279. 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), 68-95. doi: 10.1006/jdeq.2000.3812.  Google Scholar

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