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

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

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.
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.

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