Multiple solutions to a Neumann problem withequi-diffusive reaction term

Giuseppina D’Aguì; Salvatore A. Marano; Nikolaos S. Papageorgiou

doi:10.3934/dcdss.2012.5.765

Article Contents

2012, Volume 5, Issue 4: 765-777. Doi: 10.3934/dcdss.2012.5.765

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

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

1.
Engineering Faculty, University of Messina, 98166 Messina
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: December 31, 2010

Revised: January 31, 2011

Published: July 2012

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J25, 35J70; Secondary: 58E05.

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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.