Positive solutions to indefinite Neumann problems when the weight has positive average

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October 2016, 36(10): 5231-5244. doi: 10.3934/dcds.2016028

Positive solutions to indefinite Neumann problems when the weight has positive average

Alberto Boscaggin ^1, and Maurizio Garrione ^2,

1.	Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, I-10123 Torino, Italy
2.	Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi 55, I-20125 Milano, Italy

Received October 2015 Revised April 2016 Published July 2016

Abstract
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We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE $$ u'' + q(t)g(u) = 0, \quad t \in [0, T], $$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(u) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type $$ x' = y, \qquad y' = h(x)y^2 + q(t), $$ with $h(x)$ a continuous function defined on the whole real line.

Keywords: average condition, Neumann problem, Indefinite weight, shooting method..

Mathematics Subject Classification: Primary: 34B15, 34B0.

Citation: Alberto Boscaggin, Maurizio Garrione. Positive solutions to indefinite Neumann problems when the weight has positive average. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5231-5244. doi: 10.3934/dcds.2016028

References:

[1]	S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475. doi: 10.1007/BF01206962.
[2]	H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Functional Analysis, 11 (1972), 346-384. doi: 10.1016/0022-1236(72)90074-2.
[3]	H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440.
[4]	F. V. Atkinson, W. N. Everitt and K. S. Ong, On the $m$-coefficient of Weyl for a differential equation with an indefinite weight function, Proc. London Math. Soc. (3), 29 (1974), 368-384.
[5]	C. Bandle, M. A. Pozio and A. Tesei, Existence and uniqueness of solutions of nonlinear Neumann problems, Math. Z., 199 (1988), 257-278. doi: 10.1007/BF01159655.
[6]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.
[7]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 553-572. doi: 10.1007/BF01210623.
[8]	A. Boscaggin, G. Feltrin and F. Zanolin, Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: A topological degree approach for the super-sublinear case, Proc. Roy. Soc. Edinburgh Sect. A., 146 (2016), 449-474. doi: 10.1017/S0308210515000621.
[9]	A. Boscaggin and M. Garrione, Multiple solutions to Neumann problems with indefinite weight and bounded nonlinearities, J. Dynam. Differential Equations, 28 (2016), 167-187. doi: 10.1007/s10884-015-9430-5.
[10]	A. Boscaggin and F. Zanolin, Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight, J. Differential Equations, 252 (2012), 2900-2921. doi: 10.1016/j.jde.2011.09.011.
[11]	A. Boscaggin and F. Zanolin, Second order ordinary differential equations with indefinite weight: the Neumann boundary value problem, Ann. Mat. Pura Appl. (4), 194 (2015), 451-478. doi: 10.1007/s10231-013-0384-0.
[12]	G. Feltrin and F. Zanolin, Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems, Adv. Differential Equations, 20 (2015), 937-982.
[13]	P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030. doi: 10.1080/03605308008820162.
[14]	J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.
[15]	P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations, Indiana Univ. Math. J., 23 (1973/74), 173-186. doi: 10.1512/iumj.1974.23.23014.

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References:

[1]	S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475. doi: 10.1007/BF01206962.
[2]	H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Functional Analysis, 11 (1972), 346-384. doi: 10.1016/0022-1236(72)90074-2.
[3]	H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440.
[4]	F. V. Atkinson, W. N. Everitt and K. S. Ong, On the $m$-coefficient of Weyl for a differential equation with an indefinite weight function, Proc. London Math. Soc. (3), 29 (1974), 368-384.
[5]	C. Bandle, M. A. Pozio and A. Tesei, Existence and uniqueness of solutions of nonlinear Neumann problems, Math. Z., 199 (1988), 257-278. doi: 10.1007/BF01159655.
[6]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.
[7]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 553-572. doi: 10.1007/BF01210623.
[8]	A. Boscaggin, G. Feltrin and F. Zanolin, Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: A topological degree approach for the super-sublinear case, Proc. Roy. Soc. Edinburgh Sect. A., 146 (2016), 449-474. doi: 10.1017/S0308210515000621.
[9]	A. Boscaggin and M. Garrione, Multiple solutions to Neumann problems with indefinite weight and bounded nonlinearities, J. Dynam. Differential Equations, 28 (2016), 167-187. doi: 10.1007/s10884-015-9430-5.
[10]	A. Boscaggin and F. Zanolin, Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight, J. Differential Equations, 252 (2012), 2900-2921. doi: 10.1016/j.jde.2011.09.011.
[11]	A. Boscaggin and F. Zanolin, Second order ordinary differential equations with indefinite weight: the Neumann boundary value problem, Ann. Mat. Pura Appl. (4), 194 (2015), 451-478. doi: 10.1007/s10231-013-0384-0.
[12]	G. Feltrin and F. Zanolin, Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems, Adv. Differential Equations, 20 (2015), 937-982.
[13]	P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030. doi: 10.1080/03605308008820162.
[14]	J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.
[15]	P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations, Indiana Univ. Math. J., 23 (1973/74), 173-186. doi: 10.1512/iumj.1974.23.23014.

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