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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations,, Indiana Univ. Math. J., 23 (): 173.  doi: 10.1512/iumj.1974.23.23014.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations,, Indiana Univ. Math. J., 23 (): 173.  doi: 10.1512/iumj.1974.23.23014.  Google Scholar

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