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

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

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.
Citation: Alberto Boscaggin, Maurizio Garrione. Positive solutions to indefinite Neumann problems when the weight has positive average. Discrete & Continuous Dynamical Systems - A, 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.  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.  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.  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.   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.  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.   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.  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.  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.  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.  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.  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.   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.  doi: 10.1080/03605308008820162.  Google Scholar

[14]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,, Applied Mathematical Sciences, 74 (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.  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.  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.  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.   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.  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.   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.  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.  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.  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.  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.  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.   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.  doi: 10.1080/03605308008820162.  Google Scholar

[14]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,, Applied Mathematical Sciences, 74 (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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