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

Alberto  Boscaggin; Maurizio  Garrione

doi:10.3934/dcds.2016028

Article Contents

2016, Volume 36, Issue 10: 5231-5244. Doi: 10.3934/dcds.2016028

This issue Previous Article Strong coincidence and overlap coincidence Next Article Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems

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
2.
Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi 55, I-20125 Milano

Received: September 30, 2015

Revised: March 31, 2016

Published: May 2017

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 34B15, 34B08.

Citation:

Related Papers

Cited by

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.