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

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

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
Related Papers

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 - 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.
[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.
[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.
[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.
[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.
[6]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems,, Topol. Methods Nonlinear Anal., 4 (1994), 59.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[6]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems,, Topol. Methods Nonlinear Anal., 4 (1994), 59.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

[1]	Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315
[2]	Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436
[3]	Vladimir Lubyshev. Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem. Communications on Pure & Applied Analysis, 2009, 8 (3) : 999-1018. doi: 10.3934/cpaa.2009.8.999
[4]	Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595
[5]	Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 257-277. doi: 10.3934/dcdss.2018014
[6]	Zuzana Došlá, Mauro Marini, Serena Matucci. Global Kneser solutions to nonlinear equations with indefinite weight. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3297-3308. doi: 10.3934/dcdsb.2018252
[7]	Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
[8]	Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann problems with indefinite potential and concave terms. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2561-2616. doi: 10.3934/cpaa.2015.14.2561
[9]	Leszek Gasiński, Nikolaos S. Papageorgiou. Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1985-1999. doi: 10.3934/cpaa.2013.12.1985
[10]	Alexandre Rocha, Mário Jorge Dias Carneiro. A dynamical condition for differentiability of Mather's average action. Journal of Geometric Mechanics, 2014, 6 (4) : 549-566. doi: 10.3934/jgm.2014.6.549
[11]	M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure & Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411
[12]	Wenxian Shen, Xiaoxia Xie. Spectraltheory for nonlocal dispersal operators with time periodic indefinite weight functions and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1023-1047. doi: 10.3934/dcdsb.2017051
[13]	Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2305-2328. doi: 10.3934/dcds.2016.36.2305
[14]	Virginie Bonnaillie-Noël. Harmonic oscillators with Neumann condition on the half-line. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2221-2237. doi: 10.3934/cpaa.2012.11.2221
[15]	Henrik Garde, Stratos Staboulis. The regularized monotonicity method: Detecting irregular indefinite inclusions. Inverse Problems & Imaging, 2019, 13 (1) : 93-116. doi: 10.3934/ipi.2019006
[16]	Xin Yang Lu. Regularity of densities in relaxed and penalized average distance problem. Networks & Heterogeneous Media, 2015, 10 (4) : 837-855. doi: 10.3934/nhm.2015.10.837
[17]	Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190
[18]	Mustapha Ait Rami, John Moore. Partial stabilizability and hidden convexity of indefinite LQ problem. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 221-239. doi: 10.3934/naco.2016009
[19]	Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130
[20]	Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845

2017 Impact Factor: 1.179

American Institute of Mathematical Sciences