American Institute of Mathematical Sciences

Advanced
May  2017, 16(3): 1083-1102. doi: 10.3934/cpaa.2017052

Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities

Guglielmo Feltrin

SISSA -International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy

* Current address: Département de Mathématique, Université de Mons, Place du Parc 20, B-7000 Mons, Belgium.

Received  July 2016 Revised  January 2017 Published  February 2017

Fund Project: Work supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2016: Problemi differenziali non lineari: esistenza, molteplicità e proprietà qualitative delle soluzioni"

We study the second order nonlinear differential equation
$ u'' + \sum\limits_{i = 1}^m {} {\alpha _i}{a_i}(x){g_i}(u) - \sum\limits_{j = 1}^{m + 1} {} {\beta _j}{b_j}(x){k_j}(u) = 0,{\rm{ }} $
where $\alpha_{i}, \beta_{j}>0$, $a_{i}(x), b_{j}(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen{[}0, L\mathclose{]}$, and the nonlinearities $g_{i}(s), k_{j}(s)$ are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation $u"+a(x)u.{p} = 0$, with $p>1$.When the positive parameters $\beta_{j}$ are sufficiently large, we prove the existence of at least $2.{m}-1$positive solutions for the Sturm-Liouville boundary value problems associated with the equation.The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets.Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.
Keywords: Superlinear indefinite problems, positive solutions, Sturm-Liouville boundary conditions, multiplicity results, Leray-Schauder topological degree.
Mathematics Subject Classification: 34B15, 34B18, 47H11.
Citation: Guglielmo Feltrin. Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1083-1102. doi: 10.3934/cpaa.2017052
References:
[1]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215.  doi: 10.1006/jfan.1996.0125.  Google Scholar

[2]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave, convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[3]

D. L. T. Anderson, Stability of time-dependent particlelike solutions in nonlinear field theories. Ⅱ, J. Math. Phys., 12 (1971), 945-952.   Google Scholar

[4]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[5]

D. Bonheure, J. M. Gomes and P. Habets, Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differential Equations, 214 (2005), 36-64.  doi: 10.1016/j.jde.2004.08.009.  Google Scholar

[6]

A. Boscaggin, G. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree, Trans. Amer. Math. Soc., to appear. Google Scholar

[7]

L. H. Erbe, S. C. Hu and H. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184 (1994), 640-648.  doi: 10.1006/jmaa.1994.1227.  Google Scholar

[8]

L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120 (1994), 743-748.  doi: 10.2307/2160465.  Google Scholar

[9]

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

[10]

G. Feltrin and F. Zanolin, Multiple positive solutions for a superlinear problem: a topological approach, J. Differential Equations, 259 (2015), 925-963.  doi: 10.1016/j.jde.2015.02.032.  Google Scholar

[11]

G. Feltrin and F. Zanolin, Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree, J. Differential Equations, 262 (2017), 4255-4291.  doi: 10.1016/j.jde.2017.01.009.  Google Scholar

[12]

M. Gaudenzi, P. Habets and F. Zanolin, An example of a superlinear problem with multiple positive solutions, Atti Sem. Mat. Fis. Univ. Modena, 51 (2003), 259-272.   Google Scholar

[13]

M. Gaudenzi, P. Habets and F. Zanolin, Positive solutions of superlinear boundary value problems with singular indefinite weight, Commun. Pure Appl. Anal., 2 (2003), 411-423.  doi: 10.3934/cpaa.2003.2.411.  Google Scholar

[14]

P. M. Girão and J. M. Gomes, Multi-bump nodal solutions for an indefinite non-homogeneous elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 797-817.  doi: 10.1017/S0308210508000474.  Google Scholar

[15]

R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations, 167 (2000), 36-72.   Google Scholar

[16]

K. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421.  doi: 10.1006/jdeq.1998.3475.  Google Scholar

[17]

R. Manásevich, F. I. Njoku and F. Zanolin, Positive solutions for the one-dimensional pLaplacian, Differential Integral Equations, 8 (1995), 213-222.   Google Scholar

[18]

R. D. Nussbaum, The fixed point index and some applications, vol. 94 of Séminaire de Mathématiques Suprieures [Seminar on Higher Mathematics], Presses de l'Université de Montréal, Montreal, QC, 1985.  Google Scholar

[19]

R. D. Nussbaum, The fixed point index and fixed point theorems, in Topological methods for ordinary differential equations (Montecatini Terme, 1991), vol. 1537 of Lecture Notes in Math. , Springer, Berlin, 1993, pp. 143-205. doi: 10.1007/BFb0085077.  Google Scholar

[20]

H.-J. Ruppen, Multiplicity results for a semilinear. elliptic differential equation with conflicting nonlinearities, J. Differential Equations, 147 (1998), 79-122.  doi: 10.1006/jdeq.1998.3419.  Google Scholar

show all references

References:
[1]

S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., 141 (1996), 159-215.  doi: 10.1006/jfan.1996.0125.  Google Scholar

[2]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave, convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[3]

D. L. T. Anderson, Stability of time-dependent particlelike solutions in nonlinear field theories. Ⅱ, J. Math. Phys., 12 (1971), 945-952.   Google Scholar

[4]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[5]

D. Bonheure, J. M. Gomes and P. Habets, Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differential Equations, 214 (2005), 36-64.  doi: 10.1016/j.jde.2004.08.009.  Google Scholar

[6]

A. Boscaggin, G. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree, Trans. Amer. Math. Soc., to appear. Google Scholar

[7]

L. H. Erbe, S. C. Hu and H. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184 (1994), 640-648.  doi: 10.1006/jmaa.1994.1227.  Google Scholar

[8]

L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120 (1994), 743-748.  doi: 10.2307/2160465.  Google Scholar

[9]

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

[10]

G. Feltrin and F. Zanolin, Multiple positive solutions for a superlinear problem: a topological approach, J. Differential Equations, 259 (2015), 925-963.  doi: 10.1016/j.jde.2015.02.032.  Google Scholar

[11]

G. Feltrin and F. Zanolin, Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree, J. Differential Equations, 262 (2017), 4255-4291.  doi: 10.1016/j.jde.2017.01.009.  Google Scholar

[12]

M. Gaudenzi, P. Habets and F. Zanolin, An example of a superlinear problem with multiple positive solutions, Atti Sem. Mat. Fis. Univ. Modena, 51 (2003), 259-272.   Google Scholar

[13]

M. Gaudenzi, P. Habets and F. Zanolin, Positive solutions of superlinear boundary value problems with singular indefinite weight, Commun. Pure Appl. Anal., 2 (2003), 411-423.  doi: 10.3934/cpaa.2003.2.411.  Google Scholar

[14]

P. M. Girão and J. M. Gomes, Multi-bump nodal solutions for an indefinite non-homogeneous elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 797-817.  doi: 10.1017/S0308210508000474.  Google Scholar

[15]

R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations, 167 (2000), 36-72.   Google Scholar

[16]

K. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421.  doi: 10.1006/jdeq.1998.3475.  Google Scholar

[17]

R. Manásevich, F. I. Njoku and F. Zanolin, Positive solutions for the one-dimensional pLaplacian, Differential Integral Equations, 8 (1995), 213-222.   Google Scholar

[18]

R. D. Nussbaum, The fixed point index and some applications, vol. 94 of Séminaire de Mathématiques Suprieures [Seminar on Higher Mathematics], Presses de l'Université de Montréal, Montreal, QC, 1985.  Google Scholar

[19]

R. D. Nussbaum, The fixed point index and fixed point theorems, in Topological methods for ordinary differential equations (Montecatini Terme, 1991), vol. 1537 of Lecture Notes in Math. , Springer, Berlin, 1993, pp. 143-205. doi: 10.1007/BFb0085077.  Google Scholar

[20]

H.-J. Ruppen, Multiplicity results for a semilinear. elliptic differential equation with conflicting nonlinearities, J. Differential Equations, 147 (1998), 79-122.  doi: 10.1006/jdeq.1998.3419.  Google Scholar

Figure 1.  The figure shows an example of $ 3 $ positive solutions to the Dirichlet problem associated with (1.1) on $ \mathopen{[}0, 3\pi\mathclose{]} $, where $ \tau = \pi $, $ \sigma = 2\pi $, $ L = 3\pi $, $ a (x) = \sin^{+}(x) $, $ b (x) = \sin^{-}(x) $ (as in the upper part of the figure), $ g (s) = s^{2} $, $ k (s) = s^{3} $ (for $ s>0 $).For $ \mu = 1 $, Theorem 1.1 ensures the existence of $ 3 $ positive solutions, whose graphs are located in the lower part of the figure
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The figure shows an example of $ 3 $ positive solutions to the equation $ u''+\alpha_{1}a_{1}(x) g_{1}(u)-\beta_{1}b_{1}(x) k_{1}(u)+\alpha_{2}a_{2}(x) g_{2}(u) = 0 $ on $ \mathopen{[}0, 5\mathclose{]} $ with $ u (0) = u'(5) = 0 $, whose graphs are located in the lower part of the figure.For this simulation we have chosen $ \alpha_{1} = 10 $, $ \alpha_{2} = 2 $, $ \beta_{1} = 20 $ and the weight functions as in the upper part of the figure, that is $ a_{1}(x) = 1 $ in $ \mathopen{[}0, 2\mathclose{]} $, $ -b_{1}(x) = -\sin (\pi x) $ in $ \mathopen{[}2, 3\mathclose{]} $, $ a_{2}(x) = 0 $ in $ \mathopen{[}3, 4\mathclose{]} $, $ a_{2}(x) = -\sin (\pi x) $ in $ \mathopen{[}4, 5\mathclose{]} $.Moreover, we have taken $ g_{1}(s) = g_{2}(s) = s\arctan (s) $ and $ k_{1}(s) = s/(1+s^{2}) $ (for $ s>0 $).Notice that $ k_{1}(s) $ has not a superlinear behavior, since $ \lim_{s\to 0^{+}}k_{1}(s)/s = 1>0 $ and $ \lim_{s\to +\infty}k_{1}(s)/s = 0 $.Then [10,Theorem 5.3] does not apply, contrary to Theorem 4.1
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

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
[2]

Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405
[3]

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
[4]

Julián López-Góme, Andrea Tellini, F. Zanolin. High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 1-73. doi: 10.3934/cpaa.2014.13.1
[5]

N. A. Chernyavskaya, L. A. Shuster. Spaces admissible for the Sturm-Liouville equation. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1023-1052. doi: 10.3934/cpaa.2018050
[6]

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
[7]

Andrea Tellini. Imperfect bifurcations via topological methods in superlinear indefinite problems. Conference Publications, 2015, 2015 (special) : 1050-1059. doi: 10.3934/proc.2015.1050
[8]

Ryuji Kajikiya, Daisuke Naimen. Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1593-1612. doi: 10.3934/cpaa.2014.13.1593
[9]

Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061
[10]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004
[11]

Peter Howard, Alim Sukhtayev. The Maslov and Morse indices for Sturm-Liouville systems on the half-line. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 983-1012. doi: 10.3934/dcds.2020068
[12]

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
[13]

Santiago Cano-Casanova. Bifurcation to positive solutions in BVPs of logistic type with nonlinear indefinite mixed boundary conditions. Conference Publications, 2013, 2013 (special) : 95-104. doi: 10.3934/proc.2013.2013.95
[14]

Rushun Tian, Zhi-Qiang Wang. Bifurcation results on positive solutions of an indefinite nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 335-344. doi: 10.3934/dcds.2013.33.335
[15]

Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-18. doi: 10.3934/jimo.2019066
[16]

Elimhan N. Mahmudov. Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints. Journal of Industrial & Management Optimization, 2020, 16 (1) : 169-187. doi: 10.3934/jimo.2018145
[17]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729
[18]

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
[19]

Antonio Iannizzotto, Nikolaos S. Papageorgiou. Existence and multiplicity results for resonant fractional boundary value problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 511-532. doi: 10.3934/dcdss.2018028
[20]

Patrick Winkert. Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (2) : 785-802. doi: 10.3934/cpaa.2013.12.785

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (19)
  • HTML views (67)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences