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

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

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.
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. AmbrosettiH. 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. BonheureJ. 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. ErbeS. 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. GaudenziP. 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. GaudenziP. 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. ManásevichF. 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. AmbrosettiH. 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. BonheureJ. 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. ErbeS. 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. GaudenziP. 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. GaudenziP. 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. ManásevichF. 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 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
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