# American Institute of Mathematical Sciences

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 and 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. [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. [3] D. L. T. Anderson, Stability of time-dependent particlelike solutions in nonlinear field theories. Ⅱ, J. Math. Phys., 12 (1971), 945-952. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] R. Manásevich, F. I. Njoku and F. Zanolin, Positive solutions for the one-dimensional pLaplacian, Differential Integral Equations, 8 (1995), 213-222. [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. [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. [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.

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##### 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. [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. [3] D. L. T. Anderson, Stability of time-dependent particlelike solutions in nonlinear field theories. Ⅱ, J. Math. Phys., 12 (1971), 945-952. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] R. Manásevich, F. I. Njoku and F. Zanolin, Positive solutions for the one-dimensional pLaplacian, Differential Integral Equations, 8 (1995), 213-222. [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. [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. [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.
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
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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