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

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