# 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 &amp; Applied Analysis, 2017, 16 (3) : 1083-1102. doi: 10.3934/cpaa.2017052
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##### References:
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
,Theorem 5.3] does not apply, contrary to Theorem 4.1">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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