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

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

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".
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  • 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.

    Mathematics Subject Classification: 34B15, 34B18, 47H11.


    \begin{equation} \\ \end{equation}
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  • 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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