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

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