Ordering properties of positive solutions for a class of $ \varphi $-Laplacian quasilinear Dirichlet problems

Abstract

<abstract><p>We study ordering properties of positive solutions $ u $ for the one-dimensional $ \varphi $-Laplacian quasilinear Dirichlet problem</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left \{\begin{array}{l} -\left (\varphi (u^{ \prime })\right )^{ \prime } = \lambda f (u) , \;\; -L &lt;x &lt;L, \\ u ( -L) = u (L) = 0, \end{array}\right . \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ \lambda, L &gt; 0 $ are two parameters. Assume that $ \varphi \in C (-\kappa, \kappa) \cap C^{2} ((-\kappa, 0) \cup (0, \kappa)) $ is odd for some positive $ \kappa \leq \infty, $ and $ \varphi ^{ \prime } (t) &gt; 0 $ for all $ t \in (-\kappa, 0) \cup (0, \kappa) $ and $ f \in C[0, \eta) $, $ f (0) \geq 0 $, $ f (u) &gt; 0 $ on $ (0, \eta) $ for some positive $ \eta \leq \infty $, where either $ \eta = \infty $, or $ \eta &lt; \infty $ with $ \lim_{u \rightarrow \eta ^{ -}}f (u) = \infty $ or $ \lim_{u \rightarrow \eta ^{ -}}f (u) = 0 $. Some applications are given, including $ f (u) = u^{p} $ ($ p &gt; 0 $)$, $ $ u^{p} +u^{q} $ ($ 0 &lt; p &lt; q &lt; \infty $), $ \frac{1}{(1 -u)^{p}} $ $ (p &gt; 0), $ $ \exp (u), \; \exp \left({\frac{{au}}{{a + u}}} \right) $ ($ a &gt; 0 $)$, $ and $ \frac{1}{(1 -u)^{2}} -\frac{\varepsilon ^{2}}{(1 -u)^{4}} $ ($ \varepsilon \in (0, 1) $).</p></abstract>