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

Kuo-Chih Hung; Shin-Hwa Wang; Jhih-Jyun Zeng; Kuo-Chih Hung; Shin-Hwa Wang; Jhih-Jyun Zeng

doi:10.3934/era.2022097

Electronic Research Archive

2022, Volume 30, Issue 5: 1918-1935. doi: 10.3934/era.2022097

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Research article

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

1.
Fundamental General Education Center, National Chin-Yi University of Technology, Taichung 411, Taiwan
2.
Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

Received: 28 January 2022 Revised: 22 March 2022 Accepted: 27 March 2022 Published: 08 April 2022

We study ordering properties of positive solutions $ u $ for the one-dimensional $ \varphi $-Laplacian quasilinear Dirichlet problem

$ \begin{equation*} \left \{\begin{array}{l} -\left (\varphi (u^{ \prime })\right )^{ \prime } = \lambda f (u) , \;\; -L <x <L, \\ u ( -L) = u (L) = 0, \end{array}\right . \end{equation*} $

where $ \lambda, L > 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) > 0 $ for all $ t \in (-\kappa, 0) \cup (0, \kappa) $ and $ f \in C[0, \eta) $, $ f (0) \geq 0 $, $ f (u) > 0 $ on $ (0, \eta) $ for some positive $ \eta \leq \infty $, where either $ \eta = \infty $, or $ \eta < \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 > 0 $)$, $ $ u^{p} +u^{q} $ ($ 0 < p < q < \infty $), $ \frac{1}{(1 -u)^{p}} $ $ (p > 0), $ $ \exp (u), \; \exp \left({\frac{{au}}{{a + u}}} \right) $ ($ a > 0 $)$, $ and $ \frac{1}{(1 -u)^{2}} -\frac{\varepsilon ^{2}}{(1 -u)^{4}} $ ($ \varepsilon \in (0, 1) $).
- prescribed mean curvature problem,
- $ m $-Laplacian problem,
- $ (m, n) $-Laplacian problem,
- positive solution,
- bifurcation diagram,
- ordering property
Citation: Kuo-Chih Hung, Shin-Hwa Wang, Jhih-Jyun Zeng. Ordering properties of positive solutions for a class of $ \varphi $-Laplacian quasilinear Dirichlet problems[J]. Electronic Research Archive, 2022, 30(5): 1918-1935. doi: 10.3934/era.2022097

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Abstract

We study ordering properties of positive solutions $ u $ for the one-dimensional $ \varphi $-Laplacian quasilinear Dirichlet problem

$ \begin{equation*} \left \{\begin{array}{l} -\left (\varphi (u^{ \prime })\right )^{ \prime } = \lambda f (u) , \;\; -L <x <L, \\ u ( -L) = u (L) = 0, \end{array}\right . \end{equation*} $

where $ \lambda, L > 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) > 0 $ for all $ t \in (-\kappa, 0) \cup (0, \kappa) $ and $ f \in C[0, \eta) $, $ f (0) \geq 0 $, $ f (u) > 0 $ on $ (0, \eta) $ for some positive $ \eta \leq \infty $, where either $ \eta = \infty $, or $ \eta < \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 > 0 $)$, $ $ u^{p} +u^{q} $ ($ 0 < p < q < \infty $), $ \frac{1}{(1 -u)^{p}} $ $ (p > 0), $ $ \exp (u), \; \exp \left({\frac{{au}}{{a + u}}} \right) $ ($ a > 0 $)$, $ and $ \frac{1}{(1 -u)^{2}} -\frac{\varepsilon ^{2}}{(1 -u)^{4}} $ ($ \varepsilon \in (0, 1) $).

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