A complete classification of bifurcation diagrams for a class of (p,q)-Laplace equations

Kajikiya, Ryuji; Sim, Inbo; Tanaka, Satoshi

doi:10.1016/j.jmaa.2018.02.049

Cited by 3 publications

(2 citation statements)

References 28 publications

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“…defined on W 1,r (Ω) or W 1,r 0 (Ω). Some uniqueness results have been proved in [5,14,23,27] for this class of problems under Dirichlet boundary conditions. Note that this functional corresponds to (1.1) with h(t) = 1 + t r p −1 .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Some uniqueness results in quasilinear subhomogeneous problems

Quoirin

2021

Arch. Math.

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We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in [6]. We apply it to generalized p-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative solutions. Based on a generalized hidden convexity result, we show that uniqueness holds among strongly positive solutions and nonnegative global minimizers. Problems involving nonhomogeneous operators as the socalled (p, r)-Laplacian are also treated.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Now, if X = W 1,r (Ω) then integrating the equation we see that A is empty. The existence and uniqueness of a positive critical point has been proved in [14] for the Dirichlet problem with N = 1 and a a positive constant. Assume now that a changes sign.…”

Section: Examplesmentioning

confidence: 99%

Some uniqueness results in quasilinear subhomogeneous problems

Quoirin

2021

Arch. Math.

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Positive solutions to classes of infinite semipositone (p,q)-Laplace problems with nonlinear boundary conditions

Sim

Son

2021

Journal of Mathematical Analysis and Applications

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

Hung

Wang

Zeng

2022

era

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<abstract>We study ordering properties of positive solutions $ u $ for the one-dimensional $ \varphi $-Laplacian quasilinear Dirichlet problem <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left \{\begin{array}{l} -\left (\varphi (u^{ \prime })\right )^{ \prime } = \lambda f (u) , \;\; -L <x <L, \ u ( -L) = u (L) = 0, \end{array}\right . \end{equation*} $\end{document} </tex-math></disp-formula> 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 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) $).</abstract>

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A complete classification of bifurcation diagrams for a class of (p,q)-Laplace equations

Cited by 3 publications

References 28 publications

Some uniqueness results in quasilinear subhomogeneous problems

Some uniqueness results in quasilinear subhomogeneous problems

Positive solutions to classes of infinite semipositone (p,q)-Laplace problems with nonlinear boundary conditions

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

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