Taiwanese Journal of Mathematics

EXPLICIT NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF NONNEGATIVE SOLUTIONS OF A p-LAPLACIAN BLOW-UP PROBLEM

Pei-Yu Huang, Ming-Ting Shieh, and Shin-Hwa Wang

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Abstract

We establish explicit necessary and sufficient conditions for the existence of nonnegative solutions of the $p$-Laplacian boundary blow-up problem \begin{equation*} \left\{ \begin{array}{l} \left( \varphi _{p}(u^{\prime }(x))\right) ^{\prime }=\lambda f(u(x)), \, 0 \lt x \lt 1, \\ \lim\limits_{x\rightarrow 0^{+}}u(x)=\infty =\lim\limits_{x\rightarrow 1^{-}}u(x), \end{array} \right. \end{equation*} where $p>1$, $\varphi _{p}\left( y\right) =\left\vert y\right\vert ^{p-2}y$ and $\left( \varphi _{p}(u^{\prime })\right) ^{\prime }$ is the one-dimensional $p$-Laplacian, $\lambda $ is a positive bifurcation parameter and $f$ is a locally Lipschitz continuous function on $[0,\infty)$. The gap is extremely small between the explicit necessary condition and the explicit sufficient condition for the existence of nonnegative solutions. Our results improve and extend some main results of Anuradha, Brown and Shivaji [2] and of Wang [30] from $p=2$ to any $p\gt 1$.

Article information

Source
Taiwanese J. Math., Volume 13, Number 3 (2009), 1077-1093.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405461

Digital Object Identifier
doi:10.11650/twjm/1500405461

Mathematical Reviews number (MathSciNet)
MR2526360

Zentralblatt MATH identifier
1194.34027

Subjects
Primary: 34B15: Nonlinear boundary value problems 34C23: Bifurcation [See also 37Gxx]

Keywords
$p$-Laplacian boundary blow-up problem nonnegative solution existence multiplicity

Citation

Huang, Pei-Yu; Shieh, Ming-Ting; Wang, Shin-Hwa. EXPLICIT NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF NONNEGATIVE SOLUTIONS OF A p-LAPLACIAN BLOW-UP PROBLEM. Taiwanese J. Math. 13 (2009), no. 3, 1077--1093. doi:10.11650/twjm/1500405461. https://projecteuclid.org/euclid.twjm/1500405461


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