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2004 Multiplicity of positive solutions for $N$-Laplace equation in a ball
S. Prashanth, K. Sreenadh
Differential Integral Equations 17(5-6): 709-719 (2004). DOI: 10.57262/die/1356060357


Let $N\ge 2,$ $ B_{R}\subset \mathbb R^{N}$ denote the open ball of radius $R$ about the origin. Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous map which behaves like $e^{s^{p}}$ for some $p\in [1,\frac{N}{N-1}]$ as $s\rightarrow \infty$ and like $s^{q}$ for some $q\in (0, N-1)$ as $s\rightarrow 0$. Then we show that there exists $\Lambda \in (0,\infty)$ such that the following problem, $$(P_{\lambda})\hspace{3cm}\left\{ \begin{array}{cllll}\left. \begin{array}{rllll} -\Delta_{N}u & = & \lambda f(u)\\ u & > & 0 \end{array}\right\} \;\; \text{in} \; B_{1}(0), \\[2mm] u\;\;=\;\; 0 \;\;\text{on}\;\; \partial B_{1}(0), \end{array}\right. $$ admits at least two solutions for all $\lambda \in (0,\Lambda)$ and no solution for $\lambda >\Lambda$.


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S. Prashanth. K. Sreenadh. "Multiplicity of positive solutions for $N$-Laplace equation in a ball." Differential Integral Equations 17 (5-6) 709 - 719, 2004.


Published: 2004
First available in Project Euclid: 21 December 2012

zbMATH: 1174.35389
MathSciNet: MR2054944
Digital Object Identifier: 10.57262/die/1356060357

Primary: 34B15
Secondary: 34B18 , 35J60 , 35P30

Rights: Copyright © 2004 Khayyam Publishing, Inc.

Vol.17 • No. 5-6 • 2004
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