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Sufficient Conditions for Uniqueness of Positive Solutions and Non Existence of Sign Changing Solutions for Elliptic Dirichlet Problems

In this paper we study the uniqueness of positive solutions as well as the non existence of sign changing solutions for Dirichlet problems of the form $$\eqalign{\Delta u + g(\lambda,\ u) &= 0\quad\rm in\ \Omega,\cr u &= 0\quad\rm on\ \partial\Omega,}$$where $\Delta$ is the Laplace operator, $\Omega$ is a region in $\IR\sp{N}$, and $\lambda>0$ is a real parameter. For the particular function $g(\lambda,\ u)=\vert u\vert\sp{p}u+\lambda$, where $p={4\over N-2}$, and $\Omega$ is the unit ball in $\IR\sp{N}$ for $N\ge3$, we show that there are no sign changing solutions for small $\lambda$ and also we show that there are no large sign changing solutions for $\lambda$ in a compact set. We also prove uniqueness of positive solutions for $\lambda$ large when $g(\lambda,\ u)=\lambda f(u)$, where f is an increasing, sublinear, concave function with f(0) $<$ 0, and the exterior boundary of $\Omega$ is convex. In establishing our results we use a number of methods from non-linear functional analysis such as rescaling arguments, methods of order, estimation near the boundary, and moving plane arguments.

Identiferoai:union.ndltd.org:unt.edu/info:ark/67531/metadc279227
Date08 1900
CreatorsHassanpour, Mehran
ContributorsCastro, Alfonso, 1950-, Warchall, Henry Alexander, DeLatte, David, Iaia, Joseph A., Acevedo, Miguel F.
PublisherUniversity of North Texas
Source SetsUniversity of North Texas
LanguageEnglish
Detected LanguageEnglish
TypeThesis or Dissertation
Formativ, 41 leaves, Text
RightsPublic, Copyright, Copyright is held by the author, unless otherwise noted. All rights reserved., Hassanpour, Mehran

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