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

### Description

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 ... continued below

iv, 41 leaves

### Creation Information

Hassanpour, Mehran August 1995.

### Context

This dissertation is part of the collection entitled: UNT Theses and Dissertations and was provided by UNT Libraries to Digital Library, a digital repository hosted by the UNT Libraries. It has been viewed 108 times . More information about this dissertation can be viewed below.

## Who

People and organizations associated with either the creation of this dissertation or its content.

### Rights Holder

For guidance see Citations, Rights, Re-Use.

• Hassanpour, Mehran

### Provided By

#### UNT Libraries

The UNT Libraries serve the university and community by providing access to physical and online collections, fostering information literacy, supporting academic research, and much, much more.

## What

Descriptive information to help identify this dissertation. Follow the links below to find similar items on the Digital Library.

### Description

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.

iv, 41 leaves

### Identifier

Unique identifying numbers for this dissertation in the Digital Library or other systems.

### Collections

This dissertation is part of the following collection of related materials.

#### UNT Theses and Dissertations

Theses and dissertations represent a wealth of scholarly and artistic content created by masters and doctoral students in the degree-seeking process. Some ETDs in this collection are restricted to use by the UNT community.

What responsibilities do I have when using this dissertation?

## When

Dates and time periods associated with this dissertation.

• August 1995

### Added to The UNT Digital Library

• March 26, 2014, 9:30 a.m.

### Description Last Updated

• April 7, 2015, 12:16 p.m.

Yesterday: 0
Past 30 days: 1
Total Uses: 108

## Interact With This Dissertation

Here are some suggestions for what to do next.