Positive Solutions for Nonlinear Elliptic Equations Depending on a Parameter with Dirichlet Boundary Conditions

TL;DR

This paper establishes new existence results for positive radial solutions of nonlinear elliptic equations with parameter dependence in annular domains, using fixed point theorems under various growth conditions of the nonlinearity.

Contribution

It introduces novel existence criteria for positive solutions of elliptic equations depending on parameters, utilizing a fixed point approach under superlinear and sublinear growth conditions.

Findings

Existence of positive radial solutions under superlinear growth conditions.

Existence of positive radial solutions under sublinear growth conditions.

Application of a fixed point theorem to nonlinear elliptic equations.

Abstract

We prove new results on the existence of positive radial solutions of the elliptic equation $-\Delta u= \lambda h(|x|,u)$ in an annular domain in $\mathbb{R}^{N}, N\geq 2$. Existence of positive radial solutions are determined under the conditions that the nonlinearity function $h(t,u)$ is either superlinear or sublinear growth in $u$ or satisfies some upper and lower inequalities on $h$. Our discussion is based on a fixed point theorem due to a revised version of a fixed point theorem of Gustafson and Schmitt.