Existence of positive solutions for nonlinear systems

TL;DR

This paper investigates the existence of positive solutions for a class of nonlinear systems related to elliptic equations, providing conditions and eigenvalue characterizations using fixed point theorems in cones.

Contribution

It introduces new existence criteria for positive solutions of nonlinear systems and characterizes eigenvalue intervals, extending previous results in the study of elliptic systems.

Findings

Established existence of positive solutions under certain conditions.

Characterized eigenvalue intervals for the nonlinear systems.

Applied fixed point theorem in cones for proofs.

Abstract

This paper deals with the existence of positive solutions for the nonlinear system q(t)\phi(p(t)u'_{i}(t)))'+f^{i}(t,\textbf{u})=0,\quad 0<t<1,\quad i=1,2,...,n. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here $\textbf{u}=(u_{1},...,u_{n})$ and $f^{i}, i=1,2,...,n$ are continuous and nonnegative functions, $p(t), q(t)\hbox{\rm :} [0,1]\to (0,\oo)$ are continuous functions. Moreover, we characterize the eigenvalue intervals for (q(t)\phi(p(t)u'_{i}(t)))'+\lambda h_{i}(t)g^{i} (\textbf{u})=0, \quad 0<t<1,\quad i=1,2,...,n. The proof is based on a well-known fixed point theorem in cones.