A Dirichlet problem on the half-line for nonlinear equations with indefinite weight

TL;DR

This paper investigates the existence of positive solutions for a nonlinear second order differential equation on the half-line, allowing indefinite weights and asymptotically linear nonlinearities, with applications to periodic and unbounded cases.

Contribution

It establishes new existence results for positive solutions under indefinite weight functions and asymptotically linear nonlinearities, extending previous work to broader conditions.

Findings

Existence of positive solutions with indefinite weights.

Solutions exist even when weight functions are unbounded or periodic.

Results apply to nonlinear equations with sign-changing coefficients.

Abstract

We study the existence of positive solutions on the half-line $[0,\infty)$ for the nonlinear second order differential equation \[ \bigl(a(t)x^{\prime}\bigr)^{\prime}+b(t)F(x)=0,\quad t\geq0, \] satisfying Dirichlet type conditions, say $x(0)=0$, $\lim_{t\rightarrow\infty}x(t)=0$. The function $b$ is allowed to change sign and the nonlinearity $F$ is assumed to be asymptotically linear in a neighborhood of zero and infinity. Our results cover also the cases in which $b$ is a periodic function for large $t$ or it is unbounded from below.