On a nonlinear eigenvalue problem in ODE (Q1766709)

The author studies the following variant of the logistic differential equation with diffusion \[ -du''(x)=g(x)u(x)-u^2(x)\text{ for }x\in\mathbb{R}, \] where \(d\) is a positive parameter and \(g\) is a smooth function. As the main result he proves that if \(g(x)<0\) in a neighborhood of infinity, then every positive solution \(u\) must tend to zero as \(| x|\to\infty\).