Using a fixed point index the author proves several existence theorems for positive solutions \(u\in D^{1,p}_0\) of the equation \[ -\text{div}(|\nabla u|^{p-2}\nabla u)= \lambda a(x)| u|^{p- 2}u+ f(x,u,\lambda), \] where \(\Omega\) is an unbounded domain in \(\mathbb{R}^N\) with smooth boundary, \(1 0\) with \(a\in L^\infty(\Omega)\cap L^1(\Omega)\), \(\lambda>0\) is a real parameter and \(f\) is a Carathéodory function satisfying certain growth and limiting conditions.