Infinite semipositone problems with asymptotically linear growth forcing terms

Abstract

We study the existence of positive solutions to the singular problem \begin{equation*} \begin{cases} -\Delta_p u = \lambda f(u)-\frac{1}{u^{\alpha}} & \mbox{ in } \Omega, \\ u = 0 & \mbox{ on } \partial \Omega, \end{cases} \end{equation*} where $\lambda$ is a positive parameter, $\Delta_p u =\operatorname{div}(|\nabla{u}|^{p-2}\nabla{u})$, $p > 1$, $\Omega $ is a bounded domain in $\mathbb{R}^{n}, n \geq 1$ with smooth boundary $\partial\Omega$, $0 < \alpha < 1$, and $f:[0,\infty) \rightarrow \mathbb{R}$ is a continuous function which is asymptotically $p$-linear at $\infty$. We prove the existence of positive solutions for a certain range of $\lambda$ using the method of sub-supersolutions. We also extend our study to classes of systems which have forcing terms satisfying a combined asymptotically p-linear condition at $\infty$ and to corresponding problems on exterior domains.