Positive solutions for infinite semipositone problems on exterior domains

Abstract

We study positive radial solutions to the problem \begin{equation} \label{maineqn1} \begin{cases} -\Delta u = \lambda K(\left|x \right|) f(u) \quad & x \in \Omega, \\ \hskip 15pt u = 0 & \mbox {if } \left|x \right| = r_0, \\ \hskip 15pt u \rightarrow0 & \mbox{as } \left|x \right|\rightarrow \infty , \end{cases} \end{equation} where $\lambda$ is a positive parameter, $\Delta u=\mbox{div} \big(\nabla u\big)$ is the Laplacian of $u$, $\Omega=\{x\in\ \mathbb{R} ^{n},n>2 : \left|x \right|>r_0\}$ is an exterior domain and $f:(0,\infty)\rightarrow \mathbb{R}$ belongs to a class of sublinear functions at $\infty$ such that they are continuous and $f(0^+)= \lim_{s \to 0^+} f(s) < 0$. In particular we also study infinite semipositone problems where $\lim_{s\rightarrow 0^{+}} f(s) =-\infty$. Here $K: [r_0,\infty)\rightarrow(0,\infty)$ belongs to a class of continuous functions such that $\lim_{r\rightarrow \infty}K(r)=0$. We establish various existence results for such boundary value problems and also extend our results to classes of systems. We prove our results by the method of sub-/supersolutions.