## Positive solutions for infinite semipositone problems on exterior domains.(English)Zbl 1249.35081

Summary: We study positive radial solutions to the problem $$-\Delta u=\lambda K(| x| )f(u)$$ for $$x\in \Omega$$, $$u=0$$ if $$| x| =r_0$$, $$u\rightarrow 0$$ as $$| x| \rightarrow \infty$$, where $$\lambda$$ is a positive parameter, $$\Delta u=\text{div}(Vu)$$ is the Laplacian of $$u,\Omega =\{x\in \mathbb {R}^n,n > 2\: | x| > r_0\}$$ 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 $$f(0^{+})=\lim _{s\rightarrow 0^{+}}f(s)<0$$. In particular, we also study infinite semipositone problems where $$f(0^{+})=\lim _{s\rightarrow 0^{+}}f(s)=-\infty$$. 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-/super-solutions.

### MSC:

 35J25 Boundary value problems for second-order elliptic equations 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35B09 Positive solutions to PDEs 35B07 Axially symmetric solutions to PDEs