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.


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