Connected sets of positive solutions of elliptic systems in exterior domains

The existence of infinitely many connected sets of positive solutions for a certain elliptic system is investigated in this paper. We consider semilinear equations with perturbed Laplace operators described in an exterior domain. We show that each of these solutions $$\mathbf {u}=( u_{1},u_{2})$$u=(u1,u2) has the minimal asymptotic decay, namely $$ u_{i}(x)=O(||x||^{2-n})$$ui(x)=O(||x||2-n) as $$||x||\rightarrow \infty ,$$||x||→∞,$$i=1,2,$$i=1,2, and finite energy in a neighborhood of infinity. Our main tool is the sub and super-solutions theorem which is based on the Sattinger’s iteration procedure. We do not need any growth assumptions concerning nonlinearities.