Lane-Emden equations perturbed by nonhomogeneous potential in the super critical case

Our purpose of this paper is to study positive solutions of Lane-Emden equation -Delta u = Vu(p) in R-N\{0} (0.1) perturbed by a non-homogeneous potential V when p 2 [p(c), N+2/N-2), where p(c) is the Joseph-Ludgren exponent. When p is an element of(N/N-2, p(c)), the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k-fast decaying solution w(k) of -Delta u = u(p) in R-N \ {0} by authors in [9]. While the fast decaying solution w(k) is unstable for p is an element of(p(c), N+2/N-2), so these fast decaying solutions seem not able to disturbed like (0.1) by non-homogeneous potential V. A surprising observation that there exists a bounded sub solution of (0.1) from the extremal solution of -Delta u = u(N+2/N-2) in R-N and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V.