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 -Δu=Vup in RN0 perturbed by a non-homogeneous potential V when p ϵ [pc,N+2N-2), where pc is the Joseph-Ludgren exponent. When p ϵ (NN-2,pc) the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k-fast decaying solution wk of -Δu = up in RN \ 0 by authors in [9]. While the fast decaying solution wk is unstable for p ϵ (pc,N+2N-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 -Δu=uN+2N-2 in RN and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V. © 2021 Yong Ma et al., published by De Gruyter 2021.Y. Ma is supported by Key R&D plan of Jiangxi Province, No:20181ACE50029, Y. Wang is supported by NNSF of China, No:12001252 and 12071189, by the Jiangxi Provincial Natural Science Foundation, No:20202ACBL201001 and 20202BAB201005, by the Science and Technology Research Project of Jiangxi Provincial Department of Education, No: 200325 and 200307, C. Torres was partially supported by CONCYTEC, Peru, 379-2019-FONDECYT “ASPECTOS CUALITATIVOS DE ECUACIONES NO-LOCALES Y APLICACIONES