Existence and separation of positive radial solutions for semilinear elliptic equations

We consider the semilinear elliptic equation Delta u + K (vertical bar x vertical bar)u(P) = 0 in R-N for N > 2 and p > 1, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r(-l) K (r) with l > -2 around a positive constant is small near r = infinity and p is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent pc which is determined by N and the order of the behavior of K (r) as r = vertical bar x vertical bar -> 0 and infinity. In order to understand how subtle the structure is on K at p = pc, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p = (N + 2)/(N - 2). (C) 2014 Elsevier Inc. All rights reserved.