Existence and separation of positive radial solutions for semilinear elliptic equations

We consider the semilinear elliptic equation Δu+K(|x|)up=0Δu+K(|x|)up=0 in RNRN for N>2N>2 and p>1p>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−ℓK(r)r−ℓK(r) with ℓ>−2ℓ>−2 around a positive constant is small near r=∞r=∞ 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 pcpc which is determined by N   and the order of the behavior of K(r)K(r) as r=|x|→0r=|x|→0 and ∞. In order to understand how subtle the structure is on K   at p=pcp=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)p=(N+2)/(N−2).