Existence and separation of positive radial solutions for semilinear elliptic equations

摘要

We consider thesemilinear elliptic equation Δu +K(|x|)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~(-?)K(r) with ? >-2 around a positive constant is small near r=∞ and pis 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 p_c which is determined by Nand the order of the behavior of K(r) as r=|x| →0 and ∞. In order to understand how subtle the structure is on K at p=p_c, 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).