On a class of semilinear elliptic equations in Rn

We establish that for n greater than or equal to 3 and p > 1, the elliptic equation Deltau + K(x)u(p) = 0 in R-n possesses separated positive entire solutions of infinite multiplicity, provided that a locally Holder continuous function K greater than or equal to O in R-n\M, satisfies K(x) = O(\x\(sigma)) at x = 0 for some sigma > -2, and K(x) = c\x\(-2) + O(\x\(-n)[log \x\](q)) near infinity for some constants c > 0 and q > 0. In the radial case K(x) = \x\(1)/1+\x\(2) with l > -2 and tau greater than or equal to 0, or K(x) \x\(lambda-2)/(1+\x\(2))(lambda\2) with lambda > 0, we investigate separation phenomena of positive radial solutions, and show that if n and p are large enough, the equation possesses a positive radial solution with initial value a at 0 for each a > 0 and a unique positive radial singular solution among which any two solutions do not intersect. (C) 2001 Elsevier Science (USA).