On a Class of Semilinear Elliptic Equations in Rn

We establish that for n 3 and p>1, the elliptic equation Δu+K(x)up=0 in Rn possesses separated positive entire solutions of infinite multiplicity, provided that a locally Hölder continuous function K 0 in Rn\{0}, satisfies K(x)=O( x σ) at x=0 for some σ>−2, and K(x)=c x −2+O( x −n[log x ]q) near ∞ for some constants c>0 and q>0. In the radial case with l>−2 and τ 0, or with λ>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 α at 0 for each α>0 and a unique positive radial singular solution among which any two solutions do not intersect.