On a Class of Semilinear Elliptic Equations in Rn

Abstract

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 K(x)=∣x∣l1+∣x∣τ with l>−2 and τ⩾0, or K(x)=∣x∣λ−2(1+∣x∣2)λ/2 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.