Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient

AbstractWe establish that for n⩾3 and p>1, the elliptic equation Δu+K(x)up=0 in Rn possesses a continuum of positive entire solutions with logarithmic decay at ∞, 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 |x|2K(x)=c+O([log|x|]−θ) near ∞ for some constants c>0 and θ>1. The continuum contains at least countably many solutions among which any two do not intersect. This is an affirmative answer to an open question raised in [S. Bae, T.K. Chang, On a class of semilinear elliptic equations in Rn, J. Differential Equations 185 (2002) 225–250]. The crucial observation is that in the radial case of K(r)=K(|x|), two fundamental weights, (logr)pp−1 and rn−2(logr)−pp−1, appear in analyzing the asymptotic behavior of solutions