Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient

Abstract

We establish that for n ⩾ 3 and p > 1 , the elliptic equation Δ u + K ( x ) u p = 0 in R n possesses a continuum of positive entire solutions with logarithmic decay at ∞, provided that a locally Hölder continuous function K ⩾ 0 in R n ∖ { 0 } , satisfies K ( x ) = O ( | x | σ ) at x = 0 for some σ > − 2 , and | x | 2 K ( 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 R n , J. Differential Equations 185 (2002) 225–250]. The crucial observation is that in the radial case of K ( r ) = K ( | x | ) , two fundamental weights, ( log r ) p p − 1 and r n − 2 ( log r ) − p p − 1 , appear in analyzing the asymptotic behavior of solutions.