Radial symmetry of positive solutions of nonlinear elliptic equations

Abstract

We study the radial symmetry and asymptotic behavior at x=∞ of positive solutions of Δu=φ(single rule fence sign x single rule fence sign) uλ,x∈Rn and single rule fence sign x single rule fence sign large where λ1 and φ(r) is positive and continuous for r large. In particular we give conditions on φ which guarantee that each positive solution of (*) satisfies u(x)ū(single rule fence sign x) single rule fence sign →1as single rule fence sign x single rule fence sign →∞ where ū(r) is the average of u on the sphere single rule fence sign x single rule fence sign=r. © 1999 Academic Press.