Entire solutions with asymptotic self-similarity for elliptic equations with exponential nonlinearity

We consider the elliptic equation Δu+K(|x|)eu=0 in Rn\{0} with n>2 when for ℓ>−2, K(r) behaves like rℓ near 0 or ∞. The asymptotic behavior of radial solutions at ∞ is described by −(2+ℓ)log⁡r for ℓ>−2 and −log⁡log⁡r for ℓ=−2. When r−ℓK(r)→c>0 as r→∞ and r→0, regular radial solutions at ∞ and singular radial solutions at 0 exhibit self-similarity at ∞ and 0, respectively. Singular solutions with the asymptotic self-similarity exist uniquely in the radial class. Moreover, for n≥10+4ℓ, separation of any two radial solutions with the asymptotic self-similarity may happen, while intersection of two solutions may occur for 2−2, if K(≢0) satisfies that r2K(r)→0 as r→0 and 0≤k(r)=r−ℓK(r)≤n−24(2+ℓ)inf00, then any two radial solutions do not intersect each other and each radial solution is linearly stable. When n≥10+4ℓ, we apply the global results to prove the uniqueness of positive radial solutions for the Dirichlet problem with zero data on a ball.