Asymptotic behavior of large radial solutions of a polyharmonic equation with superlinear growth

This paper concerns the blow-up behavior of large radial solutions of polyharmonic equations with power nonlinearities and positive radial weights. Specifically, we consider radially symmetric solutions of Δmu=c(|x|)|u|pΔmu=c(|x|)|u|p on an annulus {x∈Rn|σ≤|x|<ρ}{x∈Rn|σ≤|x|<ρ}, with ρ∈(0,∞)ρ∈(0,∞) and σ∈[0,ρ)σ∈[0,ρ), that diverge to infinity as |x|→ρ|x|→ρ. Here n,m∈Nn,m∈N, p∈(1,∞)p∈(1,∞), and c   is a positive continuous function on the interval [σ,ρ][σ,ρ]. Letting ϕρ(r):=QCρ/(ρ−r)qϕρ(r):=QCρ/(ρ−r)q for r∈[σ,ρ)r∈[σ,ρ), with q:=2m/(p−1)q:=2m/(p−1), Q:=(q(q+1)⋯(q+2m−1))1/(p−1)Q:=(q(q+1)⋯(q+2m−1))1/(p−1), and Cρ:=c(ρ)−1/(p−1)Cρ:=c(ρ)−1/(p−1), we show that, as |x|→ρ|x|→ρ, the ratio u(x)/ϕρ(|x|)u(x)/ϕρ(|x|) remains between positive constants that depend only on m and p  . Extending well-known results for the second-order problem, we prove in the fourth-order case that u(x)/ϕρ(|x|)→1u(x)/ϕρ(|x|)→1 as |x|→ρ|x|→ρ and obtain precise asymptotic expansions if c is sufficiently smooth at ρ  . In certain higher-order cases, we find solutions for which the ratio u(x)/ϕρ(|x|)u(x)/ϕρ(|x|) does not converge, but oscillates about 1 with non-vanishing amplitude.