Asymptotic behavior of positive solutions of a semilinear polyharmonic problem in the unit ball

Let m be a positive integer. We investigate the existence and the asymptotic behavior of positive continuous solutions to the following semilinear polyharmonic boundary value problem in the unit ball B of Rn(n≥2): (−Δ)mu=a(x)uα,lim|x|→1u(x)(1−|x|)m−1=0, where −1<α<1 and a is a positive measurable function in B such that there exists c>0 satisfying for each x∈B, 1c≤a(x)(1−|x|)λexp(−∫1−|x|ηz(s)sds)≤c,η>1,λ≤m(1+α)+1−α and z is a continuous function on [0,η] with z(0)=0.