Radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry

Let n∈Nn∈N with n⩾2n⩾2, a∈(−1,0)∪(0,1]a∈(−1,0)∪(0,1] and f:(0,1)×(0,∞)→Rf:(0,1)×(0,∞)→R such that for each u∈(0,∞)u∈(0,∞), r↦(1+ar2)(n+2)/2f(r,(1+ar2)−(n−2)/2u):(0,1)→Rr↦(1+ar2)(n+2)/2f(r,(1+ar2)−(n−2)/2u):(0,1)→R is nonincreasing. We show that each positive solution ofΔu+f(|x|,u)=0in B,u=0on ∂B is radially symmetric, where B   is the open unit ball in RNRN.