On the Dirichlet Problem of the Equation -Δu=Ku⁵ on the Ball in R³

where $D(R)$ and a denote the ball of radius R centered at the origin 0 in Rⁿ $(n geqq 3)$ and a small positive constant respectively.If $Kequiv 1$ , then the equation for the unknown function $v=u/a-1$ becomesAccording to the result of 2.6 in [1], this equation for sufficiently small $a>0$ has at least two solutions with $v=0$ on $partial D(R)$ and $v>0$ in $D(R)$.Hence we get at least two solutions mathrmw , if $Kequiv 1$ .Because there is only the trivial solution for $a=0$ by the result of [5], these solutions may tend to 0 or diverge as $arightarrow 0$ .Let us turn to the case for nonconstant K .We can prove by the varia- 2.In this section, we formulate the problem as a nonlinear equation on a domain of the unit sphere, and investigate the associated linear operator.Let $P=(0,0,0,1)$ be the north pole of S³subset R⁴ and p:S³rightarrow R³ be the stereographic projection.Denoting by g and ds² the standard metric on S³ and R³ respectively, we haveIf deltaₑₚₛᵢₗₒₙ:R³rightarrow R³ is the dilation by $epsilon$ , i.e., deltaₑₚₛᵢₗₒₙ(x)=x/epsilon , deltaₑₚₛᵢₗₒₙ*(p⁻¹)*g=[2epsilon/(epsilon²+|x|²)1/2]⁴ds²=uₑₚₛᵢₗₒₙ⁴ds²Let us define a domain Dₑₚₛᵢₗₒₙ of S³ by p⁻¹deltaₑₚₛᵢₗₒₙ(D(R)) .Then p⁻¹deltaₑₚₛᵢₗₒₙ is an isometry between (D(R), uₑₚₛᵢₗₒₙ⁴ds²) and (Dₑₚₛᵢₗₒₙ, g) .The radius of Dₑₚₛᵢₗₒₙ is