Uniqueness of positive radial solutions of \(\Delta{} u + g(r)u + h(r)u^ p = 0\) in \(\mathbb{R}{}^ n\) (Q1175234)

The uniqueness of a solution to the equation mentioned in the title under Dirichlet boundary condition is obtained with general assumptions on the functions \(g\) and \(h\) (basically, \(g\) and \(h\) are \(C^ 1(0,\infty)\) and satisfy \(r^{2-\sigma}f(r)\to0\) as \(r\to 0^ +\) for some \(\sigma\)) for \(n>2\) and \(p>1\) by utilizing a generalized version of the Pokhozhaev identity. The result applies to the scalar field equation \((g(r)=-1\), \(h(r)=1)\) and to Matukuma's equation \((g(r)=0\), \(h(r)=(1+r^ 2)^{-1}\) and is obtained by studying solutions of \[ u_{rr}+(n-1)r^{-1}u_ r+g(r)u+h(r)u^ p=0 \hbox { on } [0,R] \] which satisfy \(u(0)<\infty\), \(u(R)=0\) and \(u(r)>0\) for \(r\in [0,R)\).