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We prove a uniqueness result for the positive solution of      Δ  u + f ( u ) = 0 \Delta u + f(u) = 0       in      R n {\mathbb {R}^n}       which goes to      0 0       at      ∞  \infty      . The result applies to a wide class of nonlinear functions      f f      , including the important model case      f ( u ) = −  u + u p f(u) = - u + {u^p}       ,      1 &gt; p &gt; ( n + 2 ) / ( n −  2 ) 1 &gt; p &gt; (n + 2)/(n - 2)      . The result is proved by reducing to an initial-boundary problem for the      ODE u + ( n −  1 ) / r + f ( u ) = 0 {\text {ODE}}\;u + (n - 1)/r + f(u) = 0       and using a shooting method.

Uniqueness of positive radial solutions of Δ𝑢+𝑓(𝑢)=0 in 𝑅ⁿ. II