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

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 > p > ( n + 2 ) / ( n − 2 ) 1 > p > (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.