 # Differential Equations

## Question

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## Transcribed Text

- If we look for a radial solution u(x) = v(r) of the nonlinear elliptic equation −∆ = u p in R n , where r = |x| and p > 1, we are led to the nonautonomous ODE (∗) v 00 + n − 1 r v 0 + v p = 0. Show that the Emden-Fowler transformation t := log r, x(t) := e 2t p−1 v(e t ) converts (*) into an autonomous ODE for the new unknown x = x(t). - Find a nonnegative scaling invariant solution having the form u(x, t) = t −α v(xt−β ) for the nonlinear heat equation ut − ∆(u γ ) = 0, where n−2 n < γ < 1. Your solution should go to zero algebraically as |x| → ∞. - Find a solution of −∆u + u n+2 n−2 = 0 in B(0, 1) having the form u = α(1 − |x| 2 ) −β for positive constants α, β. This example shows that a solution of a nonlinear PDE can be finite within a region and yet approach infinity everywhere on its boundary

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