- If we look for a radial solution u(x) = v(r) of the nonlinear elliptic equation
−∆ = u
where r = |x| and p > 1, we are led to the nonautonomous ODE
n − 1
0 + v
p = 0.
Show that the Emden-Fowler transformation
t := log r, x(t) := e
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
for the nonlinear heat equation
ut − ∆(u
) = 0,
n < γ < 1. Your solution should go to zero algebraically as |x| → ∞.
- Find a solution of
−∆u + u
n−2 = 0 in B(0, 1)
having the form u = α(1 − |x|
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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