p = 2 + 4/ d

. For

u_{a, λ} (x) = a u (λ x)

y = λ x

gives

∥\nabla u_{a, λ} ∥_{2}^{2} = ∣ a ∣^{2} λ^{2 - d} ∥\nabla u ∥_{2}^{2}, ∥ u_{a, λ} ∥_{2}^{4/ d} = ∣ a ∣^{4/ d} λ^{- 2} ∥ u ∥_{2}^{4/ d},

and

∥ u_{a, λ} ∥_{p}^{p} = ∣ a ∣^{2 + 4/ d} λ^{- d} ∥ u ∥_{p}^{p} .

The factors cancel, so the Weinstein functional satisfies

J (u_{a, λ}) = J (u)

∥ u ∥_{p}^{p} \leq C_{d} ∥\nabla u ∥_{2}^{2} ∥ u ∥_{2}^{4/ d} .

Consequently

J (u) \geq C_{d}^{- 1}

for every nonzero

u \in H^{1}

, and therefore

I > 0

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