25.8. PERTURBATION THEOREMS 925

Thus if ∥u∥V > R,

inf

{⟨z∗,u⟩+ ε ∥Lu∥2

V ′

∥u∥V +∥Lu∥V ′: z∗ ∈ Au

}≥

M ∥u∥V + ε ∥Lu∥2V ′

∥u∥V +∥Lu∥V ′

I claim that if ∥u∥X is large enough, the above is larger than M/2. If not, then there exists{un} such that ∥un∥X → ∞ but the right side is less than M/2. First say ∥Lun∥ is bounded.then there is an obvious contradiction since the right hand side then converges to M. Thusit can be assumed that ∥Lun∥V ′ →∞. Hence, for all n large enough, ε ∥Lu∥2

V ′ > M ∥Lun∥V ′ .However, this implies the right side is larger than

M ∥un∥V +M ∥Lun∥V ′∥un∥V +∥Lun∥V ′

= M > M/2

This is a contradiction. Hence the right side is larger than M/2 for all n large enough. Itfollows since M is arbitrary, that

lim∥u∥X→∞

inf

{⟨z∗,u⟩+ ε ∥Lu∥2

V ′

∥u∥V +∥Lu∥V ′: z∗ ∈ Au

}= ∞

It follows from Theorem 25.5.4 that if f ∈ V ′, there exists uε such that for all v ∈D(L) = X ,

ε⟨Lv,F−1 (Luε)

⟩V ′,V + ⟨Luε ,v⟩V ′,V + ⟨w∗ε ,v⟩V ′,V = ⟨ f ,v⟩ , w∗ε ∈ Auε (25.8.91)

First we get an estimate.

ε⟨Luε ,F−1 (Luε)

⟩V ′,V + ⟨Luε ,uε⟩V ′,V + ⟨w∗ε ,uε⟩V ′,V = ⟨ f ,uε⟩

ε ∥Luε∥2V ′ + ⟨Luε ,uε⟩V ′,V + ⟨w∗ε ,uε⟩V ′,V = ⟨ f ,uε⟩

Hence it follows from the coercivity of A that ∥uε∥V is bounded independent of ε . Thusthe w∗ε are also bounded in V ′ because it is assumed that A is bounded. Now from theequation solved 25.8.91, it follows that F−1 (Luε) ∈ D(L∗) . Thus the first term is justε⟨L∗(F−1 (Luε)

),v⟩

V ′,V . It follows, since D(L) = X is dense in V that

εL∗(F−1 (Luε)

)+Luε +w∗ε = f (25.8.92)

Then act on F−1 (Luε) on both sides. From monotonicity of L∗, this yields ∥Luε∥V ′ isbounded independent of ε > 0. Thus there is a subsequence still denoted with a subscriptof ε such that

uε ⇀ u in V

Luε ⇀ Lu in V ′

This because of the fact that the graph of L is closed, hence weakly closed. Thus u ∈ X .Also

w∗ε ⇀ w∗ in V ′.