2 < q ¤ 2n/(n ’ 2T ), 2T < n

2 < q < ∞, n ¤ 2T.

By (16.2),

[a0 (v ’ )2 + b0 (v + )2 ’ 2F(x, v)] d x

J (v) ¤ G(v) ¤ I (v, a0 , b0 ) +

|v|>δ

¤’ v +C |v|q d x

2

D

|v|>δ

¤’ v + o( v D)

2 2

D

¤ ’µ v 2

D

for r suf¬ciently small.

Lemma 16.23. If

2F(x, t) ¤ a1 (t ’ )2 + b1 (t + )2 , |t| ¤ δ,

(16.59)

for some δ > 0, with b1 < l (a1 ), l < m, then there are µ > 0, r > 0 such that

J (v) ≥ µ v D, v ∈ Nm © Ml © Br .

2

(16.60)

16.4. Some lemmas 205

Proof. Let u = v + •(v) ∈ Ml . Then

[a0 (u ’ )2 + b0 (u + )2 ’ 2F(x, u)] d x

J (v) = G(u) ≥ I (u, a1 , b1 ) +

|u|>δ

≥ ’C |u|q d x

2

u D

|u|>δ

≥ ’ o( u D)

2 2

u D

2 2

≥ v ’ o( v D)

D

≥µ v 2

D

for r suf¬ciently small, since

v ¤u ¤C v D.

D D

Lemma 16.24. If

a0 (t ’ )2 + b0 (t + )2 ¤ 2F(x, t), |t| < δ,

(16.61)

for some δ > 0, with b0 > γl (a0 ), l ≥ m, then there are µ > 0, r > 0 such that

J (w) ¤ ’µ w D, w ∈ Nl © Mm © Br .

2

(16.62)

Proof. For w ∈ Mm © Nl , let u = w + ψ(w) ∈ Nl . By (16.2),

J (w) = G(w + ψ(w)) = G(u)

[a0 (v ’ )2 + b0 (u + )2 ’ 2F(x, u)] d x

¤ I (u, a0 , b0 ) +

|u|>δ

2

|u|q d x

¤’ u +C

D

|u|>δ

2 2

¤’ u + o( u D)

D

¤ ’µ u 2

D

for r suf¬ciently small. Since

w ¤u ¤C w D,

D D

the result follows.

Lemma 16.25. If

2F(x, t) ¤ a1 (t ’ )2 + b1 (t + )2 , |t| ¤ δ

(16.63)

for some δ > 0, with b1 < l (a1 ), l > m, then there are µ > 0, r > 0 such that

J (w) ≥ µ w D, w ∈ Ml © Br .

2

(16.64)

206 16. Multiple Solutions

Proof. We have

[a0 (u ’ )2 + b0 (w+ )2 ’ 2F(x, w)] d x

G(w) ≥ I (w, a1 , b1 ) +

|w|>δ

2

|w|q d x

≥ w ’C

D

|w|>δ

2 2

≥ w ’ o( w D)

D

≥ w ’ o( w D)

2 2

D

≥µ w 2

D

for r suf¬ciently small. Since

J (w) = sup G(v + w) ≥ G(w),

v∈Nl

the result follows.

16.5 Local linking

The following theorem will also be used in the proofs of the theorems of Section 16.3.

It is also of interest in its own right.

Theorem 16.26. Let M, N be closed subspaces of a Hilbert space E such that

0 < dim N < ∞ and M = N ⊥ . Let G ∈ C 1 (E, R) satisfy the PS condition and

G(v) ¤ 0, v ∈ N © BR,

G(w) ≥ 0, w ∈ M © BR,

for some R > 0. Assume that

’∞ < ± = inf G < 0.

E

Then G has at least three critical points.