f (x, s)ds,

F(x, t) :=

0

and

a2 t 2 ’ W1 (x) ¤ 2F(x, t), |t| > K ,

(16.5)

for some K ≥ 0, where a2 > »m and W1 ∈ L 1 ( ). Then the equation

Au = f (x, u), u ∈ D,

(16.6)

has at least two nontrivial solutions.

Theorem 16.2. Equation (16.6) will have at least two nontrivial solutions if we assume

that for some integers l > m, the following inequalities hold:

t[ f (x, t1 ) ’ f (x, t0 )] ≥ at 2 , t j ∈ R, t = t1 ’ t0 ,

(16.7)

where a > »m ,

a0 t 2 ¤ 2F(x, t) ¤ a1 t 2 , |t| < δ,

(16.8)

for some δ > 0, with »l < a0 ¤ a1 < »l+1 ,

2F(x, t) ¤ a2 t 2 + W2 (x), |t| > K ,

(16.9)

for some K ≥ 0, where a2 < »m+1 and W2 ∈ L 1 ( ).

More comprehensive theorems will be presented in the next section. Proofs will be

given in Section 16.6.

16.3 Statement of the theorems

We use the notation

a(u, v) = (Au, v), a(u) = a(u, u), u, v ∈ D.

We de¬ne

:= A1/2 u

(16.10) u D

and

’2

2

G(u) := u F(x, u)d x.

D

It is known that G is a continuously differentiable functional on the whole of D (cf.,

e.g., [120]) and

(G (u), v) D = 2(u, v) D ’ 2( f (u), v),

16.3. Statement of the theorems 193

where we write f (u) in place of f (x, u(x)). In connection with the operator A, the

following quantities are very useful. For each ¬xed positive integer , we let N denote

the subspace of D spanned by the eigenfunctions corresponding to »0 , . . . , » , and let

M = N ⊥ © D. Then D = M • N . For real a, b, we de¬ne

I (u, a, b) = (Au, u) ’ a u ’ ’ b u+ 2,

2

γ (a) = sup{I (v, a, 0) : v ∈ N , v + = 1},

(a) = inf{I (w, a, 0) : w ∈ M , w+ = 1},

F1 (w, a, b) = sup{I (v + w, a, b) : v ∈ N },

F2 (v, a, b) = inf{I (v + w, a, b) : w ∈ M },

M (a, b) = inf{F1 (w, a, b) : w ∈ M , w = 1},

D

m (a, b) = sup{F2 (v, a, b) : v ∈ N , v = 1},

D

ν (a) = sup{b : M (a, b) ≥ 0},

μ (a) = inf{b : m (a, b) ¤ 0},

where u ± (x) = max{±u(x), 0}. Our ¬rst result is

Theorem 16.3. Assume that for some integers l < m, the following inequalities hold:

t[ f (x, t1 ) ’ f (x, t0 )] ¤ a(t ’ )2 + b(t + )2 , t j ∈ R, t = t1 ’ t0 ,

(16.11)

where b < m (a),

a0 (t ’ )2 + b0 (t + )2 ¤ 2F(x, t) ¤ a1 (t ’ )2 + b1 (t + )2 , |t| < δ,

(16.12)

for some δ > 0, with a0 , b0 < »l+1 , a1 , b1 > »l , b0 > μl (a0 ), and b1 < νl (a1 ),

a2 (t ’ )2 + b2 (t + )2 ’ W1 (x) ¤ 2F(x, t), |t| > K ,

(16.13)

for some K ≥ 0, where a2 , b2 < »m+1 , b2 > μm (a2 ), and W1 ∈ L 1 ( ). Then (16.6)

has at least two nontrivial solutions.

In contrast to this, we have

Theorem 16.4. Equation (16.6) will have at least two nontrivial solutions if we assume

that for some integers l > m, the following inequalities hold:

t[ f (x, t1 ) ’ f (x, t0 )] ≥ a(t ’ )2 + b(t + )2 , t j ∈ R, t = t1 ’ t0 ,

(16.14)

where b > γm (a),

a0 (t ’ )2 + b0 (t + )2 ¤ 2F(x, t) ¤ a1 (t ’ )2 + b1 (t + )2 , |t| < δ

(16.15)

for some δ > 0, with a0 , b0 < »l+1 , b0 > μl (a0 ) and a1 , b1 > »l , b1 < νl (a1 ),

2F(x, t) ¤ a2 (t ’ )2 + b2 (t + )2 + W2 (x), |t| > K ,

(16.16)

for some K ≥ 0, where a2 , b2 > »m , b2 < νm (a2 ), and W2 ∈ L 1 ( ).

194 16. Multiple Solutions

It was shown in [114] that the functions γl , μl , νl’1 , all emanate from the

l’1

point (»l , »l ) and satisfy

l’1 (a) ¤ νl’1 (a) ¤ μl (a) ¤ γl (a)

on their common domains. It would therefore give a weaker result if we assumed in

Theorems 16.3 and 16.4 that b0 > γl (a0 ) and b1 < l (a1 ). However, the functions

γl , l are de¬ned on the whole of R, while the others are not. For cases in which the

other functions are not de¬ned, we state the following.

Theorem 16.5. Theorems 16.3 and 16.4 remain true if we assume that (16.12) holds

with b0 > γl (a0 ), and b1 < l (a1 ) for some a0 , a1 ∈ R.

16.4 Some lemmas

The proofs of the theorems of the previous section will be based on a series of lemmas.

Lemma 16.6. If b < l (a), > 0 such that

then there is an

I (w, a, b) ≥ w D, w ∈ Ml .

2

(16.17)

l, there is a t < 1 such that b/t < l (a/t).

Proof. By the continuity of Then

a’ b+

2 2 2

I (w, a/t, b/t) = w ’ w ’ w ≥ 0, w ∈ Ml .

D

t t

Therefore,