Schechter Minimax Systems and Critical Point Theory (Springer, 2009)

158	13. Semilinear Wave Equations

13.6 The theorems

In solving problem (13.1)–(13.3) we take p(x, t, ξ ) to be a Carath´eodory function on× R which is 2π -periodic in t and satisfies

(13.26)	θ−ξ 2 ≤ ξ [ p(x, t, ξ1) − p(x, t, ξ0)] ≤ θ+ξ 2, ξ = ξ1 − ξ2 R.
Let	μ = (μ1, . . . , μn ), μ j ≥ 0, μ j Z, μ2 = |μ|2 = μ2j .

Let σ ( ) be the set of integers of the form λ = μ2 − k2, and let ρ( ) = R\σ ( ). We shall prove

Theorem 13.13. If
(13.27)	[θ−, θ+] ρ( ),

then there is a unique weak solution of (13.1)–(13.3).

Theorem 13.14. If

(13.28) (θ−, θ+) ρ( ),

and there are constants β± such that θ− ≤ β− ≤ β+ ≤ θ+, [β−, β+] ρ( ) and

(13.29)

β− ≤

|ξ |→∞

(

x

,

t

, ξ )/ξ

2

≤ |ξ |→∞

(

x

,

t

, ξ )/ξ

2

≤ β+

lim inf 2P

lim sup 2P

uniformly in , where

ξ

P(x, t, ξ ) = 0

p(x, t, s)ds,

then (13.1)–(13.3) has at least one weak solution.

Theorems 13.13 and 13.14 are proved in the next section.

13.7 The proofs

In this section we present the proof of Theorems 13.13 and 13.14. Let x = (x1, . . . , xn ), ϕμ(x) = sin μ1ξ1 · · · sin μn xn /(π/2)n/2.

Note that

(ϕμ, ϕν ) = δμν , where the scalar product is that of L2([0, π ]n). We take

ek (t) = eikt /(2π )1/2, t [0, 2π ].

Then

(ek , e ) = δk ,

13.7. The proofs		159
where the scalar product is that of L2([0, 2π ]). If
(13.30)	u =	αμk ϕμ(x)ek (t),
then	u ≡ utt − u = (μ2 − k2)αμk ϕμ(x)ek (t).

Let γ be a fixed number in (θ−, θ+) and define

G(u) = u 2 − ut 2 − 2 P(x, t, u), u E,

where the norm is that of L2( ), and E is the set of all u L2( ) of the form (13.30)

such that

u 2E = |μ2 − k2 − γ | · |αμk |2 < ∞.

It is easily checked that G C1(E, R) and

(G (u), v)/2 = ( u, v) − ( p(u), v), u, v E,

where we write p(u) in place of p(x, t, u). Let m+ be the smallest point of σ ( ) above γ and m− the largest point of σ ( ) below γ .

Let

M= {u E : αμk = 0 when μ2 − k2 < γ },

N= {u E : αμk = 0 when μ2 − k2 > γ }.

We have

Lemma 13.15.

(G (v

w1)

G (v

w0), w)/2

1

θ+ − γ

w

2

,

+

−

+

≥

− m+ − γ

E

where w = w1 − w0 M, v N.

Proof. The left-hand side equals

( w, w) − ( p(v + w1) − p(v + w0), w)

≥ w 2E − (θ+ − γ ) w 2

1

θ+ − γ

w

2 .

≥

− m+ − γ

E

Lemma 13.16.

(G (v1

+

w)

−

G (v0

+

w), v)/2

≤ −

1

−

γ − θ−

v

2

,

γ − m−

E

where v = v1 − v0 N, w M.

160 13. Semilinear Wave Equations

Proof. Now the left-hand side equals

( v, v) − ( p(v1 + w) − p(v0 + w), v)

≤ − v 2E + (γ − θ−) v 2

	≤ −	1	−	γ − θ−	v 2 .
				γ − m−
					E

We can now give the proof of Theorem 13.14.

Proof. By (13.28) and Lemmas 13.15 and 13.16, G(v + w) is convex in w and concave in v. Moreover,

(13.31)

G(w) → ∞ as w E → ∞,

w M,

and

(13.32)

G(v) → −∞ as v E → ∞,

v N.

To see this, let

w j = αμ( jk)ϕμek

be a sequence in M such that ρ j

= w j E

→ ∞. Take w˜ j

= w j /ρ j , and let ε > 0 be

such that β+ < m+ − ε. Then there is a constant K such that

2P(x, t, ξ )/ξ 2 < β+ + ε, |ξ | > K .

Hence,

2

P(x, t, w j )/ρ2j

≤

(β+ + ε)w˜ 2j +

P(x, t, w j )/ρ2j

|w j |>K

|w j |≤K

β+ + ε

C/ρ2

≤ m+ − γ +

j

since w˜ j E = 1. Therefore,

G(w

)/ρ2

≥

1

−

β+ + ε − γ

−

C/ρ2.

j

j

m+ − γ

j

Consequently,

β+ + ε − γ

lim inf G(w

j

)/ρ2

≥

1

−

> 0.

j

m+ − γ

This proves (13.31). A similar argument proves (13.32). Thus, G(v + w) is convex in w, is concave in v and satisfies (13.31), (13.32). The theorem now follows from Theorems 13.6, 13.7, Corollary 13.8, and Lemma 13.11.

13.8. Notes and remarks

161

Next, we prove Theorem 13.13.

Proof. First, we note that the hypotheses of Theorem 13.13 imply those of Theorem 13.14. In fact, we can take β± = θ±, for we have

		ξ	ds	+ p(x, t, 0)ξ.
P(x, t, ξ ) =		s[ p(x, t, s) − p(x, t, 0)]
	0		s

Hence,

θ−ξ 2 ≤ 2P(x, t, ξ ) − 2 p(x, t, 0)ξ ≤ θ+ξ 2

by (13.26). Thus, there exists a weak solution of (13.1)–(13.3). To show that it is unique, we note that for fixed v N, G(v + w) is strictly convex in w. Hence, if there were two distinct points such that G (u0) = G (u) = 0, we would have by Lemma 13.10 both G(u) < G(u0) and G(u0) < G(u), an impossibility.

13.8 Notes and remarks

The problem (13.1)–(13.3) has been considered by Smiley [147] and Mawhin [90]. They assume

β0ξ 2 ≤ ξ [ p(x, t, ξ1) − p(x, t, ξ0)], ξ = ξ1 − ξ0 R,

and

| p(x, t, ξ1) − p(x, t, ξ0)| ≤ β1|ξ |,

where 0 < β1 and β12 < β0. This implies 0 < β0 ≤ β1 < 1 and β1 − β0 < 12 (1 − β0). Since ρ( ) consists only of integers, our results are extensions of their results. Theorems 13.13 and 13.14 were proved in [125].

Chapter 14

Type (II) Regions

14.1 Introduction

The Fuˇc´ık spectrum described in Chapter 11 arises in the study of semilinear elliptic boundary-value problems of the form

(14.1) Au = f (x, u),

where A is a selfadjoint operator having compact resolvent on L2( ), Rn, and

f	(	,	)	¯	×	R such that
	x	t		is a Carath´eodory function on
(14.2)				f (x, t)/t → a a.e. as t → −∞
				→ b a.e. as t → +∞.

If |u(x)| is large, then (14.1) approximates the equation

(14.3)

Au = bu+ − au−,

where u± = max{±u, 0}. It was first noticed by Fuˇc´ık [68] that (14.3) plays an important part in the study of (14.1) when (14.2) holds. Fuˇc´ık studied the problem

(14.4)

−u = bu+ − au− in (0, π ), u(0) = u(π ) = 0

and showed that there is a substantial difference in the solvability of (14.1) if (14.3) has nontrivial solutions. We now call the set of those (a, b) R2 for which (14.3) has nontrivial solutions the Fuˇc´ık spectrum of A. Fuˇc´ık showed that for (14.4), consists of a sequence of decreasing curves passing through the points (k2, k2), k = 1, 2, . . . , with one or two curves emanating from each of these points. Points not on these curves are not in . He also noticed that there are two different types of regions between the curves, namely,

(I)regions between curves passing through consecutive points

(k2, k2), ([k + 1]2, [k + 1]2)

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_14, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

164	14. Type (II) Regions

and

(II)regions between curves passing through the same point (k2, k2).

In the type (I) regions one can solve

(14.5) −u = bu+ − au− + p(x), x (0, π ), u(0) = u(π ) = 0,

for arbitrary p(x) L2(0, π ), while this is not so for regions of type (II).

No complete description of for the general case (14.1) has been found. If

(14.6) 0 < λ0 < λ1 < · · · < λk < · · ·

are the eigenvalues of A, it was shown in [111] and [122] that in the square [λ −1, λ +1]2,

there are decreasing curves C,1, C,2 (which may coincide) passing through the point

(λ , λ )

such that all points above or below both curves in the square are not in [and are of type (I)] while points on the curves are in . The status of points between the curves (when they do not coincide) is unknown in general. However, it was shown in [72] that when λ is a simple eigenvalue, points between the curves are not in . On the other hand, C. and W. Margulies [91] have shown that there are boundary-value problems for which many curves in emanate from a point (λ , λ ) when λ is a multiple eigenvalue. (Clearly, these curves are contained in the region between the curves C,1 and C,2.)

As expected, the boundary-value problem (14.1) is more readily solved when (a, b) is in a region of type (I) (i.e., on the same side of both curves C 1, C 2). For such points, the following was proved in [111] and [122].

Theorem 14.1. Assume that f (x, t) is of the form

(14.7)	f (x, t) = bt+ − at− + p(x, t),
where p(x, t) satisfies
(14.8)	| p(x, t)| ≤ V (x)1−σ |t|σ + W (x),

with 0 ≤ σ < 1 and V, W L2( ). Then (14.1) has a solution. In particular,

(14.9) Au = bu+ − au− + p(x)

has a solution for each p L2( ).

However, when it comes to points on the curves C,1, C,2, no such theorem holds. In Chapter 11 we addressed this issue and presented sufficient conditions for the existence of solutions of (14.1) when (a, b) is on either C,1 or C,2. [Needless to

14.1. Introduction

165

say, much more is required of p(x, t) in Theorems 11.1 and 11.2 than in Theorem 14.1.] In the present chapter we consider the situation when (a, b) is in the region between the curves C,1, C,2 (when they do not coincide). As we saw earlier, such points may or may not belong to . We shall be concerned with those points in this region that are not in . Even in the case when none of these points is in (as in the case of a simple eigenvalue), we cannot prove a theorem as comprehensive as Theorem 14.1. However, we can prove the theorems below. We assume that

(14.10) | p(x, t)| ≤ C(|t| + 1), x , t R,

and that E(λ ), the subspace of eigenfunctions corresponding to λ , is contained in L∞( ) (i.e., the eigenfunctions corresponding to λ are bounded). We define

	F(x, t) =		t
(14.11)		0	f (x, s) ds
and
			t
(14.12)	P(x, t) =	0	p(x, s) ds.

We have

Theorem 14.2. Let (a, b) be a point in Q := (λ −1, λ +1)2 that lies below the upper curve C,2. Assume (14.2), (14.7), and

(14.13)	f (x, t1) − f (x, t0) >λ −1(t1 − t0),		t0 < t1, x ,
(14.14)	2P(x, t) ≤W1(x) L1( ),		x , t R,
(14.15)	2F(x, t) ≤λ +1t2,	x , t R,
(14.16)	λ t2 ≤2F(x, t),	|t| < δ for some δ > 0.

If (a, b) is not in , then (14.1) has a nontrivial solution.

Theorem 14.3. Let (a, b) be a point in Q \ that lies above the lower curve C,1. Assume (14.2), (14.7), and

(14.17)	2P(x, t) ≥ − W1(x) L1( ),	x , t R,
(14.18)	f (x, t1) − f (x, t0) <λ +1(t1 − t0), t0 < t1, x ,
(14.19)	λ −1t2 ≤2F(x, t), x ,	t R,
(14.20)	2F(x, t) ≤λ t2, |t| < δ for some δ > 0.

Then (14.1) has a nontrivial solution.

It should be noted that these theorems hold even when λ is a multiple eigenvalue. In proving the theorems, we examine the functional

(14.21)

G(u) = u 2D − 2 F(x, u)dx, u D = D( A1/2),

166	14. Type (II) Regions

where u D = A1/2u , and search for solutions of G (u) = 0. Even though (a, b) is in a region of type (II), we can find a manifold S on which G is bounded from below. Unfortunately, S has a boundary that cannot be linked with a subspace, and one must search for another manifold that links the boundary of S and on which G is bounded from above. Once this is achieved, we can apply the theorems of Chapter 2 to obtain a Palais–Smale sequence. We then show that there is a convergent subsequence due to the fact that (a, b) is not in . In Theorem 14.3 we reverse the procedure, finding a manifold on which G is bounded from above and then searching for a linking set on which it is bounded from below.

14.2 The asymptotic equation

In this section we show how information concerning (14.3) affects the solvability of (14.1). For each fixed positive integer , we let N denote the subspace of D =

D( A1/2) spanned by the eigenfunctions of A corresponding to the eigenvalues λ0, λ1,

. . . , λ , and we let M = N ∩ D. Then D = M N . For (a, b) R2, we define

(14.22)	u 2D =( Au, u),
(14.23)	I (u, a, b) = u 2D − a u− 2 − b u+ 2, u D,
(14.24)	(a, b) =	w M	sup I (v + w, a, b),
	M	inf
			v		N
		w D =1
(14.25)	m (a, b) = v N		w M I (v + w, a, b),
		sup		inf
		v D =1
(14.26)	ν (a) =sup{b : M (a, b) ≥ 0},
(14.27)	μ (a) = inf{b : m (a, b) ≤ 0}.

It follows from Lemma 11.3 that ν −1(a) ≤ μ (a) in Q . We let C,1 be the (lower) curve b = ν −1(a) and we let C,2 be the (upper) curve b = μ (a). Thus, by (14.27), if (a, b) Q is below the curve C,2, then

(14.28)

m (a, b) > 0.

Consequently, there is a y0 N0 = E(λ ) such that

(14.29)

sup

w M

I

(v + w +

y

0,

a

,

b

) >

0

.

v N −1

inf

Note that I (v + w + y0, a, b) is strictly convex and lower semi-continuous in w M and strictly concave and continuous in v N −1 because (a, b) Q . Moreover,

I (w + y0, a, b) → ∞ as w D → ∞, w M.

To see this, let w D = ρ → ∞. Then, for u = (w + y0)/ρ, we have

I (u, a, b) ≥ 1 − a u− 2 − b u+ 2 → 1 − a w˜ − 2 − b w˜ + 2 > 0,

14.2. The asymptotic equation	167
where w˜ ≤ 1, since a, b < λ +1. Similarly,
I (v + y0, a, b) → ∞ as v D → ∞,	v N.

Hence we may apply Theorem 13.6 to conclude that there are unique v0 N −1, w0 M such that

(14.30) I	(v0 + w0 +	y	0,	,	b	) =	v N 1 w M I (v + w + y0, a, b)
				a			sup	inf
						=	−	sup I (v + w + y0, a, b).
							w M
							inf	v N −1
In particular, this shows that y0						= 0, for otherwise v0 = 0, w0 = 0 would satisfy

(14.30) and the uniqueness would violate (14.29). We also see from (14.29) and (14.30) that

(14.31)	sup I (v + w + y0, a, b) > 0, w M .
	v N −1

Moreover, it follows that there is a continuous map θ from M −1 to N −1 such that

(14.32)	θ (w) ≤ C w D ,				w M −1,
(14.33)	θ (sw) = sθ (w),				s ≥ 0,
and v˜ = θ (w) is the only solution of
(14.34)	I (w	+ ˜		−						−
		v , a, b)		N	1, w				M	1,
and	I (w + v˜, a, b) = v N −1
(14.35)				(w + v,		a	,	b	), w		−1.
		max	I							M
Thus, (14.31) implies
(14.36)	I (w + sy0 + θ (w + sy0), a, b) ≥ 0,								w M , s ≥ 0.
Another way of writing (14.36) is
(14.37)		I (u, a, b) ≥ 0,			u S 2
where
(14.38)	S 2, = {u D : I (u, a, b) N −1, (u, y0) ≥ 0}.

Thus, we have proved

Lemma 14.4. If (a, b) Q is below the upper curve C,2, then there is a y0 E(λ )\{0} such that (14.37) holds for all u S 2, where S 2 is given by (14.38).

168 14. Type (II) Regions

In the same vein, if (a, b) Q is above the lower curve C 1, then there is a

y1 N0 such that		sup I (v + w + y1, a, b) < 0.
(14.39)	w M
	inf	v N −1

Using the same reasoning as before, we have y1 = 0 and (14.30) holds with y0 replaced

	by y1. This implies	w M I (v + w + y1, a, b) < 0, v N −1.
	(14.40)
		inf

Now we use the fact that there is a continuous map τ from N to M such that

(14.41)

τ (v) D ≤ C v ,

v N ,

(14.42)

τ (sv) = sτ (v),

s ≥ 0,

and w˜ = τ (v) is the unique solution of

(14.43)

I (v

+ ˜

v

,

w, a, b)

M ,

N

and

I (v + w,˜ a, b) = w M

(v + w,

v

(14.44)

I

a

,

b

),

.

inf

N

Thus, (14.40) implies

(14.45)

I (v + sy1 + τ (v + sy1), a, b) ≤ 0,

v N −1,

s ≥ 0,

or

(14.46)

I (u, a, b) ≤ 0,

u S 1,

where

(14.47)

S 1 = {u D : I (u, a, b) M , (u, y1) ≥ 0}.

We therefore have

Lemma 14.5. If (a, b) Q lies above the lower curve C,1, then there is a y1 E(λ )\{0} such that (14.46) holds for all u S 1, where S 1 is given by (14.47).

14.3 Local estimates

In proving Theorems 14.2 and 14.3, we shall make use of the functional

(14.48)

G(u) = u 2D − 2

F(x, u) dx = I (u, a, b) − 2

P(x, u) dx

for functions u D. When (14.10) holds, it is easily checked that G C1(D, R) and that u D is a solution of (14.1) if, and only if, it satisfies

(14.49)

G (u) = 0.

In this section we shall examine some properties of G under the hypotheses of the theorems. First, we have