15.3. Applications 189

We see from (15.89) and (15.92) that (15.48), (15.49) has a nontrivial solution. Thus,

we may assume that (15.82) holds.

Next, we note that there is an µ > 0 depending on ρ such that

G(0, w) ≥ µ, b(w) ≥ ρ > 0.

To see this, suppose that {wk } ‚ M is a sequence such that

G(0, wk ) ’ 0, b(wk ) ≥ ρ.

If

bk = b(wk ) ¤ C,

this implies

b(wk ) ’ μ0 wk ’0

2

and

[μ0 ’ μ(x)]wk d x ’ 0

2

since

2

[μ0 ’ μ(x)]w2 d x,

G(0, w) ≥ b(w) ’ μ0 w + w ∈ M.

If we write wk = wk + h k , wk ∈ M , h k ∈ M0 as before, then this tells us that

b(wk ) ’ 0. Since M0 is ¬nite-dimensional, there is a renamed subsequence such that

h k ’ h. But the two conclusions above tell us that h = 0. Since b(h) ≥ ρ, we see

that µ > 0 exists for any constant C. If the sequence {bk } is not bounded, we take

1/2

wk = wk /bk . Then

˜

G(0, wk )/bk ≥ b(wk ) ’ μ0 wk

˜ ˜ + [μ0 ’ μ(x)](wk )2 d x.

˜

2

Next, we note that there is a ν > 0 such that

G(0, w) ≥ νb(w), w ∈ M.

(15.93)

Assuming this for the moment, we see that

inf G ≥ µ1 > 0,

(15.94)

B

where

(15.95) B = {w ∈ M : b(w) ≥ ρ 2 } ∪ {u = (s•0 , w) : s ≥ 0, w ∈ M, u = ρ},

D

and µ1 = min{µ, νρ 2 }. By (15.66), there is an R > ρ such that

sup G = a0 < ∞,

(15.96)

A

where A = N. By Proposition 15.7, A, B form a weak sandwich pair. Moreover, G

satis¬es (15.7) with µ1 ¤ b0 . Hence, there is a sequence {u k } ‚ D such that (15.8)

190 15. Weak Sandwich Pairs

holds with c ≥ µ1 . Arguing as in the proof of Theorem 15.8, we see that there is a

u ∈ D such that G(u) = c ≥ µ1 > 0, G (u) = 0. Since c = 0 and G(0) = 0, we see

that u = 0, and we have a nontrivial solution of the system (15.48), (15.49).

It therefore remains only to prove (15.93). Clearly, ν ≥ 0. If ν = 0, then there is a

sequence {wk } ‚ M such that

G(0, wk ) ’ 0, b(wk ) = 1.

(15.97)

Thus, there is a renamed subsequence such that wk ’ w weakly in M, strongly in

L 2 ( ), and a.e. in . Consequently,

[μ0 ’ μ(x)]wk d x ¤ 1 ’ μ(x)wk d x ¤ G(0, wk ) ’ 0

2 2

(15.98)

and

1= μ(x)w2 d x ¤ μ0 w ¤ b(w) ¤ 1,

2

(15.99)

which means that we have equality throughout. It follows that we must have w ∈

E(μ0 ), the eigenspace of μ0 . Since w ≡ 0, we have w = 0 a.e. But

[μ0 ’ μ(x)]w2 d x = 0

(15.100)

implies that the integrand vanishes identically on , and consequently μ(x) ≡ μ0 ,

violating (15.73). This establishes (15.93) and completes the proof of the theorem.

15.4 Notes and remarks

Weak sandwich pairs were introduced in [133].

Chapter 16

Multiple Solutions

16.1 Introduction

Nonlinear problems are characterized by the fact that very often solutions are not

unique. However, it is sometimes quite dif¬cult to show that more than one solution

exists. In this respect, variational methods can be very helpful. We have chosen some

examples that illustrate this point.

16.2 Two examples

Let be a smooth, bounded domain in Rn , and let A be a self-adjoint operator on

L 2 ( ). We assume that

C0 ( ) ‚ D := D(|A|1/2 ) ‚ H T ,2( )

∞

(16.1)

holds for some T > 0 (T need not be an integer), and the eigenvalues of A satisfy

0 < »0 < »1 < · · · < »k < · · · .

Let f (x, t) be a Carath´ odory function on — R. We assume that the function f (x, t)

e

satis¬es

| f (x, t)| ¤ C(|t| + 1), x∈ , t ∈ R.

(16.2)

We shall prove

Theorem 16.1. 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 ,

(16.3)

where a < »m+1 ,

a0 t 2 ¤ 2F(x, t) ¤ a1 t 2 , |t| < δ,

(16.4)

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_16,

© Birkh¤user Boston, a part of Springer Science+Business Media, LLC 2009

192 16. Multiple Solutions

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