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

It is readily shown that G is a continuously differentiable functional on the whole of D (cf., e.g., [122]). Since the embedding of D in L2( ) is compact, the spectrum of A consists of isolated eigenvalues of finite multiplicity:

0 < λ0 < λ1 < · · · < λ < · · · .

Let λ , > 0, be one of these eigenvalues. We assume that the eigenfunctions of λ are in L∞( ) and that the following hold:

≥λ v 2 − v 2D − C v qD

To see this, note that when (8.15) holds, we have |V v| ≤ 2|V v | and

Moreover,

v q ≤ C v m,2 ≤ C v D

by the Sobolev inequality and the embedding of D in H m,2( ). From this we see that there are positive constants , ρ such that

for some positive 1 unless there is a solution of

Ay = λ y = f (x, y), y E(λ )\{0}

(cf. [122]). Since such a solution would solve (8.13), we may assume that (8.16) holds.

Since

v 2D ≤ λ v 2, v N,

and

λ +1 w 2 ≤ w 2D , w M,

we have, by (8.6),

G(w) ≤ λ w 2 + B1 − w 2D ≤ B1, w A,

where B1 = W1(x) dx. Moreover, (8.8) implies

G(v ) ≥ (ν − λ −1) v 2, v N .

Hence, there is an ε > 0 such that

G(v) ≥ ε, v B.

In view of these inequalities, we can now apply Proposition 7.3 to conclude that there is a sequence {uk } D such that

Since σ (x) < 0 a.e. in , the last two statements imply that u˜ ≡ 0. However, we see from (8.18) that

showing that u˜ ≡ 0. This contradiction tells us that the ρk must be bounded. We can now apply Theorem 3.4.1 of [122] to conclude that there is a u D satisfying

Since c ≥ ε > 0, we see that u = 0, and the proof is complete.

The proof of Theorem 8.1 implies

Corollary 8.2. If λ is a simple eigenvalue, then hypothesis (8.6) in Theorem 8.1 can be weakened to

(8.20) 2F(x, t) ≤ λ +1t2 + W1(x), x , t R,

for some W1(x) L1(R).

Remark 8.3. The proof of Theorem 8.1 is much simpler if = 0. In this case N = {0} and (8.14) immediately implies (8.16). The rest of the proof is unchanged.

We now show that we can essentially reverse the inequalities (8.6)–(8.10) and obtain the same results. In fact, we have

Theorem 8.4. Equation (8.13) has at least one nontrivial solution if we assume > 0 and

Proof. In this case we take G to be the functional (8.5). We take A = N, N = N {v0}, and consider the mapping

F(v + w + sv0) = v + [s + δ − δϕ( w 2/δ2)]v0, v N , w M, s R, where ϕ satisfies the hypotheses of Proposition 7.3. By (8.21), we have

G(v) ≤ B1, v N.

For w M1, we write w = w + y, where w M and y E(λ ). Then (8.22) implies

G(w) ≥ ε1, w D = ρ, w M1,

unless (8.13) has a nontrivial solution. Hence, by the argument given in the proof of Theorem 8.1 we have a sequence satisfying (8.17). If u˜ k and u˜ are as in the proof of

Theorem 8.1, then (8.24), (8.25) imply that u˜ ≡ 0 as in that proof. However, (8.18) implies

showing that u˜ ≡ 0. This contradiction proves the theorem as in the case of Theorem 8.1.

The proof of Theorem 8.4 implies

Corollary 8.5. If λ is a simple eigenvalue, then hypothesis (8.21) in Theorem 8.4 can be weakened to

(8.26) λ −1t2 ≤ 2F(x, t) + W1(x), x , t R, for some W1(x) L1(R).

8.3 Some useful quantities

We now show how we can improve the results of the last section. For each fixed k, let

Nk denote the subspace of D := D( A1/2) spanned by the eigenfunctions corresponding to λ0, . . . , λk , and let Mk = Nk ∩ D. Then D = Mk Nk . We define

(8.27) αk := max{( Av, v) : v Nk , v ≥ 0, v = 1},

where v denotes the L2( )-norm of v. We assume that A has an eigenfunction ϕ0 of constant sign a.e. on corresponding to the eigenvalue λ0.

by the eigenspaces of A corresponding to the eigenvalues λ0, λ1, . . . , λ . We take M = N ∩ D, where D = D( A1/2). We assume that F(x, t) satisfies

is u ≡ 0. We have

Theorem 8.6. Under the above hypotheses, (8.13) has a nontrivial solution. Proof. By (8.28),

Since ν < (a2), we see by continuity that there is an ε > 0 such that

≥ ε w 2D , w M,

by (8.30).

As in the proof of Theorem 8.1, we note that the following alternative holds: Either

v0 E(λ ), and let F be the mapping (7.10). Take A = N, B = F−1(δv0). By (8.37),

subsequence such that u˜k → u˜ weakly in D, strongly in L ( ), and a.e. in . Hence,

by (8.30), and the right-hand side of (8.44) converges to C|u˜(x)|2 in L1( ) and

we see that the convergence in (8.45) is not only pointwise a.e., but also in L1( ). Since u˜k D = 1, (8.44) implies

(8.46). Hence, ρk ≤ C. The theorem now follows from Theorem 3.4.1 of [122].

8.4 Unbounded domains

Now we allow the domain Rn to be unbounded. Assume hypothesis (A), and assume that

(8.47) H (x, t) = 2F(x, t) − t f (x, t) ≥ −W3(x) L1( ), x , t R,