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A nonlinear Schrodinger equation resonating at an essential spectrum.

1. Introduction and Statement of Results

In this paper, the following nonlinear Schrodinger problem in [[??].sup.N] (N [greater than or equal to] 3) is considered:

-[DELTA]u = V(x)u + g(u) in [[??].sup.N]. (1)

The nonlinearity g satisfies the resonance type condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Here, [[sigma].sub.ess](T) denotes the essential spectrum (see [1, Chapter 7] or [2, Chapter 7.4]) of the Schrodinger operator T defined as follows:

T : [L.sup.2] ([[??].sup.N]) [right arrow] [L.sup.2] ([[??].sup.N]), u [??] -[DELTA]u - Vu, (3)

with domain D(T) := {u [member of] [L.sup.2] ([[??].sup.N]) | Tu [member of] [L.sup.2] ([[??].sup.N])}.

Equation (1) arises in quantum mechanics and is related to the study of the nonlinear Schrodinger equation for a particle in an electromagnetic field and it has attracted considerable attention from researchers in recent decades. One can see [3-6] and the references therein. However, there are very few results on (1) with the resonance type condition (2).

The following nonlinear elliptic problem with resonance type conditions in bounded domain has been studied by many authors and numerous existence and multiplicity results have been obtained in the past forty years:

-[DELTA]u = h (x, u) in [OMEGA], u = 0 on [partial derivative][OMEGA]. (4)

Here, [OMEGA] is a bounded domain in [[??].sup.N], and g satisfies the resonance type condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

[v.sub.k] is the kth eigenvalue of -[DELTA] with 0-Dirichlet boundary condition on [partial derivative][OMEGA]. We refer to [7-10] and references therein for more detailed discussions of some historical results. This problem has deeply inspired developments in critical point theory in the last forty years, such as the Landesman-Lazer type conditions (e.g., [8, 10]), Morse theory (e.g., [9]), and the variational reduction method (e.g., [11]). The difficulty of this problem lies in the proof of the boundedness of the Palais-Smale sequence or the [(C).sub.c] sequence (see Definition 5 in Section 2) of its corresponding functional.

The resonant Schrodinger problem (1) is much more difficult than the bounded domain case and it has fewer studies. Unlike the case of bounded domains for which the linear operators -[DELTA] in bounded domain are compact, there are continuous spectra of the linear operator T. Moreover, the proofs of boundedness and compactness of the [(C).sub.c] sequence of the corresponding functional of (4) are greatly different from and more difficult than the case of bounded domain (see [4, 5, 12]).

More precisely, this paper considers the nonlinear Schrodinger equation:

-[DELTA]u + f(u) = V(x)u, u [member of] [H.sup.1] ([[??].sup.N]). (6)

Here, N [greater than or equal to] 3 and [H.sup.1]([[??].sup.N]) is the Sobolev space:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

For V: [[??].sup.N] [right arrow] [??], the following is assumed:

([V.sub.1]) Consider V [member of] [C.sup.1]([[??].sup.N] \ {0}), V [greater than or equal to] 0 in [[??].sup.N] \ {0}, [lim.sub.[absolute value of (x)][right arrow][infinity]] V(x) = 0, V [member of] [L.sup.s.sub.loc]([[??].sup.N]) with s = max{2, N/2} if N [greater than or equal to] 3 and N [not equal to] 4, V [member of] [L.sup.s.sub.loc]([[??].sup.N]) for some s > 2 if N = 4, and there exist R > 0, 0 < [beta] < 2, and C > 0 such that, for [absolute value of (x)] > R,

V(x) [greater than or equal to] C[[absolute value of (x)].sup.-[beta]]. (8)

([V.sub.2]) For any x [member of] [[??].sup.N] \ {0}, x * [nabla]V(x) [less than or equal to] 0 and there exists 0 < [gamma] < 2 such that

[gamma]V(x) + x * [nabla]V(x) [greater than or equal to] 0, [for all]x [member of] [[??].sup.N] \{0}. (9)

For f, the following is assumed:

([f.sub.1]) f : [??] [right arrow] [??] is a continuous function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

tf (t) [greater than or equal to] 0, [for all]t [member of] [??]. (12)

([f.sub.2]) There exists D > 0 such that, for any t [member of] [??],

F(t) [less than or equal to] Dt f (t). (13)

Here,

F(t) = [[integral].sup.t.sub.0] f (s) ds, t [member of] [??]. (14)

For t [member of] [??], let

g(t) = [a.sub.0]t - f(t), (15)

[??](t) = F (t) - (1/2)t f (t). (16)

([f.sub.3]) For any t [member of] [??], [??](t) [greater than or equal to] 0 and there exists [delta] [member of] (0, [a.sub.0]) such that if [absolute value of (g(t))] [greater than or equal to] ([a.sub.0] - [delta])[absolute value of (t)], then

[??](t) [greater than or equal to] [delta]. (17)

([f.sub.4]) For any t [member of] [??], f(-t) = -f(t).

The main result of this paper is as follows.

Theorem J. Suppose that ([V.sub.1])-([V.sub.2]) and ([f.sub.1])-([f.sub.4]) are satisfied; then, problem (6) has infinitely many solutions.

Remark 2. (i) Under the assumptions ([V.sub.1]) and ([V.sub.2]), the essential spectrum of T defined by (3) is [0, +[infinity]) (see Lemma 3 in Section 2). Therefore, condition (11) indicates that f satisfies a resonance type condition at the essential spectrum of T.

(ii) A typical example for V which satisfies ([V.sub.1])-([V.sub.2]) is

V(r, [theta]) = P([theta])/[r.sup.[beta]]. (18)

Here, r = [absolute value of (x)], [theta] = x/[absolute value of (x)] [member of] [S.sup.N-1] := {y [member of] [[??].sup.N] | [absolute value of (y)] = 1}, 0 < [beta] < min{2, N/2}, P [member of] [C.sup.1]([S.sup.N-1]), and P > 0 in [S.sup.N-1].

(iii) There are many functions satisfying ([f.sub.1])-([f.sub.4]). Let G(t) = [[integral].sup.t.sub.0] g(s)ds. Using

d/dt(G(t)/[t.sup.2]) = 2[??](t)/[t.sup.3], (19)

we can construct a function f satisfying ([f.sub.1])-([f.sub.4]) from a given [??]. For example, let

[??](t) = [sigma] [[absolute value of (t)].sup.3]/1 + [[absolute value of (t)].sup.5/2], t [member of] [??], (20)

where

[sigma] = [a.sub.0]/4[tau], [[integral].sup.+[infinity].sub.0] ds/1 + [[absolute value of (s)].sup.5/2]. (21)

Then, by (19), for t [greater than or equal to] 0,

G(t) = 2[sigma][t.sup.2] [[integral].sup.t.sub.0] ds/[[absolute value of (s)].sup.5/2] (22)

and, for t < 0, G(t) = G(-t). It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

and f(t) = [a.sub.0]t - g(t) for any t [member of] [??]. Because the even function g(t)/t is strictly increasing in (0, +[infinity]), g(t)/t [right arrow] [a.sub.0] as t [right arrow] +[infinity], it can be deduced that f satisfies tf(t) > 0, [for all]t [member of] [??]\{0}. Moreover, it is easy to verify that f(t) = [a.sub.0]t + o(t) as t [right arrow] 0, f(t) = (2/3)[sigma][t.sup.-1/2] + o([t.sup.-1/2]) as [absolute value of (t)] [right arrow] [infinity], F(t) = (1/2)[a.sub.0][t.sup.2] + o([t.sup.2]) as t [right arrow] 0, and F(t) = (4/3)[sigma][[absolute value of (t)].sup.1/2] + o([[absolute value of (t)].sup.1/2]) as [absolute value of (t)] [right arrow] [infinity]. Because the even function g(t)/t is strictly increasing in (0, +[infinity]), g(t)/t [right arrow] [a.sub.0] as t [right arrow] +[infinity], and [??](t) [right arrow] +[infinity] as [absolute value of (t)] [right arrow] +[infinity], it can be deduced that, for sufficiently small [delta] > 0, if [absolute value of (g(t))]/[absolute value of (t)] [greater than or equal to] [a.sub.0] - [delta], then [??](t) [greater than or equal to] [delta]. Therefore, f satisfies ([f.sub.1])-([f.sub.4]).

Let

X = [H.sup.1] ([[??].sup.N]). (24)

The inner product and the norm in X are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

respectively. Here, [a.sub.0] is the constant in (10).

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

Then, for any u, v [member of] X, the Gateaux derivative of [PHI] is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

It follows that u [member of] X is a solution of (6) if and only if u is a critical point of [PHI]. In Section 2, it is shown that [PHI] satisfies the Cerami condition. In Section 3, an abstract critical point theorem from [13] is used to show that [PHI] has infinitely many critical points.

Notation. Let E be Banach space. We denote the dual space of E by E' and denote strong and weak convergence in E by [right arrow] and [??] respectively. For [phi] [member of] [C.sup.1] (E, [??]), we denote the Frechet derivative of [phi] at u by [phi]' (u). The Gateaux derivative of f is denoted by <[phi]' (u), v>, [for all]u, v [member of] E. [L.sup.p] ([[??].sup.N]) denotes the standard [L.sup.p] space (1 [less than or equal to] p [less than or equal to] [infinity]). We use O(h) and o(h) to mean [absolute value of (O(h))] [less than or equal to] C[absolute value of (h)] and o(h)/[absolute value of (h)] [right arrow] 0, respectively.

2. The Cerami Condition for [PHI]

Lemma 3. If V satisfies the assumptions ([V.sub.1]), then the essential spectrum of the Schrodinger operator T defined by (3) is [0, +[infinity]) and T has infinitely many negative eigenvalues accumulating at zero.

Proof. Because V satisfies ([V.sub.1]), by [14, Theorem XIII.15(b)] or [2, Theorem 10.29(b)], the essential spectrum of the Schrodinger operator T defined by (3) is [0, +[infinity]). The second result of this lemma comes from [2, Theorem 10.31] or [14, Theorem XIII.6(a)].

Lemma 4. If V satisfies the assumptions ([V.sub.1]) and ([V.sub.2]), then T has no eigenvalue in [0, +[infinity]).

Proof. Because V satisfies the assumptions ([V.sub.1] and ([V.sub.2]), by [15, Corollary 2], to prove this lemma, it suffices to prove that V satisfies the conditions I and II in [15, Corollary 2]. For this, it suffices to prove that, for 0 < [alpha] < 2, the functions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

are bounded functions in [[??].sup.N]. Because [lim.sub.[absolute value of (x)][right arrow][infinity]]V(x) = 0 and 0 < [alpha] < 2, there exists R' > 0 such that, for [absolute value of (x)] > R',

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Because V [member of] [L.sup.N/2.sub.loc]([[??].sup.N]), for [absolute value of (x)] [less than or equal to] R', the following is true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

Because 0 < [alpha] < 2, the following holds:

[[integral].sub.[absolute value of (y)][less than or equal to]1] [[absolute value of (y)].sup.N(4-N-[alpha])/(N-2)] < +[infinity]. (31)

Combining (29) and (30) yields that M is a bounded function in [[??].sup.N]. From ([V.sub.2]), we have L(x) [less than or equal to] [gamma]M(x), [for all]x [member of] [[??].sup.N]. It follows that L is also a bounded function in [[??].sup.N].

Definition 5 (see [16]). Let E be a Banach space. Let J [member of] [C.sup.1](E, [??]) and c [member of] [??]. One can call the fact that J satisfies the Cerami condition at c, denoted by [(C).sub.c] condition, if for any {[u.sub.n]} [subset] E satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

there exists u [member of] E such that, up to a subsequence, [parallel][u.sub.n] - u[parallel] [right arrow] 0 as n [right arrow] [infinity]. If J satisfies the [(C).sub.c] condition for every c [member of] [??], then J is said to satisfy the Cerami condition. A sequence {[u.sub.n]} [subset] E satisfying (32) is called a [(C).sub.c] sequence of J.

It was shown in [13] that the Cerami condition actually suffices to get a deformation theorem (see [8]) and then, by standard minimax arguments, it allows rather general minimax results.

Lemma 6. Suppose that V satisfies that V [member of] [L.sup.N/2.sub.loc]([[??].sup.N]) and [lim.sub.[absolute value of (x)][right arrow][infinity]] V(x) = 0. If [u.sub.n] [??] u in X, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Proof. Because [u.sub.n] [??] u in X, {[parallel][u.sub.n][parallel]} is bounded. It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

From [lim.sub.[absolute value of (x)][right arrow][infinity]] V(x) = 0, we deduce that, for any [EPSILON] > 0, there exists [R.sub.[EPSILON]] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

For y [member of] [[??].sup.N] and r > 0, let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in X, by Lemma 2.13 of [6], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

Combining (36) and (38) yields (33). ?

Lemma 7. Let {[u.sub.n]} [subset] X be a [(C).sub.c] sequence of [PHI]. Then, up to a subsequence, {[parallel][u.sub.n][parallel]} is bounded.

Proof. By Definition 5, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

Arguing indirectly, assume that [parallel][u.sub.n][parallel] [right arrow] [infinity] and set [w.sub.n] = [u.sub.n]/[parallel][u.sub.n][parallel].

Case 1. {[w.sub.n]} is nonvanishing; that is, there exist [y.sub.n] [member of] [[??].sup.N] and [alpha] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

Set [[??].sub.n] = [w.sub.n](* + [y.sub.n]). Then, by (40), [[??].sub.n] [??] w [not equal to] 0 in X.

By (39) and (27), for any [phi] [member of] [C.sup.[infinity].sub.0] ([[??].sup.N]), we have, as n [right arrow] [infinity],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (41)

It follows that, as n [right arrow] [infinity],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (42)

where [[??].sub.n] = [u.sub.n](* + [y.sub.n]). If lim [absolute value of ([y.sub.n])] = +[infinity], sending n [right arrow] [infinity] in (42), by (11), [lim.sub.[absolute value of (x)][right arrow][infinity]] V(x) = 0 and the fact that, for x [member of] {x [member of] [[??].sup.N] | w [not equal to] 0}, [absolute value of ([[??].sub.n](x))] +[infinity], we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (43)

This is impossible, because 0 [not equal to] w [member of] X. If {[y.sub.n]} has a subsequence bounded in [[??].sup.N], up to a subsequence, it can be assumed that [y.sub.0] = [lim.sub.n[right arrow][infinity]][y.sub.n], and sending n [right arrow] [infinity] in (42), we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (44)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (45)

This implies that w(* - [y.sub.0]) is a nonzero eigenfunction of T corresponding to eigenvalue 0. It contradicts Lemma 4.

Case 2. {[w.sub.n]} is vanishing; that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (46)

The Lions lemma (see, e.g., [6, Lemma 1.21]) shows that, for any p [member of] (2, 2N/(N - 2)),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (47)

Here, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes [L.sup.p]([[??].sup.N]) norm. Equations (39) and (27) show that, as n [right arrow] [infinity],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (48)

where g is defined in (15). It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

By Lemma 6 and [w.sub.n] [??] 0 in X, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (50)

Together with (49), this implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (51)

By (39),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (52)

Let

[E.sub.n] = {x [member of] [[??].sup.N] | [absolute value of (g([u.sub.n](x))/[u.sub.n](x))] [greater than or equal to] [a.sub.0] - [delta]}. (53)

By [??] [greater than or equal to] 0, (17), and (52), the following is true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (54)

Here, mes(A) denotes the Lebesgue measure of A. It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (55)

Let

[F.sub.n] = {x [member of] [[??].sup.N] | [absolute value of (g([u.sub.n](x))/[u.sub.n](x))] < [a.sub.0] - [delta]}. (56)

Because [delta] [member of] (0, [a.sub.0]) (see ([f.sub.3])), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (57)

Under the assumption ([f.sub.1]), there exists C > 0 such that

[absolute value of (f(t))] [less than or equal to] C [absolute value of (t)], [for all]t [member of] [??]. (58)

It follows that [absolute value of (g([u.sub.n]))] [less than or equal to] C[absolute value of ([u.sub.n])] for some constant C > 0. Then, by (55) and (47), for 2 < p < 2N/(N - 2),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (59)

Combining (57) and (59) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (60)

This contradicts (51).

From the two cases given above, it can be deduced that {[u.sub.n]} is bounded in X.

Lemma 8. The functional [PHI] satisfies the [(C).sub.c] condition for every c [member of] [??].

Proof. Let {[u.sub.n]} be a [(C).sub.c] sequence of [PHI]. By Lemma 7, {[parallel][u.sub.n][parallel]} is bounded. It follows that there exists u [member of] X such that, up to a subsequence, [u.sub.n] [??] u in X. Because the embedding X [??] [L.sup.2.sub.loc]([[??].sup.N]) is compact, we get that, up to a subsequence,

[u.sub.n] [right arrow] u a.e. in [[??].sup.N]. (61)

As the proof of Lemma 6.15 of [6], we have [PHI]'(u) = 0. It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (62)

We are going to prove that, up to a subsequence, [parallel][u.sub.n] u[parallel] [right arrow] 0 as n [right arrow] [infinity].

By (39), as n [right arrow] [infinity],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (63)

By Lemma 6,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (64)

Combining (62)-(64) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (65)

Because [u.sub.n] [??] u in X, by [17, Theorem 1.6], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (66)

Because tf(t) [greater than or equal to] 0 for any t [member of] [??] (see (12)), by (61) and the Fatou lemma, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (67)

Using (65)-(67), it can be deduced that, up to a subsequence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (68)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (69)

By (13), {D[u.sub.n]f([u.sub.n]) - F([u.sub.n])} is a sequence of nonnegative functions. Because

D[u.sub.n]f ([u.sub.n]) - F ([u.sub.n]) [right arrow] Duf (U) - F(u), a.e. in [[??].sup.N], (70)

by the Fatou lemma and (69), the following is true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (71)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (72)

By (12), we deduce that F [greater than or equal to] 0 in R. Then, by

F ([u.sub.n]) [right arrow] F (u), a.e. in [[??].sup.N], (73)

and the Fatou lemma, the following holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (74)

Combining (72) and (74) leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (75)

By (10) and (11), it can be deduced that there exists C > 0 such that

[absolute value of (f(t))] [less than or equal to] C[absolute value of (t)], [for all] [member of] R. (76)

Then, by the mean value theorem, the Holder inequality, and (76), we get that, for any a, b [member of] [??] and for any sufficiently small [EPSILON] > 0, there exists 0 < [theta] < 1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (77)

Because (61), (75), and (77) hold and {[parallel][u.sub.n][parallel]} is bounded, Theorem 2 of [18] shows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (78)

It follows that, for every sufficiently small [EPSILON] > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (79)

Here,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (80)

Equations (10) and (12) show that

F(t) > 0, t [not equal to] 0, (81)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (82)

Combining (81) and (82) yields that, for any sufficiently small [EPSILON] > 0, there exists [C.sub.[EPSILON]] > 0 such that

[t.sup.2] [less than or equal to] [C.sub.[EPSILON]]F(t), [for all]t [member of] [-[[EPSILON].sup.-1], [[EPSILON].sup. -1]]. (83)

By (79) and (83), we get that, for every sufficiently small [EPSILON] > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (84)

From

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (85)

we get that

mes ([[??].sup.N] \ [E.sub.n,[EPSILON]]) [less than or equal to] M[[EPSILON].sup.2], n = 1, 2, ... .(86)

By (86), the following holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (87)

where 2* = 2N/(N - 2) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (88)

Combining (84) and (87) leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (89)

Together with (68), this implies [u.sub.n] [right arrow] u in X. This completes the proof. ?

3. Proof of Theorem 1

For the proof of Theorem 1, the following abstract theorem, which is a corollary of [13, Theorem 2.4], is used.

Theorem 9. Let E be a real Hilbert space with norm [parallel] * [parallel]. Suppose that J is a [C.sup.1] -functional defined in E and satisfies the following conditions:

([A.sub.1]) J satisfies the [(C).sub.c] condition for every c > 0, and J(0) [greater than or equal to] 0.

([A.sub.2]) J is an even functional; that is, J(-u) = J(u) for every u [member of] E.

([A.sub.3]) There exist [c.sub.0] > J(0) and a closed subspace Z of E with finite codimension such that there exists [rho] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (90)

Here, [S.sub.[rho]] = {u [member of] E \ [parallel]u[parallel] = [rho]}.

([A.sub.4]) For any n [member of] N, there exists a subspace V [subset] X with dimension larger than or equal to n such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (91)

Then, J has infinitely many pairs of critical points.

Proof. Arguing indirectly, assume that J has only a finite number of critical points, say [n.sub.0]. The codimension of Z is here denoted by [m.sub.0]. By ([A.sub.4]), there exists a subspace V c Y with dimension larger than or equal to [n.sub.0] +[m.sub.0] +1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (92)

This indicates that there exists [c.sub.[infinity]] > [c.sub.0] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (93)

Then, by [13, Theorem 2.4], J has at least ([n.sub.0] + [m.sub.0] + 1)-[m.sub.0] = [n.sub.0] + 1 critical points. This contradicts the assumption that J has only [n.sub.0] critical points.

Let

[[lambda].sub.1] < [[lambda].sub.2] [less than or equal to] [[lambda].sub.3] [less than or equal to] *** [less than or equal to] [[lambda].sub.n] [less than or equal to] *** (94)

be the eigenvalues of T below the essential spectrum of T. Each [[lambda].sub.n] has been repeated in the sequence according to its finite multiplicity. Then, by Lemma 3, the following is true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (95)

Let

[n.sub.*] = min {n [member of] [??] \ [a.sub.0] + [[lambda].sub.n] > 0}. (96)

Here, [a.sub.0] is the constant in (10). From this definition,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (97)

Let {[E.sub.[lambda]] \ [lambda] [member of] [??]} denote the spectral family of the operator T defined by (3).

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (98)

Here, id : [L.sup.2]([[??].sup.N]) [right arrow] [L.sup.2]([[??].sup.N]) is the identity map and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (99)

It is easy to see that the codimension of Z equals [n.sub.*] - 1.

Lemma 10. There exists [zeta] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (100)

Proof. From (96) and (98), the following holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (101)

Here,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (102)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (103)

If [zeta] = 0, then there exists {[u.sub.n]} [subset] Z satisfying [parallel][u.sub.n][parallel] = 1, n= 1, 2, ... , and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (104)

Up to a subsequence, it can be assumed that [u.sub.n] [??] [u.sub.0] in X and [u.sub.n] [right arrow] [u.sub.0] a.e. in [[??].sup.N]. Since Z is a closed subspace of X, by [u.sub.n] [??] [u.sub.0] in X, we have [u.sub.0] [member of] Z. Because [u.sub.n] [??] [u.sub.0] in X, by [17, Theorem 1.6], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (105)

Because [u.sub.n] [right arrow] [u.sub.0] a.e. in [[??].sup.N], by the Fatou lemma, the following holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (106)

Moreover, by Lemma 6,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (107)

Combining (104)-(107) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (108)

Because [u.sub.0] [member of] Z, by (101) and (108), we have [u.sub.0] = 0. Then, by (104) and (107),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (109)

This contradicts [parallel][u.sub.n][parallel] = 1, n= 1, 2, ... .Therefore, [eta] > 0.

Proof of Theorem 1. We are going to prove that the functional O satisfies the conditions ([A.sub.1])-([A.sub.4]) in Theorem 9.

By Lemma 8, the functional [PHI] satisfies the condition ([A.sub.1]). By ([f.sub.4]), [PHI] satisfies the condition ([A.sub.2]).

By (10) and (11), it can be deduced that, for any [EPSILON] > 0, there exists [C.sub.[EPSILON]] > 0 such that

[absolute value of (F(t) - ([a.sub.0][t.sup.2]/2))] [less than or equal to] [EPSILON][t.sup.2] + [C.sub.[EPSILON]] [[absolute value of (t)].sup.[mu]], [for all]t [member of] [??]. (110)

Here,

[mu] = (2N - 2/N - 2) [member of] (2, 2*). (111)

Let Z be the space defined by (98). It has a finite codimension. By Lemma 10 and (110), we have, for any u [member of] Z,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (112)

where the positive constant C' comes from the Sobolev inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Choosing [EPSILON] = [eta]/4 and [rho] = [([eta]/8C'[C.sub.[EPSILON]]).sup.1/([mu]-2)], by (112), the following holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (113)

It follows that [PHI] satisfies the assumption ([A.sub.3]).

Let [??] [member of] [C.sup.[infinity].sub.0]([[??].sup.N]) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, for the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, by (8), it follows that, for [lambda] > R,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (114)

where C is a positive constant independent of X. Because [beta] < 2, it follows that there exists [[lambda].sub.0] > R such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (115)

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. These functions have mutually disjoint compact supports. Let

[V.sub.n] = span {[[psi].sub.1], [[psi].sub.2], ... , [[psi].sub.n]}. (116)

Then, [V.sub.n] is an n-dimensional subspace of X. By (10) and (11), it can be deduced that, for any [EPSILON] > 0, there exist [L.sub.[EPSILON]] > 0 and [K.sub.[EPSILON]] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (117)

Because {[[psi].sub.i] | i [member of] N} have mutually disjoint compact supports, by (115), we have, for u = [[sigma].sup.n.sub.i=1] [t.sub.i][[psi].sub.i] [member of] [V.sub.n],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (118)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (119)

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (120)

Then, [[OMEGA].sub.n] is a bounded set in [[??].sup.N], since every [[psi].sub.i] has a compact support. By (117),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (121)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (122)

Choosing

[EPSILON] = min{-[[alpha].sub.i] | 1 [less than or equal to] i [less than or equal to] n}/4 max {[[omega].sub.i] | 1 [less than or equal to] i [less than or equal to] n}, (123)

by (118) and (121), we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (124)

Since [[alpha].sub.i] < 0, for every i, (124) implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (125)

Therefore, [PHI] satisfies the condition ([A.sub.4]) of Theorem 9.

Because [PHI] satisfies the conditions ([A.sub.1])-([A.sub.4]) of Theorem 9, by Theorem 9, it has infinitely many critical points. It follows that (6) has infinitely many solutions.

http://dx.doi.org/10.1155/2016/3042493

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Shaowei Chen was supported by Science Foundation of Huaqiao University (13BS208) and Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY119).

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Shaowei Chen and Haijun Zhou

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

Correspondence should be addressed to Shaowei Chen; swchen6@163.com

Received 12 December 2015; Accepted 14 February 2016

Academic Editor: Jacopo Bellazzini
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Title Annotation:Research Article
Author:Chen, Shaowei; Zhou, Haijun
Publication:Advances in Mathematical Physics
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Date:Jan 1, 2016
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