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Multiple Nontrivial Solutions for a Class of Biharmonic Elliptic Equations with Sobolev Critical Exponent.

1. Introduction and Main Results

In the present paper, we are concerned with the existence of multiple solutions to the following biharmonic elliptic equation with perturbation

[[DELTA].sup.2]u = [[absolute value of (u)].sup.p-2] u + f, x [member of] [OMEGA], u = [nabla]u = 0, x [member of] [partial derivative][OMEGA], (1)

where [OMEGA] is a bounded domain in [R.sup.N] (N [greater than or equal to] 5), [[DELTA].sup.2] is the biharmonic operator, and p = [2.sup.**] = 2N/(N - 4) is the Sobolev critical exponent.

The second-order semilinear and quasilinear problems have been object of intensive research in the last years. Brezis and Nirenberg [1] have studied the existence of positive solutions of (1). Particularly, when f = [lambda]u, where [lambda] [member of] R is a constant, they have discovered the following remarkable phenomenon: the qualitative behavior of the set of solutions of (1) is highly sensitive to N, the dimension of the space. Precisely, Brezis and Nirenberg [1] have shown that, in dimension N [greater than or equal to] 4, there exists a positive solution of (1), if and only if [lambda] [member of] (0, [[lambda].sub.1]); while, in dimension N = 3 and when [OMEGA] = [B.sub.1] is the unit ball, there exists a positive solution of (1), if and only if [lambda] [member of] ([[lambda].sub.1]/4, [[lambda].sub.1]), where [[lambda].sub.1] > 0 is the first eigenvalue of -[DELTA] in [OMEGA]. For more results on this direction we refer the readers to [2-5] and the references therein.

During the last decades many works have been orientated to the analysis of biharmonic nonlinear Schrodinger equation (BHNSE)

i[[phi].sub.t] - [[DELTA].sup.2][phi] + g (x, [absolute value of ([phi])]) [phi] = 0, [phi] (0, x) = [[phi].sub.0] (x) [member of] [H.sup.2.sub.0] ([OMEGA]), (2)

[OMEGA] [subset] [R.sup.N] is an open domain N [greater than or equal to] 5. For instance, paper [6] proved that some of the properties and characteristics for the second-order semilinear problems can be extended to BHLSE. Paper [7] proved the existence of blow-up solutions. In papers [8-10], the authors proved the existence of global solutions, in particular, looking for standing wave solutions for (2) of the form

[phi] (t, u) = [e.sup.i[lambda]t] u (3)

such that u is a solution satisfying the equation

[[DELTA].sup.2]u + [lambda]u = [??] (x, u), u [member of] [H.sup.2.sub.0] ([OMEGA]). (4)

If [lambda] = 0 and [??] (x, u) = [[absolute value of (u)].sup.p-2], we know that (4) admits no positive solutions if [OMEGA] is star shaped under the Navier or Dirichlet boundary conditions (see [11, Theorem 3.3] and [12, Corollary 1]). If [lambda] > 0 and [OMEGA] is a ball, paper [13] proved the existence of positive radially symmetric solutions. For more general results on this direction one can refer to [14, 15, 15-21] and the references therein.

Motivated by the above results, we study the case that [lambda] = 0, [??](x, u) = [[absolute value of (u)].sup.p-2] u + f(x), and [OMEGA] [subset] [R.sup.N] is abounded domain. Precisely, we shall generalize the results of Tarantello [22] to the biharmonic and critical exponent case. Our main tool here is the Nehari manifold method which is similar to the fibering method of Pohozaev's.

In order to state the main results, we shall give some notation and assumptions. Let D = [H.sup.2.sub.0]([OMEGA]), and [[parallel]u[parallel].sub.p] = [([[integral].sub.[OMEGA]] [[absolute value of (u)].sup.p] dx).sup.1/p] be the usual [L.sup.p]([OMEGA]) norm. Obviously, D is a Hilbert space under the inner product <u, v> = [[integral].sub.[OMEGA]] [DELTA]u[DELTA]vdx. Correspondingly, the norm is denoted by [parallel] x [parallel]; i.e., [[parallel]u[parallel].sup.2] = [[integral].sub.[OMEGA]] [[absolute value of ([DELTA]u)].sup.2] dx. Assume that f [member of] [L.sup.q]([OMEGA])(q = 2N/(N + 4))(f [not equal to] 0) satisfies

[[parallel]f[parallel].sub.q] [less than or equal to] [C.sub.N][S.sup.(N+4)/8], (5)

where S is the best Sobolev embedding constant of D [??] [L.sup.p]([OMEGA]) (p = [2.sup.**] = 2N/(N - 4)), and

[mathematical expression not reproducible] (6)

Let

[mathematical expression not reproducible] (7)

be an extremal function for the Sobolev inequality in [R.sup.N]. For a [member of] [OMEGA], let [u.sub.[epsilon],a](x) = [u.sub.[epsilon]](x - a) and [[xi].sub.a] [member of] [C.sup.[infinity].sub.0]([OMEGA]) with [[xi].sub.a] [greater than or equal to] 0 and [[xi].sub.a] = 1 near a. We point out that the embedding D [??] [L.sup.P]([OMEGA]) is not compact. This leads to the lack of compactness for the proved existence and multiplicity of nontrivial solutions of (1). Motivated by [1, 22], we recover the local compactness by dividing the Nehari manifold into three parts and give some estimates for the least energy of (1)

It is easy to see that the energy functional of (1) is denoted by

[mathematical expression not reproducible] (8)

Hence, I is well defined (under (5)) and of the class [C.sup.2]([OMEGA]). Moreover, all the critical points of I are precisely the solutions of (1). We define the Nehari manifold N associated with the functional by

N = {u [member of] D | <I' (u), u> = 0}. (9)

It is clear that all critical points lie in the Nehari manifold, and it is usually effective to consider the existence of critical points in this smaller subset of the Sobolev space. For fixed u [member of] D {0}, we set

[mathematical expression not reproducible] (10)

The mapping is called fibering map. Such maps are often used to investigate Nehari manifolds for various semilinear problems. From the relationship between I and [phi](t), we can divide N into three parts

[mathematical expression not reproducible] (11)

It turns out that under the assumption (5), we infer that [N.sup.0] = {0} (see Lemma 5 below). Now the main result in this paper can be stated as follows.

Theorem 1. Assume that f [not equal to] 0 satisfies (5). Then

[mathematical expression not reproducible] (12)

is achieved at a point [u.sub.0] [member of] N. Furthermore, [u.sub.0] is a critical point of I, and [u.sub.0] [greater than or equal to] 0 when f [greater than or equal to] 0.

In the following we study the second infimum problem

[mathematical expression not reproducible] (13)

In this case we have the following results.

Theorem 2. Assume that f [not equal to] 0 satisfies (5). Then [c.sub.1] > [c.sub.0] and the infimum in (13) is achieved at a point [u.sub.1] [member of] [N.sup.-], which is a critical point of I.

The proofs of Theorems 1-2 rely on the Ekeland's variational principle and careful estimates (see [1]) of minimizing sequence.

2. Some Preliminary Results

In this section we prove some preliminary results for the proof of Theorems 1-2. The main ideas are coming from [1, 22]. We begin with the following lemma which states the purpose of assumption (5).

Lemma 3. Supposed that f [not equal to] 0 satisfies (5). For every u [member of] D\{0}, there exists a unique [t.sub.1] = [t.sub.1](u) > 0 such that [t.sub.1]u [member of] [N.sup.~]. Particularly, we have

[mathematical expression not reproducible] (14)

and [mathematical expression not reproducible]. Moreover, if [[integrl].sub.[OMEGA]] fudx > 0, then there exists a unique [t.sub.2] = [t.sub.2](u) > 0 such that [t.sub.2]u [member of] [N.sup.+]. In particular, one has

[t.sub.2] < [[[[parallel][DELTA]u[parallel].sup.2.sub.2]/(p - 1) [[parallel]u[parallel].sup.p.sub.p]].sup.1/(p-2)] (15)

and I([t.sub.2]u) [less than or equal to] I(tu), [for all]t [member of] [0, [t.sub.1]].

Proof. Recall that the fibering map is defined by

[mathematical expression not reproducible] (16)

Then

[mathematical expression not reproducible] (17)

We deduce from g' (t) = 0 that

[mathematical expression not reproducible] (18)

If 0 < t < [t.sub.max], we have [phi]"() = g'(t) > 0, and if t > [t.sub.max], one sees [phi]"(t) = g'(t) < 0. A direct computation shows that g(t) achieves its maximum at [t.sub.max], and

[mathematical expression not reproducible] (19)

We divide the following two cases to accomplish our results.

Case 1. If [[integral].sub.[OMEGA]] fudx [less than or equal to] 0, then [phi]'([t.sub.max]) = g([t.sub.max]) - [[integral].sub.[OMEGA]] fudx > 0. It is easy to see that if t [right arrow] +[infinity], we have [phi]'(t) < 0. So, there exists unique [t.sub.1] > [t.sub.max] such that [phi]'([t.sub.1]) = 0 and g([t.sub.1]) = [[integral].sub.[OMEGA]] fudx. We infer from the monotonicity of g(t) that, for [t.sub.1] > [t.sub.max],

[mathematical expression not reproducible] (20)

This shows that [t.sub.1]u [member of] [N.sup.-].

Case 2. If [[integral].sub.[OMEGA]] fudx > 0, we infer from assumption (5) that [[integral].sub.[OMEGA]] fudx < g([t.sub.max])[for all]u [member of] D. Then [phi]'([t.sub.max]) = g([t.sub.max]) - [[integral].sub.[OMEGA]] fudx > 0. Since [phi]'(0) = -[[integral].sub.[OMEGA]] fudx < 0, there exists a unique [t.sub.2] [member of] [0, [t.sub.max]] such that [phi]'([t.sub.2]) = 0 and g([t.sub.2]) = [[integral].sub.[OMEGA]] fudx. A direct computation shows that [t.sub.2]u [member of] [N.sup.+] and I([t.sub.2]u) [less than or equal to] I(tu), [for all]t [member of] [0, [t.sub.1]].

Lemma 4. Assume that f [not equal to] 0 satisfies (5). We infer that the infimum

[mathematical expression not reproducible] (21)

is achieved, where [[mu].sub.0] > 0.

The proof of Lemma 4 is technical and the idea of the proof is mainly motivated by paper [23]. We shall prove it in the appendix. Next we study the property of the set [N.sup.0].

Lemma 5. Let f [not equal to] 0 satisfy (5). Then for every u [member of] N, u [not equal to] 0, we can get the conclusion that [N.sup.0] = {0}.

Proof. We use the contradiction arguments. Assume that, for some u [member of] N, u [not equal to] 0, we have u [member of] [N.sup.0]. That is,

[[parallel][DELTA]u[parallel].sup.2.sub.2] - (p - 1) [[parallel]u[parallel].sup.p.sub.o] = 0. (22)

Since u [member of] N, it follows that [[parallel][DELTA]u[parallel].sup.2.sub.2] - [[parallel]u[parallel].sup.p.su.p] - [[integral].sub.[OMEGA]] fudx = 0. Hence, we get

[[parallel]u[parallel].sup.p] - [[integral].sub.[OMEGA]] fudx = 0. (23)

By Sobolev inequality, we deduce that (p - 2) [[parallel]u[parallel].sub.p] [greater than or equal to] [(S/(p - 1)).sup.1/(p-2)]. For u [not equal to] 0, we set

[mathematical expression not reproducible] (24)

For t [less than or equal to] 0 and [[parallel]u[parallel].sub.p] = 1, a direct computation shows that

A(tu) = t [[C.sub.N] [[parallel][DELTA]u[parallel].sup.(N+4)/4.sub.2] - [[integral].sub.[OMEGA]] fudx]. (25)

We derive from Lemma 4 that, for [gamma] > 0,

[mathematical expression not reproducible] (26)

Let [gamma] = [(S/(p - 1)).sup.1/1(p-2)] > 0. We infer from (26) that

[mathematical expression not reproducible] (27)

which is a contradiction.

Lemma 6. Let f [not equal to] 0 satisfy (5). For each u [member of] N \ {0}, there exist [xi] >0 and a differentiable function t = t(w) > 0, w [member of] D, [parallel]w[parallel] < [epsilon], satisfying the following:

[mathematical expression not reproducible] (28)

Proof. We define P : R x D [right arrow] R by

F (t, w) = t [[parallel][DELTA] (u - w)[parallel].sup.2.sub.2] - [t.sup.(p-1)] [[parallel]u - w[parallel].sup.p.sub.p] - [[integral].sub.[OMEGA]] f (u - w) dx. (29)

Since F(1,0) = 0 and [F.sub.t](1,0) = [[parallel][DELTA]u[parallel].sup.2.sub.2] - (p - 1) [[parallel]u[parallel].sup.p.sub.p] [not equal to] 0 (Lemma 5), by using the implicit function theorem at the point (1, 0) we know that the results of the lemma hold.

3. Proof of Theorem 1

In this part we shall give the proof of Theorem 1.

Proof of Theorem 1. We first claim that the functional I is bounded from below in N. For u [member of] N, we have <l' (u), u> = 0. That is, [[parallel][DELTA]u[parallel].sup.2.sub.2] - [[parallel]u[parallel].sup.p.sub.p] - [[integral].sub.[OMEGA]] fudx = 0. One deduces from (2) and Holder inequality that

[mathematical expression not reproducible] (30)

Hence, we know that the infimum [c.sub.0] is also bounded from below. Second, we can get an upper bound for [c.sub.0]. Let v [member of] D be the solution for [[DELTA].sup.2]u = f. For f [not equal to] 0, one obtains that

[[integral].sub.[OMEGA]] fvdx = [[parallel][DELTA]v[parallel].sup.2.sub.2] > 0. (31)

Set [t.sub.0] = [t.sub.2](v) > 0 as defined by Lemma 3. Thus, we have that [t.sub.0]v [member of] [N.sup.+] and

[mathematical expression not reproducible] (32)

For any minimizing sequence {[u.sub.n]} [subset] N, we can use Ekeland's variational principle (see [24]) to get following properties:

(i) I([u.sub.n]) < [c.sub.0] + 1 / n,

(ii) I(w) [greater than or equal to] I([u.sub.n]) - (1/n) [[parallel][DELTA](w - [u.sub.n])[parallel].sub.2]. [for all]w [member of] N.

Hence for n large enough, we obtain

[mathematical expression not reproducible] (33)

This implies

[[integral].sub.[OMEGA]] [fu.sub.n]dx [greater than or equal to] 4[t.sup.2.sub.0]/N + 4 [[parallel][DELTA]v[parallel].sup.2.sub.2] > 0, and [u.sub.n] [not equal to] 0. (34)

Since I([u.sub.n]) < 0, we infer from Holder's inequality that

[there exists]M > 0, [[parallel][DELTA][u.sub.n][parallel].sup.2.sub.b] [less than or equal to] M. (35)

At the same time, we observe that

4[t.sub.2.sub.0]/N + 4 [[parallel][DELTA]v[parallel].sup.2.sub.2] [less than or equal to] [[integral].sub.[OMEGA]] [fu.sub.n]dx. (36)

One deduces from (5) and Holder's and Sobolev's inequalities that

[there exists]m > 0, [[parallel][DELTA][u.sub.n][parallel].sup.2.sub.b] [greater than or equal to] m > 0. (37)

So we derive from (35) and (37) that

0 < m [greater than or equal to] [[parallel][DELTA][u.sub.n][parallel].sup.2.sub.2] [less than or equal to] M, (38)

where m and M only depend on f and [OMEGA].

Next we shall prove that [parallel]I'([u.sub.n])[parallel] [right arrow] [infinity], as n [right arrow] [infinity]. Applying Lemma 6 with u = [u.sub.n] and w = [delta](I' ([u.sub.n])/ [parallel]I' ([u.sub.n])[parallel])([delta] > 0), we can find some [t.sub.n]([delta]) = t[[delta](I'([u.sub.n])/ [parallel]I'([u.sub.n])[parallel])] such that

[w.sub.[delta]] = [t.sub.n] ([delta]) [[u.sub.n] - [delta] I' (u.sub.n])/ [parallel]I' ([u.sub.n])[parallel]] [member of] N. (39)

By condition (ii) we have

[mathematical expression not reproducible] (40)

Dividing by [delta] and letting [delta] [right arrow] 0, we get

[mathematical expression not reproducible] (41)

where [t'.sub.n](0) = <t' (0), I'([u.sub.n])/[parallel]I'([u.sub.n])[parallel]>. So, we conclude that

[mathematical expression not reproducible] (42)

where C is a constant. In order to complete the proof we need to prove that [t'.sub.n](0) is bounded uniformly on n. By Lemma 6 we can get

[mathematical expression not reproducible] (43)

Thus, there exists subsequence {[u.sub.n]} (still denote by {[u.sub.n]}) such that

[[parallel][DELTA][u.sub.n][parallel].sup.2.sub.2] - (p - 1) [[parallel][u.sub.n][parallel].sup.p.sub.p] = o(1). (44)

We infer from {[u.sub.n]} [subset] N that

[mathematical expression not reproducible] (45)

By the estimate of [[parallel][u.sub.n][parallel].sub.2] from (38), we have that [[parallel][u.sub.n][parallel].sub.p] [greater than or equal to] [gamma] > 0 and

[mathematical expression not reproducible] (46)

This is imp ossible. So, [[parallel][DELTA][u.sub.n][parallel].sup.2.sub.2] - (p-1) [[parallel][u.sub.n][parallel].sup.p.sub.b] is away from zero. Thus, we conclude that

[parallel]I' ([u.sub.n])[parallel] [right arrow] 0 as n [right arrow] [infinity]. (47)

Let [u.sub.0] [member of] D be the weak limit in D of {[u.sub.n]}. From (47) we can get that [u.sub.0] is a weak solution for (1). In fact, [u.sub.0] [member of] N and

[mathematical expression not reproducible] (48)

So, we have that [u.sub.n] [right arrow] [u.sub.0] strongly in D and I([u.sub.0]) = [c.sub.0] = [inf.sub.u[member of]N] I(u). Moreover, [u.sub.0] [member of] [N.sup.+]. By using standard method, we can prove that [u.sub.0] is a global minimum for I in D (See [25]).

4. Proof of Theorem 2

In this section, we shall give the proof of Theorem 2. Since the embedding D [??] [L.sup.2N/(N-4)] ([OMEGA]) is not compact, we need to find some way to recover this compactness. Motivated by previous works of [1, 22, 23], we will seek the level in which [(PS).sub.c]-condition will recover. Then we shall use the Mountain-Pass principle to get the second nontrivial solution of (1). The related problems have been studied in [23], and such an approach has been used. The threshold is found in the following lemma to obtain the compactness.

Lemma 7. Assume that the sequence {[u.sub.n]} [subset] D satisfying

(i) I([u.sub.n]) [right arrow]c with c < [c.sub.0] + (2/N)[S.sup.N/4], where [c.sub.0] is defined in (12).

(ii) [parallel]I' ([u.sub.n)[parallel] [right arrow] 0 as n [right arrow] [infinity].

Then {[u.sub.n]} has a convergent subsequence.

Proof. It is clear that [[parallel][DELTA][u.sub.n][parallel].sup.2.sub.2] is uniformly bounded from condition (i) and (ii). For a subsequence of [u.sub.n], we can get a [w.sub.0] [member of] D such that

[u.sub.n] [??] [w.sub.0] in D. (49)

So, from (ii), we obtain that

<I' ([w.sub.0]), w> = 0, for [for all]w [member of] D. (50)

Then [w.sub.0] is a weak solution of (1), [w.sub.0] [not equal to] 0, and [w.sub.0] [member of] N, I([w.sub.0]) [greater than or equal to] [c.sub.0]. Let [u.sub.n] = [w.sub.0] + [v.sub.n]. So, [v.sub.n] [??] 0 in D. By Brezis-Lieb lemma (see [24]), we conclude that

[mathematical expression not reproducible] (51)

Thus, for n large enough, we get

[mathematical expression not reproducible] (52)

which means

1/2 [[parallel][DELTA][v.sub.n][parallel].sup.2.sub.2] - 1/p [[parallel][v.sub.n][parallel].sup.p.sub.p] < 2/N [S.sup.N/4] + o (1). (53)

Moreover, we infer from condition (ii) that

[mathematical expression not reproducible] (54)

and then we obtain

[[parallel][DELTA][v.sub.n][parallel].sup.2.sub.2] - [[parallel][v.sub.n][parallel].sup.p.sub.p] = o(1). (55)

Next we shall prove that if (53) and (55) hold, then there exists the subsequence of {[v.sub.n]} (still denoted by {[v.sub.n]}), which satisfies

[[parallel][DELTA][v.sub.n][parallel].sub.2] [right arrow] 0, n [right arrow] +[infinity]. (56)

We assume {[v.sub.n]} is bounded away from 0; that is

[there exists]C > 0, [[parallel][DELTA][v.sub.n][parallel].sub.2] [greater than or equal to] C, [for all]n [member of] N. (57)

So from (55) we can get

[mathematical expression not reproducible] (58)

We infer from (53) and (55) that

[mathematical expression not reproducible] (59)

for n large. This is contradiction. So, we can get [u.sub.n] [right arrow] [w.sub.0] strongly in D.

Note that [u.sub.0] [not equal to] 0. Following [23], we set [SIGMA] [subset] [OMEGA] to be a set of positive measures such that [u.sub.0] > 0 on [SIGMA]. Let us define

[U.sub.[epsilon],a] (x) = [[xi].sub.a] (x) [u.sub.[epsilon],a] (x), x [member of] [R.sup.N], (60)

where [u.sub.[epsilon],a] (x) and [[xi].sub.a](x) are defined in Section 1. Without loss of generality, we take [u.sub.[epsilon],a] (x) = [[epsilon].sup.(N-4)/2]/ [([[epsilon].sup.2] + [[absolute value of (x - a)].sup.2]2).sup.(N-4)/2]. Then we have the following estimates for [U.sub.[epsilon],a].

Lemma 8. [for all]R > 0 and a.e. a [member of] [SIGMA], there exists [[epsilon].sub.0] > 0 such that

I([u.sub.0] + R[U.sub.[epsilon],a]) < [c.sub.0] + [2/N] [S.sup.N/4] (61)

for every 0 < [epsilon] < [[epsilon].sub.0].

Proof. Let [mathematical expression not reproducible] and [mathematical expression not reproducible]. By the definition of [u.sub.[epsilon]](x), we can get the Sobolev embedding exponent S = B/[A.sup.2/p]. A direct computation shows that

[mathematical expression not reproducible] (62)

Now we take the [C.sup.[infinity].sub.0] (Q) function [[xi].sub.a] (x) such that

[mathematical expression not reproducible] (63)

where [r.sub.0] > 0. On the other hand, we see that

[mathematical expression not reproducible] (64)

So, by direct computation we infer that

[mathematical expression not reproducible] (65)

where [[omega].sub.N-1] is the measure of the unit sphere in [R.sup.N]. Moreover, we have that

[mathematical expression not reproducible] (66)

Thus, we infer from [23] that

[mathematical expression not reproducible] (67)

From all of the above, noticing that [u.sub.0] [member of] N, one has that

[mathematical expression not reproducible] (68)

By using an estimate obtained by G. Folland [26] and setting [u.sub.0] = 0 outside [OMEGA], one gets that

[mathematical expression not reproducible] (69)

where

[mathematical expression not reproducible] (70)

Consequently, we have

[mathematical expression not reproducible] (71)

We set

h (s) = B/2 [s.sup.2] - A/P [s.sup.p] - [u.sub.0] (a) [E[epsilon](N-4)/2] [s.sup.p-1], s > 0, (72)

and assume h(s) achieves its maximum at [s.sub.1] > 0, which satisfies

[s.sub.1]B - [s.sup.p-1.sub.1] A = (p - 1) [u.sub.0] (a) [E[epsilon].sup.(N-4)/2] [s.sup.p-2]. (73)

We define

[s.sub.0] = [(B/A).sup.1/(p-2)], (74)

which is the maximum point of [h.sub.1](s) = (B/2)[s.sup.2] - (A/p)[s.sup.p]. We can conclude that 0 < [s.sub.1] < [s.sub.0], and [s.sub.1] [right arrow] [s.sub.0] ([epsilon] [right arrow] 0). Let [s.sub.1] = [s.sub.0] (1 - [delta]). It is easy to see that [delta] [right arrow] 0 ([epsilon] [right arrow] 0). From (73) we can get

[mathematical expression not reproducible] (75)

and then expanding for [delta], we can get

[mathematical expression not reproducible] (76)

So, one sees that

[mathematical expression not reproducible] (77)

When we take small [[epsilon].sub.0] > 0, we arrive at

I ([u.sub.0] + R[U.sub.[epsilon],a]) < [c.sub.0] + 2/N [S.sup.N/4], [for all]0 < [epsilon] < [[epsilon].sub.0]. (78)

This finishes the proof.

Now we are ready to give the proof of Theorem 2.

Proof of Theorem 2. It is clear that the uniqueness of [t.sub.1] (u) satisfies the following condition:

[mathematical expression not reproducible] (79)

At the same time, [t.sub.1] (u) is a continuous function of u. And [N.sup.-] divides D into two components [D.sub.1] and [D.sub.2], which are disconnected from each other. Let

[mathematical expression not reproducible] (80)

Obviously, D - [N.sup.-] = [D.sub.1] [union] [D.sub.2], and we can check [N.sup.+] [subset] [D.sub.1], [u.sub.0] [member of] [D.sub.1]. We can choose a constant [C.sub.0], which satisfies

0 < [t.sub.1] (u) [less than or equal to] [C.sub.0], [for all][parallel]u[parallel] = 1, (81)

and claim that

w = [u.sub.0] + [R.sub.0][U.sub.[epsilon],a] [member of] [D.sub.2], (82)

where [R.sub.0] = [((1/B)[absolute value of ([C.sup.2.sub.0] - [[parallel][u.sub.0][parallel].sup.2])]).sup.1/2] + 1. In fact, a direct computation shows that

[mathematical expression not reproducible] (83)

for [epsilon] > 0 small enough. Thus, claim (82) holds.

We fix [epsilon] > 0 such that both (61) and (82) hold by the choice of [R.sub.0] and a [member of] [SIGMA]. We set

[GAMMA] = {[gamma] [member of] C ([0, 1], D) : [gamma] (0) = [u.sub.0], [gamma] (1) = [u.sub.0] + [R.sub.0][U.sub.[epsilon],a]}, (84)

and take h(t) = [u.sub.0] + [tR.sub.0][U.sub.[epsilon],a], which belongs to [GAMMA]. From Lemma 7, we conclude that

[mathematical expression not reproducible] (85)

Since every h [member of] [GAMMA] intersects [N.sup.-], we get that

[mathematical expression not reproducible] (86)

Next we use Mountain-Pass lemma to prove Theorem 2. Let {[u.sub.n]} [subset] [N.sup.-] be such that

I ([u.sub.n]) [right arrow] [c.sub.1],

[parallel]I' ([u.sub.n])[parallel] [right arrow] 0. (87)

We deduce from Lemma 7 that there exists a subsequence (still denoted by {[u.sub.n]}) of {[u.sub.n]}, and [u.sub.1] [member of] D such that

[u.sub.n] [right arrow] [u.sub.1] in D. (88)

So, [u.sub.1] is a critical point for I, [u.sub.1] [member of] [N.sup.-] and I([u.sub.1]) = [c.sub.1].

Remark 9. We point out that the results of Theorems 1-2 can be generalized to polyharmonic problem. Precisely, we can consider the semilinear polyharmonic problem

[(-[DELTA]).sup.m] u = [[absolute value of (u)].sup.p-2] u + f, x [member of] [OMEGA],

u = Du = ... = [D.sup.m-1] u = 0, x [member of] [partial derivative][OMEGA], (89)

where [OMEGA] is a smooth bounded domain in [R.sup.N](N [greater than or equal to] 2m + 1). m [member of] [N.sup.+], p = 2N/(N - 2m) denotes the critical Sobolev exponent for [(-[DELTA]).sup.m], and f [member of] [L.sup.q]([OMEGA])(q = 2N/(N + 2m))(f [not equal to] 0) is small enough. We can define the energy functional:

[mathematical expression not reproducible] (90)

where

[mathematical expression not reproducible] (91)

H is Hilbert space and endowed with the scalar product

[mathematical expression not reproducible] (92)

and [[parallel]x[parallel].sub.m] is the corresponding norm. Let

[mathematical expression not reproducible] (93)

be an extremal function for the Sobolev inequality in [R.sup.N], and the constant [C.sub.N,m] be independent of [epsilon]. By dividing the Nehari manifold, we can prove [(PS).sub.C] condition when c < [c.sub.0] + (m/N)[S.sup.m/2N], where [c.sub.0] = I([u.sub.0]) and [u.sub.0] is the first solution. By using the same idea of this article, one can obtain that (89) has at least two nontrivial solutions.

https://doi.org/10.1155/2018/8212785

Appendix

In this appendix we mainly focus on the proof of Lemma 4.

Proof of Lemma 4. For u e D, we define

G (u) = [C.sub.N] [[parallel][DELTA]u[parallel].sup.(N+4)/4.sub.2] - [[integral].sub.[OMEGA]] fudx. (A.1)

Let {[u.sub.n]} be the minimizing sequence of (21) with [[parallel][u.sub.n][parallel].sub.p] = 1. That is, we have that

G ([u.sub.n]) = [[mu].sub.0] + o (1), (A.2)

and [u.sub.n] [??] [u.sub.0] in D, [u.sub.n] [right arrow] [u.sub.0] a.e in [OMEGA] and [[parallel][u.sub.0][parallel].sub.p] [less than or equal to] 1. If [[parallel][u.sub.0][parallel].sub.p] = 1, then the conclusion holds. In the following we consider the case [[parallel][u.sub.0][parallel].sub.p] < 1 by using contradiction argument. Let [u.sub.n] = [u.sub.0] + [w.sub.n]. So, [w.sub.n] [??] 0 in D. From Brezis-Lieb lemma [27], we obtain that

[mathematical expression not reproducible] (A.3)

By Sobolev's inequality, we conclude that

[mathematical expression not reproducible] (A.4)

Hence we get

[mathematical expression not reproducible] (A.5)

From paper [23], we know that for every u [member of] D, [[parallel]u[parallel].sub.p] < 1, and a [member of] [OMEGA], there exists [C.sub.[epsilon]] = [C.sub.[epsilon]](a) > 0 such that

[[parallel]u + [C.sub.[epsilon]][U.sub.[epsilon],a][parallel].sub.p] = 1, (A.6)

where [U.sub.[epsilon],a] is defined in (60). We infer from (A.6) that

[mathematical expression not reproducible] (A.7)

[mathematical expression not reproducible] (A.8)

Thus, for each u [member of] D and [[parallel]u[parallel].sub.p] < 1, we obtain that

[mathematical expression not reproducible] (A.9)

Combining (A.5) and (A.9), we get

[mathematical expression not reproducible] (A.10)

Moreover, for each w [member of] D one has

[mathematical expression not reproducible] (A.11)

That is,

[mathematical expression not reproducible] (A.12)

Let k = ((N + 4)I4)[C.sub.N][[[[parallel][DELTA][u.sub.0][parallel].sup.2.sub.2] + S[(1 - [[parallel][u.sub.0][parallel].sup.p.sub.p]).sup.2/p])].sup.(N-4)/8] > 0 and [lambda] = S[(1 - [[parallel][u.sub.0][parallel].sup.p.sub.p]).sup.(2-p)/p]. Then (A.12) implies that [u.sub.0] is the weak solution of

[[DELTA].sup.2]u = [lambda] [[absolute value of (u)].sup.p-2] u + 1/k f. (A.13)

Since f [not equal to] 0, we can conclude that [u.sub.0] [not equal to] 0. Recall that [u.sub.0](a) > 0, [for all]a [member of] [SIGMA], and [SIGMA] [subset] [OMEGA]. Replace [u.sub.0] with -[u.sub.0], and f with -f if necessarily. For a [member of] [SIGMA], we take [c.sub.[epsilon]] = [c.sub.[epsilon]](a) such that

[[parallel][u.sub.0] + [c.sub.[epsilon]][U.sub.[epsilon],a][parallel].sub.p] = 1 (A.14)

We obtain the contradiction if we prove that

G ([u.sub.0] + [c.sub.[epsilon]][U.sub.[epsilon],a]) < [[mu].sub.0] (A.15)

for a suitable choice of a [member of] [SIGMA] and small [epsilon].

From (A.7), we infer that [c.sub.[epsilon]] [??] [c.sub.0] as [epsilon] [right arrow] 0, where [c.sub.0] = [(1 - [[parallel][u.sub.0][parallel].sup.p.sub.p]).sup.1/p]/[A.sup.1/p]. Let [c.sub.[epsilon]] = [c.sub.0](1 - [[delta].sub.[epsilon]]), where [[delta].sub.[epsilon]] [right arrow] 0 as [epsilon] [right arrow] 0. A direct computation shows that

[mathematical expression not reproducible] (A.16)

where [mathematical expression not reproducible]. We deduce from (A.10) and (A.16) and the definition of [c.sub.0] that

[mathematical expression not reproducible] (A.17)

and, furthermore, we infer from (A.16) that

[mathematical expression not reproducible] (A.18)

Also, we notice that

[mathematical expression not reproducible] (A.19)

Hence it follows that

[mathematical expression not reproducible] (A.20)

This finishes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

Xiaoyong Qian was devoted to prove the first solution of the equation. Jun Wang proved the existence of the second solution of the equation. Maochun Zhu participated in the proof of the section solution of the equation. All authors read and approved the final manuscript.

Acknowledgments

X.-Y. Qian was supported by Jiangsu Province ordinary university graduate student scientific research innovation projects (KYLX 16_0898). J. Wang was supported by NSF of China (Grants 11571140, 11371090), NSF for Outstanding Young Scholars of Jiangsu Province (BK20160063), and NSF of Jiangsu Province (BK20150478) and the Six big talent peaks project in Jiangsu Province (XYDXX-015). M.-C. Zhu was supported by NSF of China (11601190), NSF of Jiangsu Province (BK20160483), and Jiangsu University Foundation Grant (16JDG043).

References

[1] H. Brezis and L. Nirenberg, "Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents," Communications on Pure and Applied Mathematics, vol. 36, no. 4, pp. 437-477, 1983.

[2] S. Wang, "The existence of a positive solution of semilinear elliptic equations with limiting Sobolev exponent," Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, vol. 117, no. 1-2, pp. 75-88, 1991.

[3] V. Benci and G. Cerami, "Existence of positive solutions of the Equation -[DELTA]+a(x)u = [u.sup.(N+2)/(N-2)] in [R.sup.N]," Journal of Functional Analysis, vol. 88, no. 1, pp. 90-117, 1990.

[4] M. Cuesta and C. De Coster, "Superlinear critical resonant problems with small forcing term," Calculus of Variations and Partial Differential Equations, vol. 54, no. 1, pp. 349-363, 2015.

[5] F. Faraci and C. Farkas, "A quasilinear elliptic problem involving critical Sobolev exponents," Collectanea Mathematica, vol. 66, no. 2, pp. 243-259, 2015.

[6] G. Fibich, B. Ilan, and G. Papanicolaou, "Self-focusing with fourth-order dispersion," SIAM Journal on Applied Mathematics, vol. 62, no. 4, pp. 1437-1462, 2002.

[7] G. Baruch, G. Fibich, and E. Mandelbaum, "Singular solutions of the biharmonic nonlinear Schrodinger equation," SIAM Journal on Applied Mathematics, vol. 70, no. 8, pp. 3319-3341, 2010.

[8] C. Miao, G. Xu, and L. Zhao, "Global well-posedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case," Journal of Differential Equations, vol. 246, no. 9, pp. 3715-3749, 2009.

[9] B. Pausader, "Global well-posedness for energy critical fourth-order Schrodinger equations in the radial case," Dynamics of Partial Differential Equations, vol. 4, no. 3, pp. 197-225, 2007

[10] B. Pausader, "The cubic fourth-order Schrodinger equation," Journal of Functional Analysis, vol. 256, no. 8, pp. 2473-2517, 2009.

[11] E. Mitidieri, "A Rellich type identity and applications," Communications in Partial Differential Equations, vol. 18, no. 1-2, pp. 125-151, 1993.

[12] P. Oswald, "On a priori estimates for positive solutions of a semilinear biharmonic equation in a ball," Commentationes Mathematicae, vol. 26, no. 3, pp. 565-577, 1985.

[13] F. Gazzola, H. Grunau, and M. Squassina, "Existence and nonexistence results for critical growth biharmonic elliptic equations," Calculus of Variations and Partial Differential Equations, vol. 18, no. 2, pp. 117-143, 2003.

[14] C. O. Alves and J. M. do O, "Positive solutions of a fourth-order semilinear problem involving critical growth," Advanced Nonlinear Studies, vol. 2, no. 4, pp. 437-458, 2002.

[15] T. Bartsch, T. Weth, and M. Willem, "A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator," Calculus of Variations and Partial Differential Equations, vol. 18, no. 3, pp. 253-268, 2003.

[16] Y.-X. Ge, "Positive solutions in semilinear critical problems for polyharmonic operators," Journal de Mathematiques Pures et Appliquees, vol. 84, no. 2, pp. 199-245, 2005.

[17] Y. Ge, J. Wei, and F. Zhou, "A critical elliptic problem for polyharmonic operators," Journal of Functional Analysis, vol. 260, no. 8, pp. 2247-2282, 2011.

[18] Y. Deng and W. Shuai, "Non-trivial solutions for a semilinear biharmonic problem with critical growth and potential vanishing at infinity," Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, vol. 145, no. 2, pp. 281-299, 2015.

[19] T. Bartsch, M. Schneider, and T. Weth, "Multiple solutions of a critical polyharmonic equation," Journal fur die Reine und Angewandte Mathematik. [Crelle's Journal], vol. 571, pp. 131-143, 2004.

[20] F. Bernis, J. Garcia Azorero, and I. Peral, "Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order," Advances in Differential Equations, vol. 1, no. 2, pp. 219-240, 1996.

[21] H.-C. Grunau, "Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents," Calculus of Variations and Partial Differential Equations, vol. 3, no. 2, pp. 243-252, 1995.

[22] G. Tarantello, "On nonhomogeneous elliptic equations involving critical Sobolev exponent," Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, vol. 9, no. 3, pp. 281-304, 1992.

[23] H. Brezis and L. Nirenberg, "A minimization problem with critical exponent and nonzero data," in Symmetry in Nature, Scuola Norm. Sup, vol. 1, pp. 129-140, Scuola Norm. Sup, Pisa, 1989.

[24] M. Willem, Minimax Theorems, Birkhauser, Boston, Mass, USA, 1996.

[25] M. Badiale and E. Serra, Semilinear elliptic equations for beginners, Universitext, Springer, London, 2011.

[26] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons, New York, NY, USA, 1984.

[27] H. Brezis and E. Lieb, "A relation between pointwise convergence of functions and convergence of functionals," Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486-490, 1983.

Xiaoyong Qian, Jun Wang [ID], and Maochun Zhu

Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Correspondence should be addressed to Jun Wang; wangmath2011@126.com

Received 11 September 2018; Accepted 1 November 2018; Published 21 November 2018

Academic Editor: Mariano Torrisi
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Title Annotation:Research Article
Author:Qian, Xiaoyong; Wang, Jun; Zhu, Maochun
Publication:Mathematical Problems in Engineering
Date:Jan 1, 2018
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