MULTIPLICITY OF SOLUTIONS FOR A KIRCHHOFF ... - PPGME

Differential and Integral Equations

Volume xx, Number xxx, , Pages xx–xx

MULTIPLICITY OF SOLUTIONS FOR A KIRCHHOFF EQUATION WITH SUBCRITICAL OR CRITICAL GROWTH ˜ o R. Santos Junior2 Giovany M. Figueiredo1 and Joa Universidade Federal do Pará, Faculdade de Matemática CEP: 66075-110 Belém - Pa, Brazil (Submitted by: Jean Mawhin) Abstract. This paper is concerned with the multiplicity of nontrivial solutions to the class of nonlocal boundary value problems of the Kirchhoff type, h “Z ”i − M |∇u|2 dx ∆u = λ|u|q−2 u + |u|p−2 u in Ω, and u = 0 on ∂Ω, Ω

where Ω ⊂ RN , for N = 1, 2, and 3, is a bounded smooth domain, 1 < q < 2 < p ≤ 2∗ = 6 in the case N = 3 and 2∗ = ∞ in the case N = 1 or N = 2. Our approach is based on the genus theory introduced by Krasnoselskii [22].

1. Introduction The purpose of this article is to investigate the multiplicity of nontrivial solutions to the class of nonlocal boundary value problems of the Kirchhoff type  h Z i  − M |∇u|2 dx ∆u = λ|u|q−2 u + |u|p−2 u in Ω, ≤ (Pλ )  u = 0 onΩ ∂Ω, where, throughout this work, Ω ⊂ RN , for N = 1, 2, and 3, is a bounded smooth domain, 1 < q < 2 < p ≤ 2∗ , λ is a positive parameter, and M : R+ → R+ is a continuous function that satisfies some conditions which will be stated later on. Problem (Pλ ) is called nonlocal because of the presence of the term Z 2 M |∇u| dx , Ω

Accepted for publication: March 2012. AMS Subject Classifications: 45M20, 35J25, 34B18, 34C11, 34K12. 1 Supported by CNPq/PQ 300705/2008-5 2 Supported by Fapespa - Par´ a 2009/182968 1

2

˜ o R. Santos Junior Giovany M. Figueiredo and Joa

which implies that the equation in (Pλ ) is no longer a pointwise identity. This phenomenon provokes some mathematical difficulties, which makes the study of such a class of problem particularly interesting. Besides, we have its physical motivation. Indeed, the operator Z |∇u|2 dx ∆u M Ω

appears in the Kirchhoff equation, which arises in nonlinear vibrations, namely,  Z 2   |∇u| dx ∆u = f (x, u) in Ω × (0, T ) u − M tt  Ω (1.1) u = 0 on ∂Ω × (0, T )    u(x, 0) = u0 (x), ut (x, 0) = u1 (x). Such a hyperbolic equation is a general version of the Kirchhoff equation Z L ∂2u ∂ 2 u P0 E ∂u 2 ρ 2 − + =0 (1.2) dx ∂t h 2L 0 ∂x ∂x2 presented by Kirchhoff [21]. This equation extends the classical d’Alembert’s wave equation by considering the effects of the changes in the length of the strings during the vibrations. The parameters in equation (1.2) have the following meanings: L is the length of the string, h is the area of crosssection, E is the Young modulus of the material, ρ is the mass density and P0 is the initial tension. Problem (1.1) began to catch the attention of several researchers mainly after the work of Lions [23], where a functional analysis approach was proposed to attack it. We have to point out that nonlocal problems also appear in other fields such as, for example, biological systems where u describes a process which depends on the average of itself (for example, population density). See, for example, [3] and its references. The reader may consult [1], [3], [13], [26], and [30] and the references therein for more information on nonlocal problems. We will always work in the three-dimensional space R3 , because in the other dimensions N = 1 and 2 everything follows by making standard modifications. Note that in the case that we will consider we have 2∗ = 6, which is the well-known critical Sobolev exponent. In this paper, we will consider the subcritical case, that is, 2 < p < 6, and the critical case, that is, p = 6.

Multiplicity of solutions for a Kirchhoff equation

3

The hypotheses on the function M : R+ → R+ are the following: There exists m0 > 0 such that M (t) ≥ m0 , for all t ≥ 0. There exists b > 0 such that M (t) → b as t → +∞. t There exists 4 < θ < 6 such that h1 i c(t2 ) − 1 M (t2 )t2 ≥ 0 for all t ≥ 0, M 2 θ where Z t c(t) = M (s)ds, M

(M1 )

(M2 )

(M3 )

0

and M (t) ≥ bt, fot all t ≥ 0. (M4 ) The hypotheses (M1 ) and (M2 ) will be used only in the subcritical case with 2 < p < 4. In the subcritical case with 4 < p < 6, only the hypothesis (M1 ) will used. In the subcritical case with p = 4, only the hypothesis (M4 ) will used. The hypotheses (M1 ) and (M3 ) will be used in the critical case. A typical example of a function satisfying the conditions (M1 )–(M4 ) is given by M (t) = m0 + bt with b > 0 and for all t ≥ 0, which is the one considered in the Kirchhoff equation (1.2). The main results of this paper are as follows: Theorem 1.1. Assume that conditions (M1 )–(M2 ) hold and 2 < p < 4. Then, problem (Pλ ) has infinitely many solutions, for all λ > 0. Theorem 1.2. Assume that condition (M1 ) holds and 4 < p < 6. Then, there exists λ∗ > 0, such that problem (Pλ ) has infinitely many solutions, for all λ ∈ (0, λ∗ ). Theorem 1.3. Assume that condition (M4 ) holds with p = 4. Then, problem (Pλ ) has infinitely many solutions, for all λ > 0, if b > S12 , where 4 R 2 dx |∇u| S4 = inf R Ω . 4 dx 2/4 |u| Ω Theorem 1.4. Assume that conditions (M1 ) and (M3 ) hold and p = 6. Then, there exists λ∗ > 0, such that problem (Pλ ) has infinitely many solutions, for all λ ∈ (0, λ∗ ).

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Recently, many papers concerned with the Kirchhoff-type problems by variational methods have been produced. See for example the works [2, 3, 5, 6, 10, 11, 14, 15, 16, 18, 19, 20, 25, 27, 28, 29, 30, 31, 32, 33] and references therein. Multiplicity results for the Kirchhoff equation with subcritical growth can be found in [16, 18, 19, 20, 32]. As regards the Kirchhoff equation with critical growth there are only existence results, as can be seen in [2] and references therein. We have to point out that our results are new and complement the articles [2, 16, 19, 20] and [32] because we have a multiplicity result for the Kirchhofftype problems with critical growth. In this paper, we will use an argument which has already appeared in [7], where the local case is handled. It is worthwhile to mention that, since we deal with the nonlocal case, the calculations are more involved and, in some sense, surprising. 2. Variational framework We recall that u ∈ H01 (Ω) is a weak solution of the problem (Pλ ) if it satisfies Z Z Z 2 q−2 M (kuk ) ∇u∇φ dx − λ |u| uφ dx − |u|p−2 uφ dx = 0 Ω

Ω

Ω

R for all φ ∈ H01 (Ω), where kuk2 = Ω |∇u|2 dx. We will look for solutions of (Pλ ) by finding critical points of the C 1 functional I : H01 (Ω) → R given by Z Z λ 1 1c 2 q |u| dx − |u|p dx. I(u) = M (kuk ) − 2 q Ω p Ω Note that Z Z Z |u|p−2 uφ dx, I 0 (u)φ = M (kuk2 ) ∇u∇φ dx − λ |u|q−2 uφ dx − Ω

Ω

Ω

H01 (Ω);

for all φ ∈ hence, critical points of I are weak solutions for (Pλ ). In order to use variational methods, we first derive some results related to the Palais–Smale compactness condition. We say that a sequence (un ) ⊂ H01 (Ω) is a Palais–Smale sequence for the functional I if I(un ) → d and I 0 (un ) → 0 in (H01 (Ω))0 , for some d ∈ R.

(2.1)


5

If (2.1) implies the existence of a subsequence (unj ) ⊂ (un ) which converges in H01 (Ω) for all d ∈ R, we say that I satisfies the Palais–Smale condition. If this strongly convergent subsequence exists only for some d values, we say that I satisfies a local Palais–Smale condition. 3. Preliminary results We will start by considering some basic notions on the Krasnoselskii genus that we will use in the proof of our main results. Let E be a real Banach space. Let us denote by A the class of all closed subsets A ⊂ E \ {0} that are symmetric with respect to the origin, that is, u ∈ A implies −u ∈ A. Definition 3.1. Let A ∈ A. The Krasnoselskii genus γ(A) of A is defined as being the least positive integer k such that there is an odd mapping φ ∈ C(A, Rk ) such that φ(x) 6= 0 for all x ∈ A. If k does not exist we set γ(A) = ∞. Furthermore, by definition, γ(∅) = 0. In the sequel we will recall only the properties of the genus that will be used throughout this work. More information on this subject may be found in the references by [4], [8], [17], and [22]. Proposition 3.2. Let E = RN and ∂Ω be the boundary of an open, symmetric, and bounded subset Ω ⊂ RN with 0 ∈ Ω. Then γ(∂Ω) = N . Corollary 3.3. γ(S N −1 ) = N . We now establish a result due to Clarke [12]. Proposition 3.4. Let Φ ∈ C 1 (X, R) be a functional satisfying the Palais– Smale condition. Furthermore, let us suppose that i) J is bounded from below and even; ii) there is a compact set K ∈ A such that γ(K) = k and supx∈K Φ(x) < Φ(0). Then Φ possesses at least k pairs of distinct critical points and their corresponding critical values are less than Φ(0). We point out that this result is a consequence of a basic multiplicity theorem involving an invariant functional under the action of a compact topological group. Proposition 3.5. If K ∈ A, 0 ∈ / K, and γ(K) ≥ 2, then K has infinitely many points.

6


4. Case 2 < p < 4 In the proof of Theorem 1.1, we need the following technical results. Lemma 4.1. The functional I is bounded from below. Proof. We will prove that I is coercive. Let (un ) ⊂ H01 (Ω) be a sequence such that kun k → ∞. From (M2 ) and Sobolev’s embedding, there exists C > 0 such that 1 b λ kun kp − C. I(un ) ≥ kun k4 − q/2 kun kq − p/2 8 qSq pSp Since 1 < q < 2 < p < 4, the result follows.

Lemma 4.2. The functional I satisfies the Palais–Smale condition. Proof. Let (un ) ⊂ H01 (Ω) be a sequence such that I(un ) → c and I 0 (un ) → 0. Since I is coercive, we conclude that (un ) is bounded in H01 (Ω) and, up to a subsequence, un * u in H01 (Ω), un → u in Ls (Ω) for 1 ≤ s < 6, un (x) → u(x) a.e. in Ω and kun k → t0 ≥ 0. Hence, I 0 (un )(un − u) = on (1); that is, Z Z M (kun k2 ) ∇un ∇(un − u) dx − λ |un |q−2 un (un − u) dx Ω Ω Z p−2 − |un | un (un − u) dx = on (1). Ω

From (4.1) we obtain Z

|un |q−2 un (un − u) dx = on (1)

λ Ω

and Z

|un |p−2 un (un − u) dx = on (1),

Ω

where we conclude that M (kun k2 )

Z ∇un ∇(un − u) dx = on (1). Ω

(4.1)


7

Since M is a continuous function, there exists C > 0, such that m0 ≤ M (kun k2 ) ≤ C. Thus, Z ∇un ∇(un − u) dx = on (1) Ω

and the proof is finished.

4.1. Proof of Theorem 1.1. Let us consider {e1 , e2 , . . . } an orthonormal basis of H01 (Ω) and for each k ∈ N consider Xk = span{e1 , e2 , . . . , ek }, the subspace of H01 (Ω) generated by k vectors e1 , e2 , . . . , ek . Note that the norms of H01 (Ω) and Lq (Ω) are equivalents on Xk and, hence, there exists a positive constant C(k) which depends on k, such that Z q −C(k)kuk ≥ − |u|q dx, Ω

for all u ∈ Xk . Thus, for all R > 0 and for all u ∈ Xk with kuk ∈ [0, R], from continuity of the function M, we conclude that there exists C > 0 such that λ I(u) ≤ Ckuk2 − C(k)kukq . q Fixing R such that CR2−q < λq C(k) and considering S = {u ∈ Xk : kuk = r} with 0 < r < R and for all u ∈ S, we get h i λ λ I(u) ≤ Cr2 − C(k)rq = rq Cr2−q − C(k) q q h i λ < Rq CR2−q − C(k) < 0 = I(0), q which implies supS I(u) < 0 = I(0). Since Xk and Rk are isomorphic and S and S k−1 are homeomorphic, we conclude that γ(S) = k. Moreover, I is even. By Proposition 3.4, I has at least k pairs of different critical points. Since k is arbitrary, we obtain infinitely many critical points of I. 5. Case 4 < p < 6 In this case, the arguments of the previous section do not apply because the functional I can not be bounded from below. To overcome this difficulty, arguing as [7], we will make a truncation in the functional I as follows:

8


From (M1 ) and Sobolev’s embedding, we get m0 1 λ I(u) ≥ kukp = g(kuk2 ), kuk2 − q/2 kukq − p/2 2 qSq pSp

(5.1)

where

1 p/2 m0 λ t . t − q/2 tq/2 − p/2 2 qSq pSp Thus, there exists λ∗ > 0 such that, if λ ∈ (0, λ∗ ), then g attains its positive maximum (see figure below). g(t) =

Let us assume λ ∈ (0, λ∗ ); choosing R0 and R1 as the first and second roots of g, we make the following truncation of I: Take φ ∈ C0∞ ([0, +∞)), 0 ≤ φ(t) ≤ 1, for all t ∈ [0, +∞), such that φ(t) = 1 if t ∈ [0, R0 ] and φ(t) = 0 if t ∈ [R1 , +∞). Now, we consider the truncated functional Z Z λ 1c 2 q 2 1 |u| dx − φ(kuk ) |u|p dx. J(u) = M (kuk ) − 2 q Ω p Ω As in (5.1), J(u) ≥ g(kuk2 ), where λ m0 1 g(t) = t − q/2 tq/2 − φ(t) p/2 tp/2 2 qSq pSp and whose graph is given below: Note that J is coercive, even, and satisfies the Palais–Smale condition, and arguing as in the Proof of Theorem 1.1 with R < R0 , we have sup J(u) = sup I(u) < 0 = I(0) = J(0) S

S


9

and J has infinitely many critical points. 5.1. Proof of Theorem 1.2. It is sufficient to show that the critical points of J are critical points of I. Observe that each critical point u of J satisfies g(kuk2 ) ≤ J(u) < 0. Thus, kuk2 < R0 and J(u) = I(u). Moreover, since J is a continuous functional, there exists a neighborhood U of u such that J(u) < 0, for all u ∈ U . Hence, J(u) = I(u) < 0, for all u ∈ U , where we conclude that J 0 (u) = I 0 (u) = 0, that is, u is a critical point of I. 6. Case p = 4 6.1. Proof of Theorem 1.3. We will prove that I is coercive. Let (un ) ⊂ H01 (Ω) be a sequence such that kun k → ∞. From (M4 ) and Sobolev’s embedding, we get 1 λ 1 I(un ) ≥ (b − 2 )kun k4 − q/2 kun kq . 4 S4 qSq Since 1 < q < 2 and b − S12 > 0, I is coercive. Now, arguing as in Lemma 4.2 4 and Theorem 1.1, we conclude that I has infinitely many critical points. 7. Case p = 6 As in the case 4 < p < 6, the arguments of Section 4 do not apply because the functional I can not be bounded below. Moreover, p = 6 is the main difficulty in solving problem (Pλ ), because there is the lack of compactness in

10


the inclusion of H01 (Ω) in L6 (Ω). This implies, in general, that the Palais– Smale condition is not satisfied. To overcome these difficulties, here we repeat the same truncation that appeared in Section 5. To overcome the lack of compactness, we use the concentration-compactness principle due to Lions [24]. Arguing as in the case 4 < p < 6, we consider the functional J which is bounded from below. Now, we will show that J satisfies the local Palais– Smale condition. For this, we need the following technical result involving the functional I. Lemma 7.1. Let (un ) ⊂ H01 (Ω) be a bounded sequence such that I(un ) → c and I 0 (un ) → 0. If (M1 ) and (M3 ) hold and h ( 1 − 1 )|Ω| 1 q θ 3/2 − (m0 S6 ) − c< 1 θ 6 ( θ − 16 ) 1

6−q 6

i

6 (6−q)

h q

6 (6−q)

6

−

q 6

q (6−q)

i

6

λ (6−q) ,

then there exists λ∗ > 0 such that, for all λ ∈ (0, λ∗ ), we have that, up to a subsequence, (un ) is strongly convergent in H01 (Ω). Proof. Since (un ) is bounded in H01 (Ω), taking a subsequence, we may suppose that |∇un |2 * |∇u|2 + µ and |un |6 * |u|6 + ν (weak∗ -sense of measures). Using the concentration-compactness principle due to Lions (cf. [24, Lemma 1.2]), we obtain an at-most-countable index set Λ, and sequences (xi ) ⊂ R3 and (µi ), (νi ) ⊂ (0, ∞), such that X X 1/3 ν= νi δxi , µ ≥ µi δxi and S6 νi ≤ µi , (7.1) i∈Λ

i∈Λ

for all i ∈ Λ, where δxi is the Dirac mass at xi ∈ Ω. Now, for every % > 0, we set ψ% (x) := ψ((x − xi )/%), where ψ ∈ C0∞ (Ω, [0, 1]) is such that ψ ≡ 1 on B1 (0), ψ ≡ 0 on Ω \ B2 (0). Since (ψ% un ) is bounded, I 0 (un )(ψ% un ) → 0; that is, Z 2 M (kun k ) un ∇un · ∇ψ% dx Ω Z Z Z 2 2 q = −M (kun k ) ψ% |∇un | dx + λ |un | ψ% dx + ψ% |un |6 dx + on (1). Ω

Ω

k2 )

M (t20 ),

Ω

Recalling that M (kun converges to we can argue as in [7] to show that Z lim [lim sup M (kun k2 ) un ∇un · ∇ψ% dx] = 0. →0

n→∞

Ω


11

Moreover, since un → u in Ls (Ω) for all 1 ≤ s < 6 and ψ% has compact support, we can let n → ∞ in the above expression to obtain Z Z M (t20 )ψ% dµ. ψ% dν ≥ Ω

Ω

Letting % → 0 we conclude that νi ≥ M (t20 )µi ≥ m0 µi . It follows from (7.1) that 1 1 νi ≥ (m0 S6 )3/2 ≥ ( − )(m0 S6 )3/2 . (7.2) θ 6 Now we shall prove that the above expression cannot occur, and therefore the set Λ is empty. Indeed, arguing by contradiction, let us suppose that 1 1 νi ≥ ( − )(m0 S6 )3/2 θ 6 for some i ∈ Λ. Thus, 1 c = I(un ) − I 0 (un )un + on (1). θ From (M3 ), 1 c 1 [M (kun k2 ) − M (kun k2 )kun k2 ] ≥ 0 2 θ for all n ∈ N; hence, Z Z 1 1 1 1 q c ≥ −λ( − ) |un | dx + ( − ) |un |6 dx + on (1) q θ Ω θ 6 Ω Z Z 1 1 1 1 q ≥ −λ( − ) |un | dx + ( − ) ψ% |un |6 dx + on (1). q θ Ω θ 6 Ω Letting n → ∞, we get Z Z 1 1 1 1 1 1 X q c ≥ −λ( − ) |u| dx + ( − ) |u|6 dx + ( − ) ψ% (xi )νi q θ Ω θ 6 Ω θ 6 i∈Λ Z Z 1 1 1 1 1 1 X q νi = −λ( − ) |u| dx + ( − ) |u|6 dx + ( − ) q θ Ω θ 6 Ω θ 6 i∈Λ Z Z 1 1 1 1 1 1 3/2 q ≥ ( − )(m0 S6 ) − λ( − ) |u| dx + ( − ) |u|6 dx. θ 6 q θ Ω θ 6 Ω By H¨ older’s inequality 6−q 1 1 1 1 c ≥ ( − )(m0 S6 )3/2 − λ( − )|Ω| 6 θ 6 q θ

Z Ω

|u|6 dx

q/6

1 1 +( − ) θ 6

Z Ω

|u|6 dx.

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Let 6−q 1 1 1 1 f (t) = ( − )t6 − λ( − )|Ω| 6 tq . θ 6 q θ This function attains its absolute minimum, for t > 0, at the point

t0 =

qλ( 1q − 1θ )|Ω| 6( 1θ

−

6−q 6

1/(6−q)

1 6)

.

Thus, we conclude that h ( 1 − 1 )|Ω| 3 1 1 q θ c ≥ ( − )(m0 S6 ) 2 − θ 6 ( 1θ − 16 )

6−q 6

i

6 6−q

h q i 6 q ( )6/(6−q) − ( )q/(6−q) λ 6−q . 6 6

But this is a contradiction. Thus Λ is empty and it follows that un → u in L6 (Ω). Since un → u in Lq (Ω), we conclude Z Z lim M (kun k2 )kun k2 = λ |u|q dx + |u|6 dx. n→∞

Ω

Ω

On the other hand, by using a well-known argument, we reach Z Z Z 2 q−2 M (t0 ) ∇u∇φ dx = λ |u| uφ dx + |u|4 uφ dx ∀φ ∈ H01 (Ω) Ω

Ω

Ω

and so M (t20 )kuk2 = λ

Z

|u|q dx +

Ω

Z

|u|6 dx

Ω

from which it follows that M (kun k2 )kun k2 → M (t20 )kuk2 . Since M (kun k2 ) → M (t20 ), the proof of the lemma is complete. By the Lemma 7.1 we conclude, for λ > 0 sufficiently small, that h ( 1 − 1 )|Ω| 1 1 q θ ( − )(m0 S6 )3/2 − θ 6 ( 1θ − 61 )

6−q 6

i

6 6−q

h q 6 q q i 6 ( ) 6−q − ( ) 6−q λ 6−q > 0. 6 6

Lemma 7.2. If J(u) < 0, then kuk2 < R0 and J(v) = I(v), for all v in a sufficiently small neighborhood of u. Moreover, J satisfies a local Palais– Smale condition for c < 0. Proof. Since g(kuk2 ) ≤ J(u) < 0, arguing as in Theorem 1.2, we conclude that J(v) = I(v), for all v ∈ BR0 /2 (0). Moreover, if (un ) is a sequence such that J(un ) → c < 0 and J 0 (un ) → 0, then, for n sufficiently large,


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I(un ) = J(un ) → c < 0 and I 0 (un ) = J 0 (un ) → 0. Since J is coercive, we get that (un ) is bounded in H01 (Ω). From Lemma 7.1, for λ sufficiently small, h ( 1 − 1 )|Ω| 1 1 q θ c < 0 < ( − )(m0 S6 )3/2 − θ 6 ( 1θ − 16 )

6−q 6

i

6 6−q

h q 6 q q i 6 ( ) 6−q − ( ) 6−q λ 6−q 6 6

and hence, up to a subsequence, (un ) is strongly convergent in H01 (Ω). Now, we will construct an appropriate mini-max sequence of negative critical values for the functional J. Lemma 7.3. Given k ∈ N, there exists = (k) > 0 such that γ(J − ) ≥ k, where J − = {u ∈ H01 (Ω) : J(u) ≤ −}. Proof. Fix k ∈ N and let Xk be a k-dimensional subspace of H01 (Ω). Thus, there exists C(k) > 0 such that Z −C(k)kukq ≥ − |u|q dx, Ω

for all u ∈ Xk . Considering ρ > 0 such that 0 < kuk = ρ and 0 < kuk2 < R0 , we get J(u) = I(u). Arguing as in the proof of Theorem 1.1, we can take R > 0 such that I(u) < −, for all u ∈ Xk and with u ∈ S. Hence S ⊂ J − and, since J − is symmetric and closed, from Corollary 3.3, γ(J − ) ≥ γ(S) = k.

We define now, for each k ∈ N, the sets Γk = {C ⊂ H01 (Ω) : C is closed, C = −C and γ(C) ≥ k}, Kc = {u ∈ H01 (Ω) : J 0 (u) = 0 and J(u) = c} and the number ck = inf C∈Γk supu∈C J(u). Lemma 7.4. Given k ∈ N, the number ck is negative. Proof. From Lemma 7.3, for each k ∈ N there exists > 0 such that γ(J − ) ≥ k. Moreover, 0 ∈ / J − and J − ∈ Γk . On the other hand supu∈J − J(u) ≤ −. Hence, −∞ < ck = inf sup J(u) ≤ sup J(u) ≤ − < 0. C∈Γk u∈C

u∈J −

The next lemma allows us to prove the existence of a critical point of J. Lemma 7.5. If c = ck = ck+1 = · · · = ck+r for some r ∈ N, then there exists λ∗ > 0 such that γ(Kc ) ≥ r + 1,

for λ ∈ (0, λ∗ ).

14


Proof. Since c = ck = ck+1 = · · · = ck+r < 0, from Lemma 7.1 and Lemma 7.4, we get that Kc is a compactness set. Moreover, Kc = −Kc . If γ(Kc ) ≤ r, there exists a closed and symmetric set U with Kc ⊂ U such that γ(U ) = γ(Kc ) ≤ r. Note that we can choose U ⊂ J 0 because c < 0. By the deformation lemma [9] we have an odd homeomorphism η : H01 (Ω) → H01 (Ω) such that η(J c+δ − U ) ⊂ J c−δ for some δ > 0 with 0 < δ < −c. Thus, J c+δ ⊂ J 0 , and by definition of c = ck+r there exists A ∈ Γk+r such that sup J(u) < c + δ; that is, A ⊂ J c+δ and u∈A

η(A − U ) ⊂ η(J c+δ − U ) ⊂ J c−δ .

(7.3)

But γ(A − U ) ≥ γ(A) − γ(U ) ≥ k and γ(η(A − U )) ≥ γ(A − U ) ≥ k. Then η(A − U ) ∈ Γk and this contradicts (7.3). Hence, this lemma is proved. 7.1. Proof of Theorem 1.4. If −∞ < c1 < c2 < · · · < ck < · · · < 0 and since each ck is a critical value of J, then we obtain infinitely many critical points of J, and hence the problem (Pλ ) has infinitely many solutions. On the other hand, if there are two constants ck = ck+r , then c = ck = ck+1 = · · · = ck+r , and from Lemma 7.5 there exists λ∗ > 0 such that γ(Kc ) ≥ r + 1 ≥ 2. From Proposition 3.5, Kc has infinitely many points; that is, problem (Pλ ) has infinitely many solutions. References [1] C.O. Alves and F.J.S.A. Corrêa, On existence of solutions for a class of problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal., 8 (2001), 43–56. [2] C.O. Alves, F.J.S.A. Corrêa, and G.M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, DEA, 2 (2010), 409–417. [3] C.O. Alves, F.J.S.A. Corrêa, and T.F Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85–93. [4] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349–381. [5] G. Anelo, A uniqueness result for a nonlocal equation of Kirchhoff equation type and some related open problem, J. Math. Anal. Appl., 373 (2011), 248–251. [6] G. Anelo, On a pertubed Dirichlet problem for a nonlocal differential equation of Kirchhoff type, BVP, ID 891430 (2011). [7] J.G. Azorero and I.P. Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877–895. [8] A. Castro, “Metodos Variacionales y Analisi Functional no Linear,” X Coloquio colombiano de Matematicas, 1980.


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