Multiple positive solutions for a class of Kirchhoff type problems

MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF KIRCHHOFF TYPE PROBLEMS INVOLVING GENERAL CRITICAL GROWTH

arXiv:1607.01923v1 [math.AP] 7 Jul 2016

LIEJUN SHEN AND XIAOHUA YAO

Abstract. In this paper, we study the following nonlinear Kirchhoff problems involving critical growth: R −(a + b Ω |∇u|2 dx)∆u = |u|4 u + λ|u|q−2 u, u = 0 on ∂Ω, where 1 < q < 2, λ, a, b > 0 are parameters and Ω is a bounded domain in R3 . We prove that there exists λ1 = λ1 (q, Ω) > 0 such that for any λ ∈ (0, λ1 ) and a, b > 0, the above Kirchhoff problem possesses at least two positive solutions and one of them is a positive ground state solution. We also establish the convergence property of the ground state solution as the parameter b ց 0. More generally, we obtain the same results about the following Kirchhoff problem: R −(a + b R3 |∇u|2 dx)∆u + u = Q(x)|u|4 u + λf (x)|u|q−2 u, u ∈ H 1 (R3 ), for any a, b > 0 and λ ∈ 0, λ0 (q, Q, f ) under certain conditions of f (x) and Q(x). Finally, we investigate the depending relationship between λ0 and b to show that for any (large) λ > 0, there exists a b0 (λ) > 0 such that the above results hold when b > b0 (λ) and a > 0.

1. Introduction and Main Results In recent years, the following Kirchhoff type problem Z   − a + b |∇u|2dx ∆u = f (x, u) in Ω, Ω  u = 0 on ∂Ω,

(1.1)

has been studied extensively by many researchers, here f ∈ C(Ω × R, R), Ω ⊂ RN , N ≥ 1 and a, b > 0 are constants. R Problems like (1.1) are seen to be nonlocal because of the appearance of the term b( Ω |∇u|2 dx)∆u which implies that (1.1) is not a pointwise identity any more. It is degenerate if b = 0 and non-degenerate otherwise. The non-degenerate case causes some mathematical difficulties, which make the study of (1.1) interesting. The above problem (1.1) is related to the stationary analogue of the Kirchhoff equation Z utt − a + b |∇u|2dx ∆u = f (x, u) (1.2) Ω

0

Email addresses: [email protected] (L. Shen), [email protected] (X. Yao) 1

2


which was proposed by Kirchhoff in [1] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. For some mathematical and physical background on Kirchhoff type problems, we refer the readers to [2, 3, 4, 5] and the references therein. After J. L. Lions in his pioneer work [6] introduced an abstract functional analysis framework to (1.2), the equation (1.2) has received an increasing attention on mathematical research. Recently, many important results about the the nonexistence, existence, and multiplicity of solutions for problem (1.1) have been obtained with the nonlinear term f (x, u) behaves like |u|p−2u, here 2 < p ≤ 2∗ , 2∗ = N2N if N ≥ 3, 2∗ = ∞ if N = 1, 2. −2 Please see for example [7, 8, 9, 10, 11] and the references therein. In their celebrated paper, A. Ambrosetti, H. Brézis, G. Cerami [14] studied the following semilinear elliptic equation with concave-convex nonlinearities: ( −∆u = |u|p−2u + λ|u|q−2u in Ω, (1.3) u = 0 on ∂Ω, where Ω is a bounded domain in RN with λ > 0 and 1 < q < 2 < p ≤ 2∗ = N2N . By varia−2 tional method, they have obtained the existence and multiplicity of positive solutions to the problem (1.3). Subsequently, a series of similar results including the more general nonlinearity like g(x)|u|p−2u + λf (x)|u|q−2u are established, e.g. see [15, 16, 17, 18, 19] and their reference therein. Some other types of multiplicity of positive solutions to the Schordinger or Schordinger-Poisson equations are established, see [20, 21, 22, 23] for example. Using Nehair manifold and fibering map, C. Y. Chen, Y. C. Kuo, T. F. Wu [24] extend the analysis to the Kirchhoff type equation  Z 2  −M |∇u| dx ∆u = g(x)|u|p−2u + λh(x)|u|q−2u in Ω, (1.4) Ω  u = 0 on ∂Ω,

where Ω is a smooth bounded domain in RN with 1 < q < 2 < p < 2∗ = N2N . M(s) = −2 as+b, the parameters a, b, λ > 0 and the weight functions h, g ∈ C(Ω) satisfy some specified conditions, they show that there exist at least two positive solutions when 0 < λ < λ0 (a) and 4 < p < 6 for the problem (1.4), here λ0 (a) strongly relies on a > 0. In addition, the existence and multiplicity of solutions for Kirchhoff type problems in the whole space RN has been established in [25, 26, 27] are taken into account. ), very recently, C. Y. Lei, G. S. Liu, L. T. Guo In the critical growth case (i.e. p = N2N −2 in [28] consider the following Kirchhoff problem in three dimensions: Z   − a + ǫ |∇u|2dx ∆u = u5 + λuq−1 in Ω, (1.5) Ω  u = 0 on ∂Ω,

where Ω is a smooth bounded domain in R3 , a > 0, 1 < q < 2, ǫ > 0 is small enough, λ > 0 is a positive real number. They obtained that there exists a λ⋆ > 0 such that when ǫ > 0 is small enough and λ ∈ (0, λ⋆), the problem (1.5) has at least two positive solutions.

KIRCHHOFF TYPE EQUATIONS WITH GENERAL CRITICAL GROWTH

3

Inspired by the works mentioned above, particularly, by the results in [28], we try to get the existence of positive solutions of the Kirchhoff problem for the critical growth. To be precise, we study the problem Z   − a + b |∇u|2dx ∆u = |u|4u + λ|u|q−2u, in Ω, (1.6) Ω  u = 0 on ∂Ω.

Theorem 1.1. Let 1 < q < 2. Then there exists a λ1 = λ1 (q, Ω) > 0 such that for any λ ∈ (0, λ1) and any a, b > 0, the problem (1.6) has at least two positive solutions in H01 (Ω), and one of the solutions is a positive ground state solution. In particular, let λ ∈ (0, λ1 ) be fixed, for any sequence {bn } with bn ց 0 as n → ∞, there exists a subsequence (still denoted by {bn }) such that ubn converges to w0 strongly in H01 (Ω) as n → ∞, where ubn is a positive ground state solution of the problem (1.6) and w0 is a solution of the problem −a∆u = |u|4u + λ|u|q−2u, in Ω, u = 0 on ∂Ω.

Remark 1.2. Comparing with [28], we only assume b > 0 is a positive constant not necessary to be small enough. Therefore, we greatly relax the constraints on the parameter ǫ in [28]. Moreover, we obtain the above convergence property of ground solution of the problem (1.6) as b ց 0. In this paper we will not give the direct proof of Theorem 1.1, which can be seen as a corollary of the following general problem Z  − a + b |∇u|2dx ∆u + u = Q(x)|u|4u + λf (x)|u|q−2u, in R3 (1.7) R3  1 3 u ∈ H (R )

under some assumptions on the weight functions Q(x) and f (x). Actually, we stress here that the method dealing with problem (1.7) can be applied directly in the problem (1.6) just by simply modifying some notations such as Q(x) = f (x) ≡ 1, and Ω instead of R3 in problem (1.6). Now the main results for (1.7) can be stated as follows: Theorem 1.3. Assume 1 < q < 2, and the functions Q, f satisfy the following conditions: 6 (i) 0 < f (x) ∈ L 6−q (R3 ); (ii) Q(x) ≥ 0, and ∃x0 ∈ R3 , α ∈ ( 2q , +∞), ρ > 0 such that Q(x0 ) = max3 Q(x) := |Q|∞ < +∞ x∈R

and |Q(x) − Q(x0 )| ≤ C|x − x0 |α whenever |x − x0 | < ρ, for some constant C > 0. Then there exists a λ0 = λ0 (q, Q, f ) > 0 such that for any λ ∈ (0, λ0 ) and any a, b > 0, the problem (1.7) has at least two positive solutions in H 1 (R3 ), and one of the solutions is a positive ground state solution.

4


Remark 1.4. (1) The condition (i) in Theorem 1.3 that f (x) is positive is just for simplicity. In fact, our method can deal with the case that f (x) is sign-changing. (2) It is clear that Q(x) ≡ 1 satisfies the condition (ii) in Theorem 1.3. Also, Q(x0 ) > 0 is necessary and otherwise Q(x) ≡ 0 implies that the critical term disappears. The general critical term Q(x)|u|4 u was firstly introduced in [29] by F. Gazzola and M. Lazzarino, then this condition has already been extended to Schördinger-Possion system in [30] and Kirchhoff type problem in [31] with α ∈ [1, 3). Compared with [30] and [31], the condition (ii) has a bit improvement, since we just assume that α ∈ ( 2q , +∞) for q ∈ (1, 2). By the results in Theorem 1.3, we know that the constant λ0 is independent of b > 0. So we are interested to know what happens if b > 0 is sufficiently small in our problem (1.7). Here we can give the following theorem: Theorem 1.5. Under the assumptions of Theorem 1.3 and λ ∈ (0, λ0) is fixed, then for any sequence {bn } with bn ց 0 as n → ∞, there exists a subsequence (still denoted by {bn }) such that ubn convergent to w0 strongly in H 1 (R3 ) as n → ∞, where ubn is a positive ground state solution to problem (1.7) and w0 is a solution of the problem ( −a∆u + u = Q(x)|u|4 u + λf (x)|u|q−2u, in R3 , (1.8) u ∈ H 1 (R3 ). For λ0 (independent on b) in Theorem 1.3 and Theorem 1.5, it is unknown to us whether the constant λ0 is a best one or not for all b > 0. However, for a fixed large b > 0, it is far from the best constant. Indeed, in the following theorem we have proved that there exists ˜ 0 > λ0 such that results above holds true for any λ ∈ (0, λ ˜ 0 ) and any a > 0. λ Theorem 1.6. Let q, Q, f satisfy the assumptions as the Theorem 1.3, then for any b > 0, ˜ 0 = Cb 6−q 2 , here C = C(q, Q, f ) > 0 is independent on b, such that for any there exists λ ˜ 0) and any a > 0, the problem (1.7) has at least two positive solutions in H 1 (R3 ), λ ∈ (0, λ and one of the solutions is a positive ground state solution. 6−q

˜ 0 = Cb 2 converges to ∞ as b ր ∞. Hence, combining with Theorem 1.3 Obviously, λ and Theorem 1.6, we have the following conclusion: Corollary 1.7. Let q, Q, f satisfy the assumptions as the Theorem 1.3, then for any λ ∈ (0, +∞) there exists b0 (λ) ≥ 0 such that problem (1.7) has at least two positive solutions in H 1 (R3 ), and one of the solutions is a positive ground state solution when b > b0 (λ) and any a > 0. Remark 1.8. To the best knowledge of us, the Corollary 1.7 seems new for the Kirchhoff problem involving critical growth in the whole space R3 . Considered if b = 0, it seems failure for the degenerate case for Kirchhoff problem.


5

Remark 1.9. Corollary 1.7 is a direct conclusion of the Theorem 1.3 and Theorem 1.6, and we can give in fact the explicit definition of b0 (λ) as the following: b0 (λ) =

0, if λ < λ0 , 2 ˜ 6−q Cλ , if λ ≥ λ0 ,

˜ Q, f ) > 0 are independent on b. On the other hand, where λ0 = λ0 (q, Q, f ) and C˜ = C(q, We mention here that Corollary 1.7 remains true for problem (1.6) on bounded domain Ω. Before we turn to next section, we would like to mention some main ideas of the proof of Theorems 1.3, 1.5 and 1.6. It seems that the methods used in [24] and [28] can not be applied directly in our paper. On one hand, in [24], the method used the Nehair manifold and fibering map will be not directly applied because the norm can’t be defined R 1 as ( R3 |∇u|2 dx) 2 . On the other hand, the first solution in [28] is easy to be found, and however, the procedure by which the authors of [28] used the first solution to find the second one, seems strongly to depend on b > 0 small enough. To establish Theorem 1.3, we will used the Mountain-pass theorem and the Ekeland’s variational principle [40] which indicate that the (P S) condition of the energy functional I is necessary. However, the functional I does not satisfy the (P S) condition at every energy level c because of the appearance of critical term. To overcome this difficulty, we try to pull the energy level down below some critical to handle the R level. 2It is more R complicated 2 nonlocal effect which does not imply that R3 |∇un | dx → R3 |∇u| dx from un ⇀ u in H 1 (R3 ). Therefore we estimate a critical value 3

2 abS 3 (b2 S 4 + 4a|Q|∞ S) 2 b3 S 6 c< + − Ci λ 2−q , C1 , C2 ∈ (0, +∞). + 2 2 4|Q|∞ 24|Q|∞ 24|Q|∞

different from J. Wang et al [32] and G. Li, H. Ye [33] to recover the compactness condition. Then we can prove Theorem 1.3. To show Theorem 1.5, we follow the idea used in W. Shuai [34] and X. Tang, B. Cheng [35] to consider what happens when b ց 0. Fixing b ∈ (0, 1] and then we get a bounded sequence of positive ground state solutions to (1.7). As a consequence of that the functional I(u) satisfies the (P S) condition at some level, the proof of Theorem 1.5 is clear. In Theorem 1.6, we concern how λ0 can be determined by b. To do it, we used a different approach (see Lemma 5.1) to find a (P S) sequence. Considering the effect from b, the proof of (P S) condition of the energy functional I(u) may be different from Lemma 2.3. But the weak limit u ∈ H 1 (R3 ) of a (P S)c˜ sequence {un } is a critical point of the following functional: Z Z Z Z a + bA2 1 1 λ 2 2 6 J(u) := |∇u| dx + |u| dx − Q(x)|u| dx − f (x)|u|q , 2 2 6 q 3 3 3 3 R R R R

6


R and the {un } is a (P S)c˜+ bA2 sequence for J(u), where A2 = limn→∞ R3 |∇un |2 dx. We try 4 to prove that I(u) satisfies (P S)c condition with the help of J(u) (see Lemma 5.3). Hence, the proof Theorem 1.6 is complete. This paper is organized as follows: In Section 2, we give some notations and crucial lemmas. In Section 3, we prove the existence of the two different positive solutions and the positive ground state solution of problem (1.7). In Section 4, we analyze the convergence property of the positive ground state solution of problem (1.7) and prove the Theorem 1.5. In Section 5, we establish the proof of Theorem 1.6. 2. Some Notations and Lemmas In this section, we first give several notations and definitions. Throughout this paper, L (R3 ) (1 ≤ p ≤ ∞) is the usual Lebesgue space with the standard norm |u|p . We use “ → ” and “ ⇀ ” to denote the strong and weak convergence in the related function space, respectively. For any ρ > 0 and any x ∈ R3 , Bρ (x) denotes the ball of radius ρ centered at x, that is Bρ (x) := {y ∈ R3 : |y − x| < ρ}. C will denote a positive constant unless specified. We denote by H 1 (R3 ) the usual Sobolev space equipped with the norm 12 Z 2 2 a|∇u| + |u| dx , ∀u ∈ H 1 (R3 ), kuk = p

R3

and (H (R )) is the dual space of H 1 (R3 ). Let D 1,2 (R3 ) be the completion of C0∞ (R3 ) with respect to the Dirichlet norm, Z 12 2 kukD1,2 (R3 ) := |∇u| dx , ∀u ∈ D 1,2 (R3 ) 1

3

∗

R3

and S denote the best Sobolev constant, namely R S :=

u∈D

inf3 1,2

(R )\{0}

R3

(

R

R3

|∇u|2dx

1

|u|6dx) 3

,

Define the energy functional I : H 1 (R3 ) → R of (1.7) by Z 2 Z 1 b 2 2 2 I(u) = a|∇u| + |u| dx + |∇u| dx 2 R3 4 R3 Z Z λ 1 6 Q(x)|u| dx − f (x)|u|q dx, − 6 R3 q R3

(2.1)

the functional I is well-defined on H 1 (R3 ) and I ∈ C 1 (H 1 (R3 ), R) (see [36]). What’s more, for any u, v ∈ H 1 (R3 ), we have Z Z Z ′ 2 hI (u), vi = a∇u∇v + uvdx + b |∇u| dx ∇u∇v R3 R3 R3 Z Z 4 − Q(x)|u| uvdx − λ f (x)|u|q−2 uvdx. R3

R3


7

Clearly, the critical points are the weak solutions of problem (1.7). Furthermore, a sequence {un } ⊂ H 1 (R3 ) is a (P S) sequence of the functional I(u) at the level d ∈ R if I(un ) → d and I ′ (un ) → 0 as n → ∞. And we say a (P S)d sequence satisfies the (P S)d condition if it contains a strong convergent subsequence. The following Lemmas play vital roles in proving Theorem 1.3: Lemma 2.1. For any λ ∈ (0, λ0), the functional I(u) satisfies the Mount-pass geometry around 0 ∈ H 1 (R3 ), that is, (i) there exist η, β > 0 such that I(u) ≥ η > 0 when kuk = β; (ii) there exists e ∈ H 1 (R3 ) with kek > β such that I(e) < 0. Proof. (i) By the definition of I(u), 1 |Q|∞ −3 λ I(u) ≥ kuk2 − S kuk6 − |f | 6 kukq 2 6 p 6−q |Q|∞ −3 λ 2−q 6−q q 1 kuk − S kuk − |f | 6 = kuk 2 6 q 6−q λ ≥ kukq C0 − |f | 6 , q 6−q 3S 3 (2−q) 2−q 2 0 4 . Therefore letting λ0 = |fqC , and there exist η, β > 0 such where C0 = 6−q |Q|∞ (6−q) | 6 6−q

that I(u) ≥ η > 0 when kuk = β for any λ ∈ (0, λ0 ). (ii) It’s clear that lim I(tu0 ) = −∞ for some u0 ∈ H 1 (R3 ), choosing e = t0 u0 with t0 t→+∞

large enough, we have kek > β and I(e) < 0.

By Lemma 2.1, we can find a (P S) sequence of the functional I(u) at the level c := inf sup I(γ(t)), γ∈Γ t∈[0,1]

(2.2)

where Γ := {γ ∈ C([0, 1], H 1(R3 )) : γ(0) = 0, I(γ(1) < 0}. (2.3) 1 Now a (P S)c sequence can be constructed, that is, there exists a sequence {un } ⊂ H (R3 ) satisfies I(un ) → c, I ′ (un ) → 0 as n → ∞. (2.4) The functional I(u) do not satisfy the (P S)c condition at every energy level because of the appearance of the critical term Q(x)|u|4 u, and in order to recover the compactness condition, we would have to estimate the critical energy carefully. 1

We know that U(x) =

34 1+|x|2

is a solution of −∆u = u5 , x ∈ R3 ,

− 12

(⋆)

then for any ǫ > 0, Uǫ (x) = ǫ U(ǫ−1 x) is also a solution of (⋆) and satisfies that R R 3 |∇Uǫ |2 dx = R3 |Uǫ |6 dx = S 2 . To estimate the critical energy, we chose a cut-off R3

8


function ϕ(x) satisfying ϕ(x) ∈ C0∞ (BR (x0 )), 0 ≤ ϕ(x) ≤ 1 in R3 and ϕ(x) ≡ 1 on B R (x0 ). 2 Define 1 ϕ(x)(3ǫ2 ) 4 vǫ (x) = ϕ(x)Uǫ (x) = (2.5) 1 , (ǫ2 + |x − x0 |2 ) 2 thanks to the results in [37], we have with

K1 K2

|∇vǫ |22 = K1 + O(ǫ), |vǫ |26 = K2 + O(ǫ)

(2.6)

= S and for any s ∈ [2, 6),  s if s ∈ [2, 3),  O(ǫ 2 ), 3 s |vǫ |s = O(ǫ 2 | log ǫ|), if s = 3,  6−s O(ǫ 2 ), if s ∈ (3, 6).

(2.7)

Lemma 2.2. Assume 1 < q < 2 and λ ∈ (0, λ0 ), then the critical energy 3

2 (b2 S 4 + 4a|Q|∞ S) 2 b3 S 6 abS 3 2−q , + − C λ + c< 1 4|Q|∞ 24|Q|2∞ 24|Q|2∞ 2 (4−q)|f | 2−q 6 2−q 6−q where C1 = 4q is a positive constant, and S is the best Sobolev constant q

2S 2

given in (2.1).

Proof. Firstly, we claim that there exist t1 , t2 ∈ (0, +∞) independent of ǫ, λ such that max I(tvǫ ) = I(tǫ vǫ ) and t≥0

0 < t1 < tǫ < t2 < +∞. (2.8) Indeed, by the fact that lim I(tvǫ ) = −∞ and (i) of Lemma 2.1, there exists tǫ > 0 such t→+∞

that

max I(tvǫ ) = I(tǫ vǫ ), t≥0

which imply that Z 2 3 tǫ kvǫ k + tǫ

R3

and

2

kvǫ k +

3t2ǫ

Z

2 Z 6 |∇vǫ | dx − tǫ 2

R3

6

R3

2 Z 5 |∇vǫ | dx − 6tǫ 2

d2 d I(tvǫ ) = 0, 2 I(tvǫ ) < 0 dt dt

Q(x)|vǫ | dx − 6

R3

Q(x)|vǫ | dx −

λtqǫ

Z

qλtǫq−1

R3

Z

f (x)|vǫ |q dx = 0

R3

f (x)|vǫ |q dx < 0.

(2.9)

(2.10)

It follows from (2.9) that tǫ is bounded from above and (2.10) that tǫ is bounded from below, then (2.8) is true. The energy level c given in (2.2) tells us it’s enough to show for any sufficient small 3

ǫ > 0, max I(tvǫ ) < t≥0

abS 3 4|Q|∞

+

b3 S 6 24|Q|2∞

+

(b2 S 4 +4a|Q|∞ S) 2 24|Q|2∞

2

− C1 λ 2−q holds. In fact, for the given

vǫ in (2.5), there exists a large T > 0 such that I(T vǫ ) < 0. Let e γ (t) = tT vǫ ∈ Γ, where


9

Γ is defined as in (2.3), then c ≤ max0≤t≤1 I(e γ (t)) = max0≤t≤1 I(tT vǫ ) ≤ maxt≥0 I(tvǫ ). It follows from the definition of I(u) that Z 2 Z t2 bt4 2 2 2 I(tvǫ ) = a|vǫ | + |vǫ | dx + |∇vǫ | dx 2 R3 4 R3 Z Z λtq t6 6 Q(x)|vǫ | dx − f (x)|vǫ |q dx, − 6 R3 q R3

(2.11)

and then set

where

Z 2 Z t2 bt4 t6 2 2 g(t) := kvǫ k + |∇vǫ | dx − Q(x0 )|vǫ |6 dx 2 4 6 3 3 R R 2 4 6 f f f = C1 t + C2 t − C3 t , f2 = b f1 = 1 kuk2, C C 2 4

Z

2 Z 1 f |∇vǫ | dx , C3 = Q(x0 )|vǫ |6 dx. 6 R3 R3 2

Combing with (2.5)-(2.6) and some elementary computations we have 2

3

f1 C f3 ) 2 f2 + 3C f1 C f2 C f3 + 2C f2 + 2(C 9C max g(t) = 2 t≥0 f3 27C 3

3

(2.12)

(b2 S 4 + 4a|Q|∞ S) 2 b3 S 6 abS 3 + + O(ǫ). + = 4|Q|∞ 24|Q|2∞ 24|Q|2∞

On the other hand, for ǫ > 0 with ǫ < λtqǫ

Z

R3

q

f |vǫ | dx =

R 2

λtqǫ

(2.8)

we have Z

BR (x0 )

Z

f |vǫ |q dx q

ǫ2 ≥ Cλ f (x) 2 q dx (ǫ + |x|2 ) 2 B R (x0 ) 2 Z q 2 q q f (x)dx := C2 ǫ 2 , ≥ λ( 2 ) 2 ǫ 2 R B R (x0 )

(2.13)

2

6

where C2 ∈ (0, +∞) since f (x) ∈ L 6−q (R3 ) and f (x) ∈ L1loc (R3 ). It follows from some direct computations: Z Z |x − x0 |α 1 3 ǫ dx ≤ 3 |x − x0 |α dx ≤ Cǫα 2 2 3 ǫ Bǫ (x0 ) Bǫ (x0 ) (ǫ + |x − x0 | )

10

and


Z |x − x0 |α 3 |x − x0 |α−6 dx dx ≤ ǫ ǫ 2 + |x − x |2 )3 (ǫ 0 Bρ (x0 )\Bǫ (x0 ) Bρ (x0 )\Bǫ (x0 )  α Z ρ if α < 3,  Cǫ , α−4 3 3 Cǫ | ln ǫ|, if α = 3, r dr = = Cǫ  ǫ Cǫ3 , if α > 3, 3

Z

that the following conclusion Z Z 6 6 6 |Q(x) − Q(x0 )||vǫ | dx = tǫ tǫ R3

(2.7)

Bρ (x0 )

3

≤ Cǫ

Z

Bρ (x0 )

|Q(x) − Q(x0 )||vǫ |6 dx

 if α < 3,  Cǫα , |x − x0 | 3 Cǫ | ln ǫ|, if α = 3, dx ≤  (ǫ2 + |x − x0 |2 )3 Cǫ3 , if α > 3, α

(2.14)

holds. It is obvious that if α > 3, then there exists ǫ1 > 0 such for any ǫ ∈ (0, ǫ1 ) we have

2 O(ǫ) + Cǫ2 − C2 ǫq−1 < −C1 λ 2−q ; ǫ if α = 3, then there exists ǫ2 > 0 such for any ǫ ∈ (0, ǫ2 ) we have

2 O(ǫ) + Cǫ2 | ln ǫ| − C2 ǫq−1 < −C1 λ 2−q ; ǫ q if α ∈ ( 2 , 3), then there exists ǫ3 > 0 such for any ǫ ∈ (0, ǫ3 ) we have

(2.15)

(2.16)

2 O(ǫ) + Cǫ2 | ln ǫ| − C2 ǫq−1 < −C1 λ 2−q . (2.17) ǫ Therefore combing with (2.11)-(2.17), for any α > q2 , there exists ǫ0 = min{ǫ1 , ǫ2 , ǫ3 , R2 } such that ∀ǫ ∈ (0, ǫ0 ), Z Z 6 q 6 f (x)|vǫ |q dx |Q(x) − Q(x0 )||vǫ | dx − λtǫ max I(tvǫ ) = I(tǫ vǫ ) = g(tǫ ) + tǫ t≥0 3 R3 R α if α < 3,  Cǫ , q 3 Cǫ | ln ǫ|, if α = 3, ≤ max g(t) − C2 ǫ + t≥0  Cǫ3 , if α > 3 3

2 abS 3 (b2 S 4 + 4a|Q|∞ S) 2 b3 S 6 < + − C1 λ 2−q . + 2 2 4|Q|∞ 24|Q|∞ 24|Q|∞

Now the proof is complete.

Lemma 2.3. The functional I(u) satisfies the (P S)c condition when λ ∈ (0, λ0 ) and 3

2 (b2 S 4 + 4a|Q|∞ S) 2 b3 S 6 abS 3 + − C1 λ 2−q . + c< 2 2 4|Q|∞ 24|Q|∞ 24|Q|∞


11

Proof. When λ ∈ (0, λ0 ), by Lemma 2.1, there exists a sequence {un } assuming that I(un ) → c, I ′ (un ) → 0 as n → ∞.

Then 1 c + 1 + o(1)kun k ≥ I(un ) − hI ′ (un ), un i 4 Z Z 1 1 1 1 6 2 Q(x)|un | dx − λ( − ) f |un |q dx = kun k + 4 12 R3 q 4 R3 q 1 4−q ≥ kun k2 − λ|f | 6 S − 2 kun kq , 6−q 4 4q hence {un } is bounded in H 1 (R3 ) by the fact that 1 < q < 2. There exist a subsequence still denoted by {un } and u0 ∈ H 1 (R3 ) such that   un ⇀ u0 in H 1(R3 ), un → u0 in Lrloc (R3 ), where 1 ≤ r < 6, (2.18)  3 un → u0 a.e. in R . S The set R3 {∞} is compact for theSstand topology which means that the measures can be identified as the dual space C(R3 {∞}). For example, δ∞ is well defined and δ∞ (ϕ) = ϕ(∞). Due to the concentration-compactness principle in [38, 39], we can chose a subsequence denoted again by {un } such that X |∇un |2 ⇀ dµ ≥ |∇u0|2 + µj δxj + µ∞ δ∞ , (2.19) ˜ j∈Γ

|un |6 ⇀ dν = |u0 |6 +

X

νj δxj + ν∞ δ∞ ,

(2.20)

˜ j∈Γ

where δxj and δ∞ are the Dirac mass at xj and infinity respectively, and xj in the support ˜ is an at most countable index set. What’s more, by the Sobolev of the measures µ, ν and Γ inequality we have 1

µj , νj ≥ 0, µj ≥ Sνj3 .

(2.21)

˜ Arguing it by contradiction, for any ǫ > 0, let φǫj be a We claim that νj = 0 for any j ∈ Γ. smooth cut-off function centered at xj such that 0 ≤ φǫj ≤ 1, φǫj ∈ C0∞ (Bǫ (xj )), φǫj = 1 in B 2ǫ (xj ) and |∇φǫj | ≤ 4ǫ . It is easy to see that Z lim |∇u0 |2 φǫj dx = 0. (2.22) ǫ→0

R3

Indeed, by Hölder inequality and the Lebesgue Convergence Theorem we have Z Z Z 2 ǫ 2 ǫ |∇u0|2 dx = 0, 0 ≤ lim |∇u0| φj dx = lim |∇u0 | φj dx ≤ lim ǫ→0

R3

ǫ→0

Bǫ (xj )

ǫ→0

Bǫ (xj )

12


thus (2.22) holds. Similarly lim

ǫ→0

Z

Z

lim

ǫ→0

R3

and lim

ǫ→0

|u0|2 φǫj dx = 0,

(2.23)

Q(x)|u0 |6 φǫj dx = 0,

(2.24)

f (x)|u0|2 φǫj dx = 0.

(2.25)

R3

Z

R3

Some direct conclusions of (2.22)-(2.25) are Z Z 2 ǫ |∇un | φj dx ≥ lim lim lim ǫ→0 n→∞

lim lim

ǫ→0 n→∞

Z

R3

lim lim ǫ→0 n→∞ Z = lim ǫ→0

lim lim | ǫ→0 n→∞

Z

R3

ǫ→0 n→∞

ǫ→0

≤ C lim

ǫ→0

Z

Z

R3

ǫ→0 n→∞

|u0 |6 φǫj dx

Z

R3

Bǫ (xj )

Z

Bǫ (xj )

R3

|∇u0|2 φǫj dx + µj = µj ,

|un |2 φǫj dx

= lim lim

ǫ→0 n→∞

Z

Bǫ (xj )

ǫ→0

Bǫ (xj )

|u0 |2 φǫj dx = 0,

Q(x)|un |6 φǫj dx + Q(xj )νj

≤ lim lim

ǫ→0 n→∞

Z

R3

21 Z |∇un | dx 2

Z 21 2 ǫ 2 = C lim lim un |∇φj | dx ǫ→0 n→∞

21 Z 2 ǫ 2 u0 |∇φj | dx = C lim ǫ→0

16 Z 6 u0 dx

(2.27)

(2.28)

Bǫ (xj )

R3

Bǫ (xj )

12

u2n |∇φǫj |2 dx

21 2 ǫ 2 u0 |∇φj | dx

21

u2n |∇φǫj |2 dx

(2.29)

13 Z 16 4 3 6 ( ) dx u0 dx = C lim =0 ǫ→0 Bǫ (xj ) ǫ Bǫ (xj )

and lim lim ǫ→0 n→∞ Z = lim ǫ→0

= lim

Z

(2.26)

+ Q(xj )νj = Q(xj )νj ,

(∇un , ∇φǫj )un dx|

Bǫ (xj )

Z

= lim lim

Q(x)|un |6 φǫj dx

Bǫ (xj )

≤ C lim lim = C lim

|un |2 φǫj dx

ǫ→0

R3

Z

R3

f (x)|un |q φǫj dx

Bǫ (xj )

f (x)|u0 |q φǫj dx

= lim lim

ǫ→0 n→∞

= 0.

Z

Bǫ (xj )

f (x)|un |q φǫj dx

(2.30)


13

′

Since {un } is bounded, limǫ→0 limn→∞ hI (un ), un φǫj i = 0, that is Z Z Z 2 ǫ ǫ lim lim a |∇un | φj dx + a (∇un , ∇φj )un dx + u2n φǫj dx ǫ→0 n→∞ R3 R3 R3 Z Z Z Z |∇un |2 dx (∇un , ∇φǫj )un dx +b |∇un |2 dx |∇un |2 φǫj dx + b 3 3 R3 R3 R ZR Z − u6n φǫj dx − λ f (x)|un |q φǫj dx = 0. R3

(2.31)

R3

√ 1 bS 2 + b2 S 4 +4aQ(xj )S By (2.26)-(2.31) we have aµj +bµ2j ≤ Q(xj )νj , moreover (νj ) 3 ≥ together 2Q(xj ) with (2.21). Hence 1 c + o(1) = I(un ) − hI ′(un ), un i 4 Z Z Z 1 1 1 1 2 2 6 a|Dun | + |un | dx + Q(x)|un | dx − λ( − ) f |un |q dx = 4 R3 12 R3 q 4 R3 q a 1 1 4−q ≥ µj + Q(xj )νj + ku0 k2 − λ|f | 6 S − 2 ku0 kq 6−q 4 12 4 4q 3

2 abS 3 b3 S 6 (b2 S 4 + 4aQ(xj )S) 2 2−q ≥ + + − C λ 1 4Q(xj ) 24Q2 (xj ) 24Q2 (xj ) 3

2 (b2 S 4 + 4a|Q|∞ S) 2 b3 S 6 abS 3 + − C1 λ 2−q , + ≥ 2 2 4|Q|∞ 24|Q|∞ 24|Q|∞ 2 (4−q)|f | 2−q 6 2−q 6−q ∈ (0, +∞), which is a contradiction! Therefore νj = 0 for where C1 = 4q q

2S 2

˜ any j ∈ Γ. We now study the concentration at infinity, we define a new cut-off function ϕ ∈ ∞ 3 C0 R , [0, 1] , such that ϕ(x) = 0 if |x| < R, ϕ(x) = 1 if |x| > R and |∇ϕ| ≤ R2 . Consider the following equalities Z µ∞ = lim lim sup |∇un |2 ϕdx R→∞ n→∞

and

ν∞ = lim lim sup R→∞ n→∞

|x|>R

Z

|x|>R

|un |6 ϕdx.

Using the same technique as at the xj , we obtain the same conclusion, namely, µ∞ = ν∞ = 0. Then in view of (2.20), we have un → u0 in L6 (R3 ). Next we will show that un → u0 in H 1 (R3 ). In fact, set v := un − u0 , (2.32) Zn Z I11 (un ) := (a + b |∇un |2 dx) ∇un , ∇vn dx, (2.33) R3

R3

14


and I22 (un ) := (a + b

Z

2

R3

|∇u0 | dx)

Z

R3 6 3

∇u0 , ∇vn dx.

From the fact vn = un − u0 → 0 and un → u0 in L (R ) we have Z Z Z 2 2 |∇un | dx − |∇u0| dx = |∇vn |2 dx + o(1), 3 3 3 R R R Z I2 (un ) := (|un |4 un − |u0 |4 u0 )(un − u0 )dx = o(1), R3 Z I3 (un ) := f (x)(|un |q−2 un − |u0 |q−2 u0 )(un − u0 )dx = o(1).

(2.34)

(2.35) (2.36) (2.37)

R3

Then combing with (2.32)-(2.35) we have

I1 (un ) := I11 (un ) − I22 (un ) Z Z Z Z 2 2 2 = (a + b |∇un | dx) |∇vn | dx + b |∇vn | dx (∇u0 , ∇vn )dx + o(1) R3 R3 R3 R3 Z Z Z 2 2 ≥a |∇vn | dx + b |∇vn | dx (∇u0 , ∇vn )dx + o(1) R3 R3 R3 Z =a |∇vn |2 dx + o(1). (2.40) R3

Therefore (2.36)-(2.40) imply that

o(1) = hI ′ (un ) − I ′ (u0 ), vn i Z = I1 (un ) + |vn |2 dx − I2 (un ) − I3 (un ) R3 Z Z 2 ≥a |∇vn | dx + |vn |2 dx + o(1) =

R3 kvn k2

R3

+ o(1),

and then kvn k → 0 which is un → u0 in H 1 (R3 ) and we complete the proof.

3. The proof of Theorem 1.3 From this section, we can find out that why we can see the Theorem 1.1 as a corollary of the Theorem 1.3. Before we find the two different positive solutions of problem (1.7), we introduce the following well-known proposition: Proposition 3.1. (Ekeland’s variational principle [40], Theorem 1.1) Let V be a complete metric space and F : V → R ∪ {+∞} be lower semicontinuous, bounded from below. Then for any ǫ > 0, there exists some point v ∈ V with F (v) ≤ inf F + ǫ, F (w) ≥ F (v) − ǫd(v, w) for all w ∈ V. V

Now, we will split it into three parts to prove Theorem 1.3.


15

3.1. Existence of the first positive solution. If λ ∈ (0, λ0 ), then by Lemma 2.1 there exists a (P S)c sequence {un } of the functional I(u). Lemma 2.2 and Lemma 2.3 tell us that un → u0 in H 1 (R3 ), then I ′ (u0 ) = 0 and I(u0 ) = c > 0 which imply that u0 is nontrivial. Obviously, |u0 | is also a solution of the problem (1.7), then by maximum principle, u0 is positive and then u0 is a positive solution of the problem (1.7). 3.2. Existence of the second positive solution. We investigate the second positive solution for problem (1.7) similar to an idea from [41]. For ρ > 0 given by Lemma 2.1(i), define B β = {u ∈ H 1 (R3 ), kuk ≤ β}, ∂Bβ = {u ∈ H 1 (R3 ), kuk = β} and clearly B β is a complete metric space with the distance Lemma 2.1 tells us that

d(u, v) = ku − vk, ∀u, v ∈ B β . I(u)|∂Bβ ≥ η > 0.

(3.1)

It’s obvious that I(u) is lower semicontinuous and bounded from below on B β . We claim that c1 := inf I(u) < 0. (3.2) u∈B β

Indeed, we chose a nonnegative function ψ ∈ C0∞ (R3 ), and clearly ψ ∈ H 1 (R3 ). Since 1 < q < 2, we have 2 R q R 6 R t2 bt4 2 2 kψk + 4 − t6 R3 Q(x)|ψ|6 dx − λtq R3 f |ψ|q dx 3 |∇ψ| dx 2 R I(tψ) lim q = lim t→0 t→0 t tq Z λ =− f (x)|ψ|q dx < 0. q R3 Therefore there exists a sufficiently small t0 > 0 such that kt0 ψk ≤ β and I(t0 ψ) < 0, which imply that (3.2) holds. By Proposition 3.1, for any n ∈ N there exists uñ such that 1 (3.3) c1 ≤ I(˜ u n ) ≤ c1 + , n and 1 I(v) ≥ I(˜ un ) − k˜ un − vk, ∀v ∈ B β . (3.4) n Firstly, we claim that k˜ un k < β for n ∈ N sufficiently large. In fact, we will argue it by contradiction and just suppose that k˜ un k = β for infinitely many n, without loss of generality, we may assume that k˜ un k = β for any n ∈ N. It follows from (3.1) that I(˜ un ) ≥ η > 0,

then combing it with (3.3), we have c1 ≥ η > 0 which is a contradiction to (3.2). Next, we will show that I ′ (˜ un ) → 0 in (H 1 (R3 ))∗ . Indeed, set vn = uñ + tu, ∀u ∈ B1 = {u ∈ H 1 (R3 ), kuk = 1},

16


where t > 0 small enough such that 2t + t2 ≤ ρ2 − k˜ un k2 for fixed n large, then kvn k2 = k˜ un k2 + 2t(˜ un , u) + t2 ≤ k˜ un k2 + 2t + t2

≤ β2

which imply that vn ∈ B β . So it follows from (3.4) that

t I(vn ) ≥ I(˜ un ) − k˜ un − vn k, n

that is, un ) I(˜ un + tu) − I(˜ 1 ≥− . t n 1 ′ Letting t → 0, then we have hI (˜ un ), ui ≥ − n for any fixed n large. Similarly, chose t < 0 and |t| small enough, repeating the process above we have hI ′ (˜ un ), ui ≤ n1 for any fixed n large. Therefore the conclusion I ′ (˜ un ), ui → 0 as n → ∞, ∀u ∈ B1

implies that I ′ (˜ un ) → 0 in (H 1 (R3 ))∗ . Finally, we know that {˜ un } is a (P S)c1 sequence for the functional I(u) and by the Lemma 2.3, there exists u1 such that uñ → u1 in H 1 (R3 ) and then I ′ (u1 ) = 0, that is u1 a solution of problem (1.7) with I(u1 ) = c1 < 0. Moreover, the strong maximum principle implies that u1 is positive. 3.3. Existence of the positive ground state solution. To obtain a positive ground state solution for problem (1.7), we define m := inf I(u), N

where N := {u ∈ H 1 (R3 ) : I ′ (u) = 0}. Firstly we have the following claim: Claim 1: N = 6 ∅ and m ∈ (−∞, c]. 6 ∅ and then m ≤ max{I(u0), I(u1 )} = c. Proof of the Claim 1: It’s obvious that N = On the other hand, ∀u ∈ N , we have 1 I(u) = I(u) − hI ′ (u), ui 4 Z Z 1 1 1 1 6 2 = kuk + Q(x)|u| dx − λ( − ) f |u|q dx 4 12 R3 q 4 R3 q 4 − q 1 λ|f | 6 S − 2 kukq , ≥ kuk2 − λ 6−q 4 4q

(3.5)

hence I(u) is coercive and bounded below on N by the fact that 1 < q < 2, that is m > −∞.


17

By virtue of Claim 1, we can choose a minimizing sequence of m, that is a sequence {vn } ⊂ N satisfying I(vn ) → m as n → ∞ and I ′ (vn ) = 0.

Therefore {vn } is a (P S)m sequence of I(u) with m < 2

abS 3 4|Q|∞ 3

+

b3 S 6 24|Q|2∞

3

+

(b2 S 4 +4a|Q|∞ S) 2 − 24|Q|2∞ 1 3

C0 λ 2−q . Then similar to (3.5), vn is bounded in H 1 (R ) and there exists a v ∈ H (R ) such that vn ⇀ v in H 1 (R3 ). Claim 2: v 6= 0. Proof of the Claim 2: Argue it by contradiction and just suppose that v ≡ 0, hence Z f (x)|vn |q dx → 0 (3.6) R3

6

by f (x) ∈ L 6−q and vn → 0 in Lrloc (R3 ) (1 ≤ r < 6). Therefore we can infer from (3.6) that Z 2 Z 2 2 kvn k + b |∇vn | dx − Q(x)|vn |6 dx = o(1), (3.7) R3

and

b 1 kvn k2 + 2 4 Let

R3

Z

2 Z 1 |∇vn | dx − Q(x)|vn |6 dx = m + o(1). 6 R3 R3 2

Z and b

2

kvn k → l1

R3

thus (3.7)-(3.9) give us that

m= and |Q|∞

Z

6

R3

|vn | dx ≥

By the Sobolev imbedding theorem,

Z

kvn k2 ≥ aS and

Z b

2 |∇vn | dx → l2 , 2

l1 l2 + 3 12

R3

R3

31 |vn |6 dx ,

2 Z 2 |∇vn | dx ≥ bS 2

R3

(3.9)

(3.10)

Q(x)|vn |6 dx → l1 + l2 .

Z

(3.8)

23 |vn | dx , 6

R3

It follows from (3.11)-(3.13) that, 1 32 l1 + l2 3 2 l 1 + l2 and l2 ≥ bS . l1 ≥ aS |Q|∞ |Q|∞

(3.11)

(3.12)

(3.13)

(3.14)

18


Now we define a function 1

2

3 2 3 t − bS 2 t − aS|Q|∞ . h(t) = |Q|∞

It’s obvious that there exist t1 < 0 < t2 such that h(t1 ) = h(t2 ) = 0, and p p bS 2 + b2 S 4 + 4a|Q|∞ S bS 2 − b2 S 4 + 4a|Q|∞ S , t2 = . t1 = 2 2 3 3 2|Q|∞ 2|Q|∞ Therefore if h(t) ≥ 0, then t ≤ t1 or t ≥ t2 . On the other hand, 2 1 (3.14) 1 1 2 3 3 h (l1 + l2 ) 3 = |Q|∞ (l1 + l2 ) 3 − bS 2 (l1 + l2 ) 3 − aS|Q|∞ ≥ 0, hence 1 3

(l1 + l2 ) ≥ t2 =

bS 2 +

p

b2 S 4 + 4a|Q|∞ S 2 3

.

(3.15)

2|Q|∞

As a direct conclusion of (3.10) and (3.14)-(3.15), we have l2 1 1 2 l1 1 + ≥ aS(l1 + l2 ) 3 + bS 2 (l1 + l2 ) 3 3 12 3 12 3 2 4 3 6 3 (b S + 4a|Q|∞ S) 2 bS abS + > c, + ≥ 4|Q|∞ 24|Q|2∞ 24|Q|2∞

m=

(3.16)

a contradiction! So v 6= 0. Similar to the Lemma 2.3, vn → v in H 1 (R3 ), then I(v) = m and I ′ (v) = 0. Obviously, |v| is also a critical point, by strong maximum principle v is positive. Hence v is a nontrivial ground state solution of problem (1.7). 4. The convergence property of the ground state solution. In this section, we give the proof of Theorem 1.5. Firstly, the energy functional of problem (1.7) is written as 2 Z Z 1 b 2 2 2 Ib (u) = |∇u| dx a|∇u| + |u| dx + 2 R3 4 R3 Z Z λ 1 6 Q(x)|u| dx − f (x)|u|q dx, − 6 R3 q R3

and the least energy of which is

mb := inf Ib (u), Nb

where Nb := {u ∈ H 1 (R3 ) : Ib′ (u) = 0}. Proof of the Theorem 1.5: For any fixed λ ∈ (0, λ0 ) and any b > 0, ub ∈ Nb is a positive ground solution of problem (1.7) obtained in Section 3 and satisfies Ib (ub) = mb . Thus for


any b ∈ (0, 1],

19

3

abS 3 (b2 S 4 + 4a|Q|∞ S) 2 b3 S 6 mb < + + 4|Q|∞ 24|Q|2∞ 24|Q|2∞ 3

(S 4 + 4a|Q|∞ S) 2 S6 aS 3 + + ≤ 4|Q|∞ 24|Q|2∞ 24|Q|2∞ where M is independent on b. On one hand, 1 M > Ibn (ubn ) − hIb′n (u), ubn i 4 Z 1 1 1 = kubn k2 + Q(x)|ubn |6 dx − λ( − 4 12 R3 q q 4−q 1 λ|f | 6 S − 2 kubn kq . ≥ kubn k2 − 6−q 4 4q

:= M < +∞,

1 ) 4

Z

R3

f |ubn |q dx

This shows that {ubn } is bounded in H 1 (R3 ) due to 1 < q < 2. Hence, there exist a subsequence still denoted {bn } and w0 ∈ H 1 (R3 ) such that ubn ⇀ w0 in H 1 (R3 ). Similar to Section 3, we have ubn → w0 6= 0 in H 1 (R3 ). On the other hand, ∀ϕ ∈ C0∞ (H 1(R3 )), we have Z Z Z 2 ′ ∇ubn ∇ϕdx |∇ubn | dx a∇ubn ∇ϕ + ubn ϕdx + bn hIbn (ubn ), ϕi = R3 R3 R3 Z Z 4 f |ubn |q−1 ubn ϕdx − Q(x)|ubn | ubn ϕdx − λ 3 3 ZR ZR Z 4 → a∇w0 ∇ϕ + w0 ϕdx − Q(x)|w0 | w0 ϕdx − λ f (x)|w0 |q−1w0 ϕdx R3

R3 Ib′n (ubn )

R3

= 0 we have as bn → 0 and n → ∞. By the fact Z Z Z 4 a∇w0 ∇ϕ + w0 ϕdx − Q(x)|w0 | w0 ϕdx − λ R3

R3

1

R3

f (x)|w0 |q−2w0 ϕdx = 0

3

for any ϕ ∈ H (R ). Therefore w0 is a solution of the problem (1.8).

5. The proof of Theorem 1.6. We will give some Lemmas before we prove the Theorem 1.6. We just give the detail proof of the difference from Lemmas 2.1-2.3. The following Lemma has a great difference which seems failure in [14] from our Lemma 2.1. ˜ 0 ), the functional I(u) of (1.7) satisfies the Mount-pass Lemma 5.1. For any λ ∈ (0, λ geometry around 0 ∈ H 1 (R3 ), that is, ˜ (i) there exist η˜, β˜ > 0 such that I(u) ≥ η˜ > 0 when kuk = β; 1 3 ˜ (ii) there exists e˜ ∈ H (R ) with k˜ ek > β such that I(˜ e) < 0.

20


Proof. (i) By the definition of I(u), Z 2 Z Z λ b 1 1 2 6 2 |∇u| dx − Q(x)|u| dx − f (x)|u|q dx I(u) = kuk + 2 4 6 q 3 3 3 R R R Z 2 Z Z b λ 1 ≥ |∇u|2dx − Q(x)|u|6 dx − f (x)|u|q dx 4 6 q 3 3 3 R R R Z 2 Z 3 λ|f | 6 Z 2q |Q|∞ b 6−q 2 2 2 |∇u| dx − |∇u| dx − ≥ |∇u| dx q 4 6S 3 qS 2 R3 R3 R3 6 |Q|∞ 6−q λ|f | 6−q q b 4−q , =t t − t − q 4 6S 3 qS 2 4−q 21 q 2 R 3 6−q 2 3(4−q)S qS 2 ˜ 0 = C2 b 2 , here C2 := . Therefore letting λ , where t := |∇u| dx 3 R 2(6−q)|f | 6 2(6−q) 6−q

and then there exist η˜, β˜ > 0 such that I(u) ≥ η˜ > 0 when kuk = β˜ for any λ ∈ (0, λ˜0 ). (ii) The proof is similar to Lemma 2.1(ii), so we omit it here. Now by Lemma 5.1, we can construct a (P S) sequence of the functional I(u) at the level c˜ := inf sup I(γ(t)),

(5.1)

Γ := {γ ∈ C([0, 1], H 1(R3 )) : γ(0) = 0, I(γ(1) < 0},

(5.2)

I(un ) → c˜, I ′ (un ) → 0 as n → ∞.

(5.3)

γ∈Γ t∈[0,1]

where 1

3

that is, there exists a sequence {un } ⊂ H (R ) satisfies

˜ 0 ), then the critical energy Lemma 5.2. Assume 1 < q < 2 and λ ∈ (0, λ 3

2 abS 3 (b2 S 4 + 4a|Q|∞ S) 2 b3 S 6 c˜ < + − C3 λ 2−q , + 2 2 4|Q|∞ 24|Q|∞ 24|Q|∞ q 2 2−q 2−q (6−q)|f | 6 2−q 3q 6−q where C3 = 2 2a is a positive constant. q

6qS 2

Proof. The proof is totally same as Lemma 2.2 except changing the constant C2 to C3 in (2.15), (2.16), and (2.17). ˜ 0 ) and Lemma 5.3. The functional I(u) satisfies the (P S)c˜ condition when λ ∈ (0, λ 3

2 abS 3 (b2 S 4 + 4a|Q|∞ S) 2 b3 S 6 c˜ < + − C3 λ 2−q . + 2 2 4|Q|∞ 24|Q|∞ 24|Q|∞


21

˜ 0), by Lemma 5.1, there exists a sequence {un } assuming that Proof. For λ ∈ (0, λ I(un ) → c˜, I ′ (un ) → 0 as n → ∞.

Then similar to Lemma 2.1, {un } is bounded in H 1 (R3 ) and there exist a subsequence still denoted by {un } and u ∈ H 1 (R3 ) such that   un ⇀ u in H 1 (R3 ), un → u in Lrloc (R3 ), where 1 ≤ r < 6, (5.4)  3 un → u a.e. in R . Set

2

A := lim

n→∞

and

Z

R3

|∇un |2 dx

vn := un − u. We can infer from the Brezis-Lieb Lemma in [42] that Z Z Z 2 2 2 A = |∇un | dx + o(1) = |∇vn | dx + |∇u|2dx + o(1), R3 R3 R3 Z Z Z |un |2 dx = |vn |2 dx + |u|2dx + o(1), 3 3 3 R R R Z Z Z 6 6 Q(x)|un | dx = Q(x)|vn | dx + Q(x)|u|6 dx + o(1), R3

R3

(5.5)

(5.6)

(5.7) (5.8) (5.9)

R3

6

and since f ∈ L 6−q , we have Z Z (5.4) q f (x)|un | dx = R3

f (x)|u|q dx + o(1).

(5.10)

R3

Now we introduce a new functional, for any u ∈ H 1 (R3 ), Z Z Z Z a + bA2 1 1 λ 2 2 6 J(u) := |∇u| dx + |u| dx − Q(x)|u| dx − f (x)|u|q . (5.11) 2 2 R3 6 R3 q R3 R3 Hence 2 Z Z b bA2 2 2 |∇un | dx |∇un | dx − J(un ) = I(un ) + 2 R3 4 R3 (5.12) bA4 bA4 bA4 (5.7) = c˜ + − + o(1) = c˜ + + o(1) 2 4 4 and for any ϕ ∈ H 1 (R3 ) we have Z Z Z 2 ′ ′ 2 ∇un ∇ϕdx − b |∇un | dx ∇un ∇ϕdx hJ (un ), ϕi = hI (un ), ϕi + bA R3 R3 R3 Z Z 2 2 = b(A − |∇un | dx) ∇un ∇ϕdx R3 R3 Z =b ∇un ∇ϕdx o(1) = o(1) R3

22


show that {un } is a bounded (P S)c+ bA4 sequence of J(u). In particular 4

hJ ′ (un ), ϕi Z Z Z 2 (a + bA ) ∇un ∇ϕdx + un ϕdx − Q(x)|un |4 un ϕdx R3 R3 R3 Z −λ f (x)|un |q−2 un ϕ R3 Z Z Z 2 (a + bA ) ∇u∇ϕdx + uϕdx − Q(x)|u|4 uϕ R3 R3 R3 Z −λ f (x)|u|q−2uϕ + o(1)

=

o(1)

=

(5.4)

=

R3

′

hJ (u), ϕi + o(1)

=

implies that hJ ′ (u), ui = 0 by taking u = ϕ since ϕ is arbitrary. Therefore 1 J(u) = J(u) − hJ ′ (u), ui 6Z Z Z a + bA2 1 1 1 2 2 = |∇u| dx + |u| dx − λ( − ) f (x)|u|q dx 3 3 q 6 3 3 3 R R R q2 Z Z Z 6 λ(6 − q)|f | bA2 a 6−q 2 2 + |∇u| dx − |∇u|2dx ≥ |∇u| dx q 3 R3 4 3 3 6qS 2 R R 2 Z 2 bA ≥ −C3 λ 2−q + |∇u|2dx, 4 R3 where C3 =

2−q 2

q 2−q λ(6−q)|f |

3q 2a

q

6qS 2

o(1) = hJ ′ (un ), un i (5.7)

2 2−q

6 6−q

′

2

Z

(5.13)

. Since {un } is bounded and then, Z

2

Z

= hJ (u), ui + (a + bA ) |∇vn | dx + |vn | dx − Q(x)|vn |6 dx 3 3 3 R R Z Z ZR = (a + bA2 ) |∇vn |2 dx + |vn |2 dx − Q(x)|vn |6 dx 3 3 3 R R R Z 2 Z Z Z (5.7) 2 2 2 2 = kvn k + b |∇vn | dx + b |∇vn | dx |∇u| dx − Q(x)|vn |6 dx, R3

2

R3

R3

R3

which is Z kvn k + b 2

2 Z |∇vn | dx + b 2

R3

R3

2

|∇vn | dx

Z

R3

2

|∇u| dx −

Z

R3

Q(x)|vn |6 dx = o(1). (5.14)


Without loss of generality, we can assume Z 2 Z Z 2 2 2 ˜ kvn k → l1 , b |∇vn | dx + b |∇vn | dx R3

R3

R3

|∇u| dx → ˜l2 , 2

Z

23

Q(x)|vn |6 dx → ˜l3 ,

R3

that is ˜l3 = ˜l1 + ˜l2 by (5.14). Next we will show that un → u in H 1 (R3 ). Just suppose not, then ˜l1 > 0 and ˜l3 > 0. Similar to (3.16), we also have 3

l1 (b2 S 4 + 4a|Q|∞ S) 2 l2 abS 3 b3 S 6 + . + ≥ + 3 12 4|Q|∞ 24|Q|2∞ 24|Q|2∞

(5.15)

By (5.6)-(5.10), we have J(un )

= =

(5.14)

=

Z Z 1 1 2 |∇vn | dx + |vn | dx − Q(x)|vn |6 dx + o(1) 2 6 3 3 3 R R R Z 2 Z Z b b 1 2 2 2 |∇vn | dx + |∇vn | dx |∇u|2dx J(u) + kvn k + 2 4 4 3 3 3 R R R Z 2 Z bA 1 |vn |6 dx + |∇vn |2 dx + o(1) − 6 R3 4 R3 2 Z Z Z 1 b 2 2 2 2 |∇u| dx |∇vn | dx |∇vn | dx + J(u) + kvn k + 3 12 R3 R3 R3 Z bA2 + |∇vn |2 dx + o(1). 4 R3 a + bA2 J(u) + 2

Z

Letting n → ∞, and by (5.7) bA4 c˜ + 4

= (5.15)

≥

(5.13)

≥

l1 l2 bA2 J(u) + + + 3 12 4

2

Z

R3

|∇vn |2 dx + o(1) 3

abS 3 (b2 S 4 + 4a|Q|∞ S) 2 bA2 b3 S 6 J(u) + + + + 4|Q|∞ 24|Q|2∞ 24|Q|2∞ 4 2

−C3 λ 2−q +

3

6

3 2

4

R3

|∇vn |2 dx + o(1)

(b S + 4a|Q|∞ S) bA4 bS abS + + + 4|Q|∞ 24|Q|2∞ 24|Q|2∞ 4 3

2

Z

which implies that 3

2 abS 3 (b2 S 4 + 4a|Q|∞ S) 2 b3 S 6 2−q , c˜ ≥ + − C λ + 3 4|Q|∞ 24|Q|2∞ 24|Q|2∞

a contradiction! Then ˜l1 = 0 which is un → u in H 1 (R3 ).

Proof of Theorem 1.6: Using Lemmas 5.1-5.3 and repeating the processes as in Sections 3.1-3.3, the proof of Theorem 1.6 is clear.

24


Acknowledgements: Both authors would like to express their sincere gratitude to Professor Gongbao Li for his interest and some insightful discussions about the Kirchhoff problem. The authors was supported by NSFC (Grant No. 11371158), the program for Changjiang Scholars and Innovative Research Team in University (No. IRT13066). References [1] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [2] M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997) 4619-4627. [3] A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996) 305-330. [4] S. Bernstein, une classe d’équations fonctionnelles aux dérivées partielles. Bulletin de I’Acadmie des Sciences de I’URSS. VII. Série 1940; 4:17-26. [5] P. D’Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992) 247-262. [6] J. L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of International Symposium, Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holland Math. Stud. vol. 30, North-Holland, Amsterdam, 1978, pp. 284-346. [7] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations 6 (2001) 701-730. [8] G. M. Figueiredo, N. Ikoma, J. R. Santos Junior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal. 213 (2014) 931-979. [9] X. He, W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009) 1407-1414. [10] Y. He, G. Li, S. Peng, Concentrating Bound States for Kirchhoff type problems in R3 involving critical Sobolev exponents, Adv. Nonlinear Stud. 14 (2014) 441-468. [11] G. Li, H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3 , J. Differential Equations 257 (2014) 566-600. [12] W. Liu, X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput. 39 (2012) 473-487. [13] K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006) 246-255. [14] A. Ambrosetti, H. Brézis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519-543. [15] T. Barstch, M. Willem, On a elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc. 123 (1995) 3555-3561. [16] T. S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities, Nonlinear Anal. 71 (2009) 2688-2698. [17] J. Garcia-Azorero, I. Peral, J. D. Rossi, A convex-concave problem with a nonlinear boundary condition, J. Differential Equations 198 (2004) 91-128. [18] M. Bouchekif, A. Matallah, Multiple positive solutions for elliptic equations involving a concave term and critical Sobolev-Hardy exponent, Appl. Math. Lett. 22 (2009) 268-275. N +2 [19] V. Benci and G. Cerami, Existence of positive solutions of the equation −∆u + a(x)u = u N −2 in RN , J. Funct. Anal. 88 (1990) 90-117. [20] A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986) 397-408.


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