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Multiplicity of nontrivial solutions for a critical degenerate Kirchhoff type problem
Applied Mathematics Letters 69 (2017) 87–93
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Multiplicity of nontrivial solutions for a critical degenerate Kirchhoff type problem✩ Chunhong Zhanga , Zhisu Liu b, * a
Department of Public Courses, Hunan Vocational College of Science and Technology, Changsha, Hunan 410004, PR China b School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, PR China
article
info
abstract
Article history: Received 4 December 2016 Received in revised form 20 January 2017 Accepted 21 January 2017 Available online 14 February 2017
In this paper, we study the following Kirchhoff type problem with critical growth
Keywords: Multiplicity Degenerate Kirchhoff type equation Critical growth Variational method
where Ω is a smooth bounded domain in R3 , M ∈ C(R+ , R) and λ > 0. We prove the existence of multiple nontrivial solutions for the above problem, when parameter λ belongs to some left neighborhood of the eigenvalue of the nonlinear operator ∫ −M ( |∇u|2 dx)△. Ω © 2017 Elsevier Ltd. All rights reserved.
⎧ ⎨
(∫
) |∇u|2 dx
−M
△u = λ|u|2 u + |u|4 u
in Ω ,
Ω
⎩
u=0
on ∂Ω ,
1. Introduction In this paper we are concerned with the multiplicity of nontrivial solutions for the following non-local problem: ⎧ (∫ ) ⎨ 2 −M |∇u| dx △u = λ|u|2 u + |u|4 u in Ω , (1.1) Ω ⎩ u=0 on ∂Ω , where Ω is a smooth bounded domain in R3 . It is worthy noticing that the term |u|4 u reaches the Sobolev critical exponent since 2∗ = 6 in Dimension 3. We also observe that Problem (1.1) is related to the stationary analogue of the following equation ⎧ (∫ ) ⎨ utt − M |∇u|2 dx △u = f (x, u) in Ω , (1.2) Ω ⎩ u=0 on ∂Ω . ✩
Research supported by the NSFC (Grant No. 11626127).
* Corresponding author. E-mail addresses: [email protected] (C. Zhang), [email protected] (Z. Liu). http://dx.doi.org/10.1016/j.aml.2017.01.016 0893-9659/© 2017 Elsevier Ltd. All rights reserved.
88
C. Zhang, Z. Liu / Applied Mathematics Letters 69 (2017) 87–93
Such kinds of equations as (1.2) fall outside the scope of the theory of classical elliptic equations because ) (∫ of their obvious nonlocal feature in M Ω |∇u|2 dx . Indeed, Eq. (1.2) are no longer a pointwise identity. The solvability of the Kirchhoff type equations for any dimension has already been extensively studied since Lions [1] introduced an abstracted framework to this problem, see for instance [2–4] and the references therein. As we all know, there exist extensive literature devoted to the study of semilinear elliptic equations with critical growth after the celebrated paper by Brezis and Nirenberg [5]. In particular, people generalize those results in [5] to various kinds of elliptic equations either in bounded domain or the whole space (see [6–8] and the references therein). Recently, the study of Kirchhoff type equation with critical growth attracts lots of attention. For example, we refer to [9–12] for the problems in bounded domain, in which the variational techniques are developed when proving the existence of solutions. The main difficulty is due to the lack of compactness of the Sobolev embedding H01 (Ω ) ↪→ L6 (Ω ). In [9], the multiplicity of nontrivial solutions was obtained by using the genus theory. Combining with the method of Brezis and Nirenberg [5], Naimen [11] made use of cut-off techniques to derive the existence results. Based on [12], Yang, Liu and Ouyang also got the multiplicity of nontrivial solutions. For problems in the whole space, we refer to [13–16] and the references therein. Note that problem (1.1) is said to be degenerate if M (0) = 0, and non-degenerate if M (0) > 0 (see [17]). We also point out that the above mentioned works only cover the non-degenerate case. To the best of our knowledge, there are very few results employing variational methods to consider the degenerate case of problem (1.1). We refer the readers to [17–22] and the references therein. Motivated by the above observations, our main goal in this paper is to investigate multiplicity of nontrivial solutions of the degenerate Kirchhoff problem (1.1). Our method is based on the concentration-compactness principle (see [23]) together with an abstract critical point theorem (see [7]), which relates to the Z2 -cohomological index. Hereafter, we need the following notations. • For any ρ > 0 and for any z ∈ R3 , Bρ (z) denotes the ball of radius ρ centered at z and |Bρ (z)| denotes its Lebesgue measure. Let H := H01 (Ω ) be the Sobolev space equipped with the inner product and the ∫ 1 norm ⟨u, v⟩ = Ω ∇u∇vdx, ∥u∥ := ⟨u, u⟩ 2 , respectively. Let | · |s be the usual Ls -norm and therefore the embedding H ↪→ Ls (Ω ) is continuous for s ∈ [1, 2∗ ], and is compact for s ∈ [1, 2∗ ), where 2∗ = 6. 2 Denote by S the best Sobolev constant, where S := inf{∥u∥2 /|u|2∗ : u ∈ H01 (Ω ) \ {0}}. Now we consider the Kirchhoff type nonlinear eigenvalue problem: ⎧ (∫ ) ⎨ − in Ω , |∇u|2 dx △u = λu3 Ω ⎩ u=0 on ∂Ω .
(1.3)
In [24], the authors used Morse theory to prove the existence of unbounded sequences of eigenvalues of the above problem. However, it does not seem to be known whether such a sequence coincides with the standard sequence of eigenvalues defined by using the genus. We recall some details as follows. Define { } (∫ )2 1 1 2 Ψ (u) = ∫ , u ∈ M = u ∈ H0 (Ω ) : |∇u| dx = 1 . (1.4) |u|4 dx Ω Ω Then eigenvalues of problem (1.3) on M coincide with critical values of Ψ . Set Ψ a = {u ∈ M : Ψ (u) ≤ a},
Ψa = {u ∈ M : Ψ (u) ≥ a},
a ∈ R,
and F denotes the class of symmetric subsets of M and i(M) stands for the Z2 -cohomological index of M ∈ F which will be introduced in Section 3. Set λk :=
inf
sup Ψ (u),
M ∈F ,i(M )≥k u∈M
k ∈ N.
C. Zhang, Z. Liu / Applied Mathematics Letters 69 (2017) 87–93
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Then 0 < λ1 < λ2 ≤ λ3 ≤ · · · → +∞ is a sequence of eigenvalues of problem (1.3), and if λk < λk+1 , then i(Ψ λk ) = i(M \ Ψλk ) = k.
(1.5)
Now we note that λ1 > S/|Ω |2/3 . Indeed, let φ1 be an eigenfunction associated with λ1 . It is known that S 2 is not attained at φ1 . Hence, from the H¨ older inequality, we deduce that λ1 = ∥φ1 ∥2 /|φ1 |2 > S/|Ω |2/3 . Our main result reads as follows: 2
Theorem 1.1. Assume the function M (t) = t and let λ∗ = min{ |Ω|S1/3 , |Ω|S2/3 }. (A1) If λ1 − λ∗ < λ < λ1 , then problem (1.1) has a pair of nontrivial solutions ±u such that u → 0 as λ ↗ λ1 . (A2) If λk ≤ λ < λk+1 = · · · · · = λk+m < λk+m+1 for some k, m ∈ N and λ > λk+1 − λ∗ , then problem (1.1) has m distinct pairs of nontrivial solutions ±uj , j = 1, . . . , m such that uj → 0 as λ ↗ λk+1 . The remainder of this paper is organized as follows. Some preliminaries are presented in Section 3. In Section 2, we complete the proof of Theorem 1.1. 2. Preliminaries Recall that u ∈ H is called a weak solution of (1.1) if ∫ ∫ ∫ ∥u∥2 ∇u∇vdx = λ |u|2 uvdx + |u|4 uvdx, Ω
Ω
∀v ∈ H.
(2.1)
Ω
Seeking a weak solution of problem (2.1) is equivalent to finding a critical point of C 1 functional I(u) =
1 λ 4 1 6 ∥u∥4 − |u|4 − |u|6 , 4 4 6
∀u ∈ H.
(2.2)
The Z2 -cohomological index of Fadell and Rabinowitz [25] is defined as follows. Let X be a Banach space and A denote the class of symmetric subsets of X \ {0}. For A ∈ A, let A¯ = A/Z2 be the quotient space ¯ and assume that of A with each u and −u identified. Define g : A¯ → RP ∞ to be the classifying map of A, ∗ ∗ ∞ ∗ ¯ g : H (RP ) → H (A) is the induced homomorphism of the Alexander–Spanier cohomology rings. The cohomological index of A is defined by { sup{m ≥ 1 : g ∗ (ω m−1 ) ̸= 0}, A ̸= ∅, i(A) = 0, A = ∅, where ω ∈ H 1 (RP ∞ ) is the generator of the polynomial ring H ∗ (RP ∞ ) = Z2 [ω]. Proposition 2.1 (See [25])). If there is an odd continuous map from A to B (in particular, if A ⊂ B), then i(A) ≤ i(B). In particular, equality holds when the map is an odd homeomorphism. Let r > 0 and Sr = {u ∈ X : ∥u∥ = r}. Let Γ denote the group of odd homeomorphisms of X that are the identity outside Φ −1 (0, d) for 0 < d ≤ +∞. The pseudo-index i∗ of M ∈ A is defined as i∗ (M ) = minγ∈Γ i(γ(M ) ∩ Sr ) (see Benci [26]). We recall an abstract critical point theorem which will be used later. Theorem 2.1 ([7]). Let A0 , B0 be symmetric subsets of S1 such that A0 is compact, B0 is closed, and i(A0 ) ≥ k + m, i(S1 \ B0 ) ≤ k for some integers k ≥ 0 and m ≥ 1. Assume that there exists R > r such that sup I(A) ≤ 0 < inf I(B), sup I(X) < b, where A = {Ru : u ∈ A0 }, B = {ru : u ∈ B0 }, and
C. Zhang, Z. Liu / Applied Mathematics Letters 69 (2017) 87–93
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X = {tu : u ∈ A, 0 ≤ t ≤ 1}. For j = k + 1, . . . , k + m, let A∗j = {M ∈ A : M is compact and i∗ (M ) ≥ j} and set c∗j := inf M ∈A∗j maxu∈M I(u). Then inf I(B) ≤ c∗k+1 ≤ · · · ≤ c∗k+m ≤ sup I(X), in particular, 0 < c∗j < b. If, in addition, I satisfies the (P S)c condition for all c ∈ (0, b), then each c∗j is a critical value of I and there exist m district pairs of associated critical points. Remark 2.1. It is difficult for us to employ the above abstract critical point theorem to study higher dimensional case than three, since the relation sup I(A) ≤ 0 < inf I(B) in Theorem 2.1 is not easy to get if 2∗ ≤ 4. Therefore, in the present paper we only consider the problem in R3 .
3. Proof the main result The proof of Theorem 1.1 requires some lemmas. Lemma 3.1. The functional I satisfies the Palais–Smale condition in (0, c∗ ) with c∗ :=
1 6 12 S .
Proof . Suppose {un } ⊂ H is a sequence such that as n → ∞ I(un ) → c < c∗ and I ′ (un ) → 0 in H −1 , where H −1 is the dual space of H. Then there exist positive constants C1 , C2 such that 1 1 6 C1 + C2 ∥un ∥ ≥ I(un ) − I ′ (un )un ≥ |un |6 4 12
(3.3)
for large n ∈ N, which, together with the definition of I, implies that ∥un ∥ ≤ C for some C > 0 (see [5]). Hence, we may extract a subsequence {un } (relabeled) such that un ⇀ u in H,
un → u a.e. on Ω ,
un → u in Ls (Ω ), s ∈ (1, 6).
(3.4)
Furthermore, based on the second concentration compactness lemma [23], we can assume that there exist ¯ and values {ηk }k∈J , {νk }k∈J ⊂ R+ such that up to an at most countable set J, points {xk }k∈J ⊂ Ω subsequences, |∇un |2 ⇀ dη ≥ |∇u|2 +
∑
ηk δxk ,
|un |6 ⇀ dν = |u|6 +
k∈K
∑
1
νk δxk , ηk ≥ Sνk3
k∈K
in the measure sense, where δx is the Dirac mass at x ∈ R3 with mass 1. Now we show J = ∅. Otherwise, J ̸= ∅. Fix k ∈ J. Now for fixed ε > 0, we consider χ1 (x) := χ ˜1 (x − xk ), where χ ˜1 ∈ C0∞ (R3 , [0, 1]) is such 2 3 ′ ˜1 | ∈ [0, ε ]. Since I (un ) → 0 in H −1 (Ω ) and {un } is that χ ˜1 ≡ 1 on Bε (0), χ ˜1 ≡ 0 on R \ B2ε (0) and |∇χ bounded sequence, we have ( ) ∫ ∫ ∫ ′ 2 4 6 0 = lim I (un )un χ1 = lim ∥un ∥ ∇un ∇(un χ1 )dx − λ un χ1 dx − un χ1 dx n→∞ n→∞ Ω∫ Ω Ω ∫ (3.5) ≥ ∥un ∥2 χ1 dη − χ1 dν + o(1), ¯ Ω
¯ Ω
1
where o(1) → 0 as ε → 0. Taking ε → 0, we have νk ≥ ηk2 . It follows from ηk ≥ Sνk3 that νk ≥ S 6 . Since 6 1 6 1 |un |6 + o(1) ≥ 12 S for large n, which is impossible. {un } is the Palais–Smale sequence of I, we have c ≥ 12 6 6 Hence, J = ∅. Consequently we have |un |6 → |u|6 as n → ∞. The remaining proof is similar to Lemma 3.6 in [11], so we omit it. □
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Lemma 3.2. If λk < λk+1 , then Ψ λk has a compact symmetric subset A0 with i(A0 ) = k. Proof . The arguments will be divided into three steps. Step 1, we show that for each w ∈ L4 (Ω ), the following problem { −∥u∥2 △u = |w|2 w in Ω , u=0 on ∂Ω ,
(3.6)
has a unique weak solution u ∈ H01 (Ω ). Indeed, the existence follows from a standard minimization argument. Define the operator u ∈ H01 (Ω ) ↦→ Υ (u) ∈ H −1 by ∫ Υ (u)v = ∥u∥2 ∇u∇vdx, u, v ∈ H01 (Ω ). Ω
C(H01 (Ω ), H −1 ),
It is easy to see that Υ ∈ and that every sequence {un } ⊂ H01 (Ω ) such that un ⇀ u, Υ (un )(un − u) → 0 has a subsequence that converges strongly to u (see [24]). Moreover, we show that the operator Υ is strictly monotone, that is, (Υ (u) − Υ (v))(u − v) > 0 for all u ̸= v in H01 (Ω ). Indeed, from the definition of Υ and the H¨ older inequality, we deduce ∫ ∫ (Υ (u) − Υ (v))(u − v) = ∥u∥4 + ∥v∥4 − ∥u∥2 ∇u∇vdx − ∥v∥2 ∇u∇vdx R3
= ∥u∥4 + ∥v∥4 − (∥u∥2 + ∥v∥2 )∥u∥∥v∥ > 0
R3
for u ̸= v in H01 (Ω ), which implies that u is the unique weak solution of problem (3.6). Step 2, we claim that the map J : w ∈ L4 (Ω ) ↦→ u ∈ H01 (Ω ) is continuous. Indeed, let wn → w in L4 (Ω ) ∫ and let un = J(wn ), then Υ (un )v = Ω |wn |2 wn vdx for all v ∈ H01 (Ω ). Testing with v = un gives ∫ 3 3 4 ∥un ∥ = |wn |2 wn un dx ≤ |wn |4 |un |4 ≤ C|wn |4 ∥un ∥, Ω
which implies that {un } is bounded sequence in H01 (Ω ). Assume that un ⇀ u0 in H01 (Ω ) and un → u0 in L4 (Ω ) after extracting a subsequence, then ∫ 3 4 Υ (un )(un − u0 ) = ∥un ∥2 |wn |2 wn (un − u0 )dx ≤ ∥un ∥2 |wn |4 |un − u0 |4 → 0, Ω
which implies that un → u0 in H01 (Ω ). Therefore, the map J is continuous. u , u ∈ H01 (Ω ) be the radial projection onto Step 3, we are ready to prove the conclusion. Let π(u) = |u| 4 1 λ ¯ = {u ∈ H (Ω ) : |u|4 = 1} and set A = π(Ψ k ) = {v ∈ M ¯ : ∥v∥4 ≤ λk }. Then it follows from M 0 Proposition 2.1 and (1.5) that i(A) = i(Ψ λk ) = k. Set u = J(v) with v ∈ A. Then ∫ 3 4 Υ (u)u = ∥u∥ = v 2 vudx ≤ |v|4 |u|4 = |u|4 , 1 = Υ (u)v ≤ ∥u∥3 ∥v∥, Ω
∥u∥ |u|4
¯ which implies that ∥π(u)∥ = ≤ ∥v∥, and hence π(u) ∈ A. Set J¯ = π ◦ J and let A¯ = J(A) ⊂ A. It 4 1 ¯ follows from the compactness embedding of L (Ω ) ↪→ H0 (Ω ) and the continuity of the odd map J that A¯ is ¯ = i(A) = k. Set A0 = π(A), ¯ then A0 ⊂ Ψ λk is compact and i(A0 ) = i(A) ¯ = k. □ a compact set and i(A) Proof of Theorem 1.1. Here we only give the proof of (ii) since the arguments of (i) are similar and 3 1 simpler. In order to apply Theorem 2.1 with b = 12 S 2 to obtain our conclusion, we take B0 = Ψλk+1 such that i(S1 \ B0 ) = k by (1.5). Since λk+m < λk+m+1 , it follows from Lemma 3.2 that Ψ λk+m has a compact symmetric subset A0 with i(A0 ) = k + m. Moreover, by Lemma 3.1, I satisfies the (PS)c condition for all c < b. Set R > r > 0 and A, B, X be as in Theorem 2.1. For u ∈ B0 , ( ) r4 λr4 4 r6 6 r4 λ r6 I(ru) = ∥u∥4 − |u|4 − |u|6 ≥ 1− − , 4 4 6 4 λk+1 6S 3
C. Zhang, Z. Liu / Applied Mathematics Letters 69 (2017) 87–93
92
which, together with λ < λk+1 , implies that inf I(B) > 0 if r is sufficiently small. For u ∈ A0 ⊂ Ψ λk+m , we have by the H¨ older inequality ( ) R4 λR4 4 R6 6 R4 λ R6 , I(Ru) = ∥u∥4 − |u|4 − |u|6 ≤ 1− − 3 1 4 4 6 4 λk+m 6λ 2 |Ω | 2 k+m
which yields that there exists R > r such that I(v) ≤ 0 for v ∈ A. For u ∈ X, there exist α > 0 and 4 w ∈ Ψ λk+m such that u = αw. So, letting ρ := |u|4 , we have 6
I(u)
|u|4 α4 α4 λk+m 4 λ 4 λ 4 1 6 − |u|4 − |u|6 ≤ |w|4 − |u|4 − 1 4 4 6 4 4 6|Ω | 2 ] [ 3 4 6 (λk+m − λ)|u|4 |u|4 |Ω | (λk+m − λ)ρ ρ2 = = − − (λk+m − λ)3 , 1 ≤ sup 1 4 4 12 ρ>0 6|Ω | 2 6|Ω | 2 =
6
S 3 which implies that sup I(X) ≤ |Ω| 12 (λk+m − λ) < 12 . Therefore, Theorem 2.1 gives m distinct pairs of nontrivial critical points ±uj , j = 1, . . . , m of I so that
0 < I(uj ) ≤
|Ω | (λk+m − λ)3 → 0 as λ ↗ λk+1 , 12
(3.7)
6
1 which implies that 12 |uj |6 = I(uj ) − 14 I ′ (uj )uj = I(uj ) → 0. Hence, uj → 0 in L6 (Ω ). By the H¨older 4 1 inequality we have 2 ∥uj ∥4 + 12 |uj |4 = 6I(uj ) − I ′ (uj )uj → 0. The proof is complete.
References
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Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Multiplicity of nontrivial solutions for a critical degenerate Kirchhoff type problem✩ Chunhong Zhanga , Zhisu Liu b, * a
Department of Public Courses, Hunan Vocational College of Science and Technology, Changsha, Hunan 410004, PR China b School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, PR China
article
info
abstract
Article history: Received 4 December 2016 Received in revised form 20 January 2017 Accepted 21 January 2017 Available online 14 February 2017
In this paper, we study the following Kirchhoff type problem with critical growth
Keywords: Multiplicity Degenerate Kirchhoff type equation Critical growth Variational method
where Ω is a smooth bounded domain in R3 , M ∈ C(R+ , R) and λ > 0. We prove the existence of multiple nontrivial solutions for the above problem, when parameter λ belongs to some left neighborhood of the eigenvalue of the nonlinear operator ∫ −M ( |∇u|2 dx)△. Ω © 2017 Elsevier Ltd. All rights reserved.
⎧ ⎨
(∫
) |∇u|2 dx
−M
△u = λ|u|2 u + |u|4 u
in Ω ,
Ω
⎩
u=0
on ∂Ω ,
1. Introduction In this paper we are concerned with the multiplicity of nontrivial solutions for the following non-local problem: ⎧ (∫ ) ⎨ 2 −M |∇u| dx △u = λ|u|2 u + |u|4 u in Ω , (1.1) Ω ⎩ u=0 on ∂Ω , where Ω is a smooth bounded domain in R3 . It is worthy noticing that the term |u|4 u reaches the Sobolev critical exponent since 2∗ = 6 in Dimension 3. We also observe that Problem (1.1) is related to the stationary analogue of the following equation ⎧ (∫ ) ⎨ utt − M |∇u|2 dx △u = f (x, u) in Ω , (1.2) Ω ⎩ u=0 on ∂Ω . ✩
Research supported by the NSFC (Grant No. 11626127).
* Corresponding author. E-mail addresses: [email protected] (C. Zhang), [email protected] (Z. Liu). http://dx.doi.org/10.1016/j.aml.2017.01.016 0893-9659/© 2017 Elsevier Ltd. All rights reserved.
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C. Zhang, Z. Liu / Applied Mathematics Letters 69 (2017) 87–93
Such kinds of equations as (1.2) fall outside the scope of the theory of classical elliptic equations because ) (∫ of their obvious nonlocal feature in M Ω |∇u|2 dx . Indeed, Eq. (1.2) are no longer a pointwise identity. The solvability of the Kirchhoff type equations for any dimension has already been extensively studied since Lions [1] introduced an abstracted framework to this problem, see for instance [2–4] and the references therein. As we all know, there exist extensive literature devoted to the study of semilinear elliptic equations with critical growth after the celebrated paper by Brezis and Nirenberg [5]. In particular, people generalize those results in [5] to various kinds of elliptic equations either in bounded domain or the whole space (see [6–8] and the references therein). Recently, the study of Kirchhoff type equation with critical growth attracts lots of attention. For example, we refer to [9–12] for the problems in bounded domain, in which the variational techniques are developed when proving the existence of solutions. The main difficulty is due to the lack of compactness of the Sobolev embedding H01 (Ω ) ↪→ L6 (Ω ). In [9], the multiplicity of nontrivial solutions was obtained by using the genus theory. Combining with the method of Brezis and Nirenberg [5], Naimen [11] made use of cut-off techniques to derive the existence results. Based on [12], Yang, Liu and Ouyang also got the multiplicity of nontrivial solutions. For problems in the whole space, we refer to [13–16] and the references therein. Note that problem (1.1) is said to be degenerate if M (0) = 0, and non-degenerate if M (0) > 0 (see [17]). We also point out that the above mentioned works only cover the non-degenerate case. To the best of our knowledge, there are very few results employing variational methods to consider the degenerate case of problem (1.1). We refer the readers to [17–22] and the references therein. Motivated by the above observations, our main goal in this paper is to investigate multiplicity of nontrivial solutions of the degenerate Kirchhoff problem (1.1). Our method is based on the concentration-compactness principle (see [23]) together with an abstract critical point theorem (see [7]), which relates to the Z2 -cohomological index. Hereafter, we need the following notations. • For any ρ > 0 and for any z ∈ R3 , Bρ (z) denotes the ball of radius ρ centered at z and |Bρ (z)| denotes its Lebesgue measure. Let H := H01 (Ω ) be the Sobolev space equipped with the inner product and the ∫ 1 norm ⟨u, v⟩ = Ω ∇u∇vdx, ∥u∥ := ⟨u, u⟩ 2 , respectively. Let | · |s be the usual Ls -norm and therefore the embedding H ↪→ Ls (Ω ) is continuous for s ∈ [1, 2∗ ], and is compact for s ∈ [1, 2∗ ), where 2∗ = 6. 2 Denote by S the best Sobolev constant, where S := inf{∥u∥2 /|u|2∗ : u ∈ H01 (Ω ) \ {0}}. Now we consider the Kirchhoff type nonlinear eigenvalue problem: ⎧ (∫ ) ⎨ − in Ω , |∇u|2 dx △u = λu3 Ω ⎩ u=0 on ∂Ω .
(1.3)
In [24], the authors used Morse theory to prove the existence of unbounded sequences of eigenvalues of the above problem. However, it does not seem to be known whether such a sequence coincides with the standard sequence of eigenvalues defined by using the genus. We recall some details as follows. Define { } (∫ )2 1 1 2 Ψ (u) = ∫ , u ∈ M = u ∈ H0 (Ω ) : |∇u| dx = 1 . (1.4) |u|4 dx Ω Ω Then eigenvalues of problem (1.3) on M coincide with critical values of Ψ . Set Ψ a = {u ∈ M : Ψ (u) ≤ a},
Ψa = {u ∈ M : Ψ (u) ≥ a},
a ∈ R,
and F denotes the class of symmetric subsets of M and i(M) stands for the Z2 -cohomological index of M ∈ F which will be introduced in Section 3. Set λk :=
inf
sup Ψ (u),
M ∈F ,i(M )≥k u∈M
k ∈ N.
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Then 0 < λ1 < λ2 ≤ λ3 ≤ · · · → +∞ is a sequence of eigenvalues of problem (1.3), and if λk < λk+1 , then i(Ψ λk ) = i(M \ Ψλk ) = k.
(1.5)
Now we note that λ1 > S/|Ω |2/3 . Indeed, let φ1 be an eigenfunction associated with λ1 . It is known that S 2 is not attained at φ1 . Hence, from the H¨ older inequality, we deduce that λ1 = ∥φ1 ∥2 /|φ1 |2 > S/|Ω |2/3 . Our main result reads as follows: 2
Theorem 1.1. Assume the function M (t) = t and let λ∗ = min{ |Ω|S1/3 , |Ω|S2/3 }. (A1) If λ1 − λ∗ < λ < λ1 , then problem (1.1) has a pair of nontrivial solutions ±u such that u → 0 as λ ↗ λ1 . (A2) If λk ≤ λ < λk+1 = · · · · · = λk+m < λk+m+1 for some k, m ∈ N and λ > λk+1 − λ∗ , then problem (1.1) has m distinct pairs of nontrivial solutions ±uj , j = 1, . . . , m such that uj → 0 as λ ↗ λk+1 . The remainder of this paper is organized as follows. Some preliminaries are presented in Section 3. In Section 2, we complete the proof of Theorem 1.1. 2. Preliminaries Recall that u ∈ H is called a weak solution of (1.1) if ∫ ∫ ∫ ∥u∥2 ∇u∇vdx = λ |u|2 uvdx + |u|4 uvdx, Ω
Ω
∀v ∈ H.
(2.1)
Ω
Seeking a weak solution of problem (2.1) is equivalent to finding a critical point of C 1 functional I(u) =
1 λ 4 1 6 ∥u∥4 − |u|4 − |u|6 , 4 4 6
∀u ∈ H.
(2.2)
The Z2 -cohomological index of Fadell and Rabinowitz [25] is defined as follows. Let X be a Banach space and A denote the class of symmetric subsets of X \ {0}. For A ∈ A, let A¯ = A/Z2 be the quotient space ¯ and assume that of A with each u and −u identified. Define g : A¯ → RP ∞ to be the classifying map of A, ∗ ∗ ∞ ∗ ¯ g : H (RP ) → H (A) is the induced homomorphism of the Alexander–Spanier cohomology rings. The cohomological index of A is defined by { sup{m ≥ 1 : g ∗ (ω m−1 ) ̸= 0}, A ̸= ∅, i(A) = 0, A = ∅, where ω ∈ H 1 (RP ∞ ) is the generator of the polynomial ring H ∗ (RP ∞ ) = Z2 [ω]. Proposition 2.1 (See [25])). If there is an odd continuous map from A to B (in particular, if A ⊂ B), then i(A) ≤ i(B). In particular, equality holds when the map is an odd homeomorphism. Let r > 0 and Sr = {u ∈ X : ∥u∥ = r}. Let Γ denote the group of odd homeomorphisms of X that are the identity outside Φ −1 (0, d) for 0 < d ≤ +∞. The pseudo-index i∗ of M ∈ A is defined as i∗ (M ) = minγ∈Γ i(γ(M ) ∩ Sr ) (see Benci [26]). We recall an abstract critical point theorem which will be used later. Theorem 2.1 ([7]). Let A0 , B0 be symmetric subsets of S1 such that A0 is compact, B0 is closed, and i(A0 ) ≥ k + m, i(S1 \ B0 ) ≤ k for some integers k ≥ 0 and m ≥ 1. Assume that there exists R > r such that sup I(A) ≤ 0 < inf I(B), sup I(X) < b, where A = {Ru : u ∈ A0 }, B = {ru : u ∈ B0 }, and
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X = {tu : u ∈ A, 0 ≤ t ≤ 1}. For j = k + 1, . . . , k + m, let A∗j = {M ∈ A : M is compact and i∗ (M ) ≥ j} and set c∗j := inf M ∈A∗j maxu∈M I(u). Then inf I(B) ≤ c∗k+1 ≤ · · · ≤ c∗k+m ≤ sup I(X), in particular, 0 < c∗j < b. If, in addition, I satisfies the (P S)c condition for all c ∈ (0, b), then each c∗j is a critical value of I and there exist m district pairs of associated critical points. Remark 2.1. It is difficult for us to employ the above abstract critical point theorem to study higher dimensional case than three, since the relation sup I(A) ≤ 0 < inf I(B) in Theorem 2.1 is not easy to get if 2∗ ≤ 4. Therefore, in the present paper we only consider the problem in R3 .
3. Proof the main result The proof of Theorem 1.1 requires some lemmas. Lemma 3.1. The functional I satisfies the Palais–Smale condition in (0, c∗ ) with c∗ :=
1 6 12 S .
Proof . Suppose {un } ⊂ H is a sequence such that as n → ∞ I(un ) → c < c∗ and I ′ (un ) → 0 in H −1 , where H −1 is the dual space of H. Then there exist positive constants C1 , C2 such that 1 1 6 C1 + C2 ∥un ∥ ≥ I(un ) − I ′ (un )un ≥ |un |6 4 12
(3.3)
for large n ∈ N, which, together with the definition of I, implies that ∥un ∥ ≤ C for some C > 0 (see [5]). Hence, we may extract a subsequence {un } (relabeled) such that un ⇀ u in H,
un → u a.e. on Ω ,
un → u in Ls (Ω ), s ∈ (1, 6).
(3.4)
Furthermore, based on the second concentration compactness lemma [23], we can assume that there exist ¯ and values {ηk }k∈J , {νk }k∈J ⊂ R+ such that up to an at most countable set J, points {xk }k∈J ⊂ Ω subsequences, |∇un |2 ⇀ dη ≥ |∇u|2 +
∑
ηk δxk ,
|un |6 ⇀ dν = |u|6 +
k∈K
∑
1
νk δxk , ηk ≥ Sνk3
k∈K
in the measure sense, where δx is the Dirac mass at x ∈ R3 with mass 1. Now we show J = ∅. Otherwise, J ̸= ∅. Fix k ∈ J. Now for fixed ε > 0, we consider χ1 (x) := χ ˜1 (x − xk ), where χ ˜1 ∈ C0∞ (R3 , [0, 1]) is such 2 3 ′ ˜1 | ∈ [0, ε ]. Since I (un ) → 0 in H −1 (Ω ) and {un } is that χ ˜1 ≡ 1 on Bε (0), χ ˜1 ≡ 0 on R \ B2ε (0) and |∇χ bounded sequence, we have ( ) ∫ ∫ ∫ ′ 2 4 6 0 = lim I (un )un χ1 = lim ∥un ∥ ∇un ∇(un χ1 )dx − λ un χ1 dx − un χ1 dx n→∞ n→∞ Ω∫ Ω Ω ∫ (3.5) ≥ ∥un ∥2 χ1 dη − χ1 dν + o(1), ¯ Ω
¯ Ω
1
where o(1) → 0 as ε → 0. Taking ε → 0, we have νk ≥ ηk2 . It follows from ηk ≥ Sνk3 that νk ≥ S 6 . Since 6 1 6 1 |un |6 + o(1) ≥ 12 S for large n, which is impossible. {un } is the Palais–Smale sequence of I, we have c ≥ 12 6 6 Hence, J = ∅. Consequently we have |un |6 → |u|6 as n → ∞. The remaining proof is similar to Lemma 3.6 in [11], so we omit it. □
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Lemma 3.2. If λk < λk+1 , then Ψ λk has a compact symmetric subset A0 with i(A0 ) = k. Proof . The arguments will be divided into three steps. Step 1, we show that for each w ∈ L4 (Ω ), the following problem { −∥u∥2 △u = |w|2 w in Ω , u=0 on ∂Ω ,
(3.6)
has a unique weak solution u ∈ H01 (Ω ). Indeed, the existence follows from a standard minimization argument. Define the operator u ∈ H01 (Ω ) ↦→ Υ (u) ∈ H −1 by ∫ Υ (u)v = ∥u∥2 ∇u∇vdx, u, v ∈ H01 (Ω ). Ω
C(H01 (Ω ), H −1 ),
It is easy to see that Υ ∈ and that every sequence {un } ⊂ H01 (Ω ) such that un ⇀ u, Υ (un )(un − u) → 0 has a subsequence that converges strongly to u (see [24]). Moreover, we show that the operator Υ is strictly monotone, that is, (Υ (u) − Υ (v))(u − v) > 0 for all u ̸= v in H01 (Ω ). Indeed, from the definition of Υ and the H¨ older inequality, we deduce ∫ ∫ (Υ (u) − Υ (v))(u − v) = ∥u∥4 + ∥v∥4 − ∥u∥2 ∇u∇vdx − ∥v∥2 ∇u∇vdx R3
= ∥u∥4 + ∥v∥4 − (∥u∥2 + ∥v∥2 )∥u∥∥v∥ > 0
R3
for u ̸= v in H01 (Ω ), which implies that u is the unique weak solution of problem (3.6). Step 2, we claim that the map J : w ∈ L4 (Ω ) ↦→ u ∈ H01 (Ω ) is continuous. Indeed, let wn → w in L4 (Ω ) ∫ and let un = J(wn ), then Υ (un )v = Ω |wn |2 wn vdx for all v ∈ H01 (Ω ). Testing with v = un gives ∫ 3 3 4 ∥un ∥ = |wn |2 wn un dx ≤ |wn |4 |un |4 ≤ C|wn |4 ∥un ∥, Ω
which implies that {un } is bounded sequence in H01 (Ω ). Assume that un ⇀ u0 in H01 (Ω ) and un → u0 in L4 (Ω ) after extracting a subsequence, then ∫ 3 4 Υ (un )(un − u0 ) = ∥un ∥2 |wn |2 wn (un − u0 )dx ≤ ∥un ∥2 |wn |4 |un − u0 |4 → 0, Ω
which implies that un → u0 in H01 (Ω ). Therefore, the map J is continuous. u , u ∈ H01 (Ω ) be the radial projection onto Step 3, we are ready to prove the conclusion. Let π(u) = |u| 4 1 λ ¯ = {u ∈ H (Ω ) : |u|4 = 1} and set A = π(Ψ k ) = {v ∈ M ¯ : ∥v∥4 ≤ λk }. Then it follows from M 0 Proposition 2.1 and (1.5) that i(A) = i(Ψ λk ) = k. Set u = J(v) with v ∈ A. Then ∫ 3 4 Υ (u)u = ∥u∥ = v 2 vudx ≤ |v|4 |u|4 = |u|4 , 1 = Υ (u)v ≤ ∥u∥3 ∥v∥, Ω
∥u∥ |u|4
¯ which implies that ∥π(u)∥ = ≤ ∥v∥, and hence π(u) ∈ A. Set J¯ = π ◦ J and let A¯ = J(A) ⊂ A. It 4 1 ¯ follows from the compactness embedding of L (Ω ) ↪→ H0 (Ω ) and the continuity of the odd map J that A¯ is ¯ = i(A) = k. Set A0 = π(A), ¯ then A0 ⊂ Ψ λk is compact and i(A0 ) = i(A) ¯ = k. □ a compact set and i(A) Proof of Theorem 1.1. Here we only give the proof of (ii) since the arguments of (i) are similar and 3 1 simpler. In order to apply Theorem 2.1 with b = 12 S 2 to obtain our conclusion, we take B0 = Ψλk+1 such that i(S1 \ B0 ) = k by (1.5). Since λk+m < λk+m+1 , it follows from Lemma 3.2 that Ψ λk+m has a compact symmetric subset A0 with i(A0 ) = k + m. Moreover, by Lemma 3.1, I satisfies the (PS)c condition for all c < b. Set R > r > 0 and A, B, X be as in Theorem 2.1. For u ∈ B0 , ( ) r4 λr4 4 r6 6 r4 λ r6 I(ru) = ∥u∥4 − |u|4 − |u|6 ≥ 1− − , 4 4 6 4 λk+1 6S 3
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which, together with λ < λk+1 , implies that inf I(B) > 0 if r is sufficiently small. For u ∈ A0 ⊂ Ψ λk+m , we have by the H¨ older inequality ( ) R4 λR4 4 R6 6 R4 λ R6 , I(Ru) = ∥u∥4 − |u|4 − |u|6 ≤ 1− − 3 1 4 4 6 4 λk+m 6λ 2 |Ω | 2 k+m
which yields that there exists R > r such that I(v) ≤ 0 for v ∈ A. For u ∈ X, there exist α > 0 and 4 w ∈ Ψ λk+m such that u = αw. So, letting ρ := |u|4 , we have 6
I(u)
|u|4 α4 α4 λk+m 4 λ 4 λ 4 1 6 − |u|4 − |u|6 ≤ |w|4 − |u|4 − 1 4 4 6 4 4 6|Ω | 2 ] [ 3 4 6 (λk+m − λ)|u|4 |u|4 |Ω | (λk+m − λ)ρ ρ2 = = − − (λk+m − λ)3 , 1 ≤ sup 1 4 4 12 ρ>0 6|Ω | 2 6|Ω | 2 =
6
S 3 which implies that sup I(X) ≤ |Ω| 12 (λk+m − λ) < 12 . Therefore, Theorem 2.1 gives m distinct pairs of nontrivial critical points ±uj , j = 1, . . . , m of I so that
0 < I(uj ) ≤
|Ω | (λk+m − λ)3 → 0 as λ ↗ λk+1 , 12
(3.7)
6
1 which implies that 12 |uj |6 = I(uj ) − 14 I ′ (uj )uj = I(uj ) → 0. Hence, uj → 0 in L6 (Ω ). By the H¨older 4 1 inequality we have 2 ∥uj ∥4 + 12 |uj |4 = 6I(uj ) − I ′ (uj )uj → 0. The proof is complete.
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