Calc. Var. (2014) 49:287–290 DOI 10.1007/s00526-013-0681-x
Calculus of Variations
ERRATUM
Erratum to: Infinitely many solutions for resonant cooperative elliptic systems with sublinear or superlinear terms Guanwei Chen · Shiwang Ma
Published online: 19 November 2013 © Springer-Verlag Berlin Heidelberg 2013
Erratum to: Calc Var DOI 10.1007/s00526-012-0581-5 We mention that there are two errors in the original article.
1 The proof of Lemma 2.4 There is an error in the proof of Lemma 2.4 of the original article. In fact, the inequality |U | ≤ ε < R2 , ∀0 < ε < R2 (R2 is a constant)
(∗)
in the proof of Lemma 2.4 is incorrect (R2 > 0 is a constant). Indeed, if (∗) holds, then U = 0. But we need U = 0 in Lemma 2.4. Therefore, the proof of Lemma 2.4 fails. However, we can still complete the proof of Lemma 2.4 by making some changes in the original condition (AF2 ). We should mention that the changes will not influence the remainder framework of the original article. For the readers convenience, we state the previous conditions (AF1 )–(AF3 ) in the original article as follows: (AF1 ) F(x, U ) ≥ 0, ∀(x, U ) ∈ × R2 , and there exist constants μ ∈ [1, 2) and R1 > 0 such that (∇ F(x, U ), U ) ≤ μF(x, U ), ∀x ∈ and |U | ≥ R1 .
The online version of the original article can be found under doi:10.1007/s00526-012-0581-5. G. Chen (B) School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, Henan Province, People’s Republic of China e-mail:
[email protected] S. Ma School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China e-mail:
[email protected]
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(AF2 ) lim|U |→0 such that
F(x,U ) |U |2
= ∞ uniformly for x ∈ , and there exist constants c2 , R2 > 0 F(x, U ) ≤ c2 |U |, ∀x ∈ and |U | ≤ R2 .
(AF3 )
) lim inf |U |→∞ F(x,U |U |
≥ d > 0 uniformly for x ∈ .
In order to give a correct proof of Lemma 2.4, we change the condition (AF2 ) to (AF2 ) There exist constants a ∈ [1, 2) and c2 , c2 , R2 > 0 such that F(x, U ) ≥ c2 |U |a , ∀x ∈ and U ∈ R2
(i)
F(x, U ) ≤ c2 |U |, ∀x ∈ and |U | ≤ R2 .
(ii)
and
Remark Obviously, (AF2 )-(i) implies that (AF3 ) and F(x, U ) ≥ 0 for all (x, U ) ∈ × R2 hold. Therefore, the conditions (AF3 ) and F(x, U ) ≥ 0, ∀(x, U ) ∈ × R2 in (AF1 ) can be omitted. Next, we give the correct proof of Lemma 2.4. Lemma 2.4 (the original article) Assume that (AF1 ) and (AF2 ) hold. Then there exist a positive integer k1 and two sequences 0 < rk < ρk → 0 as k → ∞ such that αk (λ) := ξk (λ) :=
inf
U ∈Z k , U ≤ρk
inf
U ∈Z k , U =ρk
λ (U ) > 0, ∀k ≥ k1 ,
λ (U ) → 0 as k → ∞ uniformly for λ ∈ [1, 2]
(1.1) (1.2)
and βk (λ) := where Yk =
k m=1
max
U ∈Yk , U =rk
X m and Z k =
∞
m=k
λ (U ) < 0, ∀k ∈ N,
(1.3)
X m for all k ∈ N.
Proof (a) Firstly, we show that (1.1) and (1.2) hold. Obviously, Z k ⊂ W + for all k ≥ k1 := l0 + 1 (see Sect. 2.2 in the original article). It is not hard to check that (AF1 ),(AF2 ) and the imply that definition of F U ) ≤ a1 (|U | + |U |μ ), ∀x ∈ and U ∈ R2 F(x, for some constant a1 > 0. It follows from the definition of λ that for any k ≥ k1 and U ∈ Z k there holds 1 U ) d x ≥ 1 U 2 − 2a1 |U |1 − 2a1 |U |μ λ (U ) ≥ U 2 − 2 F(x, μ , ∀λ ∈ [1, 2], 2 2
(1.4) where | · | p denotes the norm of L p () × L p (). Let lk (1) :=
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|U |μ |U |1 , lk (μ) := sup , ∀k ∈ N. U U ∈Z k \{0} U ∈Z k \{0} U sup
(1.5)
Erratum to: Infinitely many solutions for resonant cooperative elliptic systems
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Then lk (1) → 0 and lk (μ) → 0 as k → ∞
(1.6)
by the Rellich embedding theorem (see [4]). Hence, for any k ≥ k1 and U ∈ Z k , (1.4) and (1.5) imply λ (U ) ≥ Let
1 μ U 2 − 2a1 lk (1) U − 2a1 lk (μ) U μ , ∀λ ∈ [1, 2]. 2
(1.7)
1/(2−μ) μ , k ∈ N. ρk := 16a1 lk (1) + 16a1 lk (μ)
(1.8)
Then by (1.6) and μ < 2, we have ρk → 0 as k → ∞.
(1.9)
For any k ≥ k1 , (1.7) together with (1.8) implies that αk (λ) :=
inf
U ∈Z k , U =ρk
λ (U ) ≥ ρk2 /2 − ρk2 /8 − ρk2 /8 = ρk2 /4 > 0.
That is, (1.1) holds. For any k ≥ k1 and U ∈ Z k with U ≤ ρk , (1.7) implies that λ (U ) ≥ −2a1 lk (1)ρk μ μ thus − 2a1 lk (μ)ρk . Observing that λ (0) = 0 by (AF2 ) and the definition of F, 0≥
μ
inf
U ∈Z k , U ≤ρk
μ
λ (U ) ≥ −2a1 lk (1)ρk − 2a1 lk (μ)ρk , ∀k ≥ k1 .
It follows from (1.6) and (1.9) that ξk (λ) :=
inf
U ∈Z k , U ≤ρk
λ (U ) → 0 as k → ∞ uniformly for λ ∈ [1, 2].
That is, (1.2) holds. (b) Now, we show that (1.3) holds. Since norms | · |a and · are equivalent on finitedimensional space Yk , there exists a constant Ck > 0 such that |U |a ≥ Ck U , ∀U ∈ Yk , ∀k ∈ N,
(1.10)
for any where a ∈ [1, 2) is given in (AF2 ). Thus, by (1.10), (AF2 ) and the definition of F, k ∈ N and U ∈ Yk , we have 1 U ) d x ≤ 1 U 2 − c2 |U |aa λ (U ) ≤ U + 2 − F(x, 2 2
≤
1 U 2 − c2 Cka U a , ∀λ ∈ [1, 2]. 2 (1.11)
If we choose
1/(2−a) 0 < rk < min ρk , c2 Cka , ∀k ∈ N,
then (1.11) implies βk (λ) :=
max
U ∈Yk , U =rk
λ (U ) ≤ −rk2 /2 < 0, ∀k ∈ N.
That is, (1.3) holds. Therefore, the proof is finished.
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2 The proof of Theorem 1.1 In the proof of Theorem 1.1 in the original article, the authors only proved the following result: Claim 1 {U j } is bounded in W . Note that the proof of Theorem 1.1 is based on Lemma 2.1 in the original article, thus we must get the following result: Claim 1 {U j } possesses a strongly convergent subsequence in W = W 0 ⊕ W − ⊕ W + , where {U j } ⊂ Y j and Y j is a finite dimensional subspace ( j ∈ N). Since j ∈ N, Claim 1 is not enough to get the result of Claim 1 . Therefore, to complete the proof of Theorem 1.1, we not only need to prove Claim 1 but also need to prove Claim 1 . Proof of Claim 1 By Claim 1 and the fact dim(W 0 ⊕ W − ) < ∞, without loss of generality, we may assume + − 0 0 + U− and U j U j → U , Uj → U , Uj U
as j → ∞
(2.1)
0 − + 0 − + . By virtue of the Riesz Representation for some U = U +U +U ∗ ∈ W = W ⊕W ⊕W ∗ Theorem, λ j Y : Y j → Y j and I : W → W can be viewed as λ j Y : Y j → Y j and j
j
I : W → W respectively, where Y j∗ and W ∗ are the dual spaces of Y j and W , respectively. Note that (2.23) in the original article implies
− 0 = λ j Y = U + U − λ + χ I (U ) , ∀ j ∈ N, j j j j j j
where χ j : W → Y j is the orthogonal projection for all j ∈ N, that is,
− U+ j = λ j U j + χ j I (U j ) , ∀ j ∈ N.
(2.2)
and the assumptions of F, by the standard argument (see [1,3]), Under the definition of F we know I : W → W ∗ is compact. Therefore, I : W → W is also compact. By the compactness of I and (2.1), we get the right-hand side of (2.2) converges strongly in W and + hence U + j → U in W . Combining this with (2.1), we have U j → U in W . Therefore, Claim 1 is true. Now from the last assertion of Lemma 2.1, we know that = 1 has infinitely many nontrivial critical points. Therefore, problem (1.1) in the original article possesses infinitely many nontrivial solutions. That is, Theorem 1.1 in the original article is true.
References 1. Benci, V., Rabinowitz, P.H.: Critical point theorems for indefinite functionals. Invent. Math. 52, 241–273 (1979) 2. Chen, G., Ma, S.: Infinitely many solutions for resonant cooperative elliptic systems with sublinear or superlinear terms. Calc. Var. doi:10.1007/s00526-012-0581-5 3. Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math., vol. 65. American Mathematical Society, Providence (1986) 4. Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
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