1 Erratum to: Collect. Math. DOI 10.1007/s13348-014-0131-x
The goal of this erratum is to correct a mistake that appears in the assumption \((M_2)\) in the original article. In the correct version, the hypothesis \((M_2)\) should be removed. In such a case, we restate the following assumption:
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\((M_1)\) There exist \(m_2\ge m_1 > 0\) and \(\alpha > 1\) such that \( m_1t^{\alpha -1} \le M(t) \le m_2t^{\alpha -1}\), for all \(t \in \mathbb {R}^+\).
We point out that the original assumption \((M_1)\) implies \(\alpha _1=\alpha _2\), so we rename constant \(\alpha \). In conditions \((F_2)\) and \((F_5)\), we replace \(\beta \) by \(\alpha \).
The correct statement of Lemma 3.2 is the following.
Lemma 3.2
Let \((u_n,v_n)\) be a Palais–Smale sequence for the Euler–Lagrange functional J. Assume that conditions \((M_1)\), \((F_2)\) are satisfied and
Then the sequence \((u_n,v_n)\) is bounded.
In the proof of Lemma 3.2, by hypotheses (0.1), \((M_1)\) and \((F_2)\), we can write for n large enough
where \(c_8\) is a positive constant. Therefore
Now, we suppose that the sequence \((u_n,v_n)\) is not bounded. Without loss of generality, we may assume \(\Vert u_n\Vert _{p(x)}\ge \Vert v_n\Vert _{q(x)}\). Therefore, for n large enough so that \(\Vert u_n\Vert _{p(x)}>1\), we obtain
But this cannot hold since \(\alpha p^->p^{-}>1\). Hence, \((u_n,v_n)\) is bounded.
Theorem 3.1 and Lemma 3.3 remain unchanged. However, Theorems 3.4, 4.1, 4.2 and Lemmas 3.2, 3.3 need to be stated without assumption \((M_2)\). Hypothesis (0.1) should be also added in the statement of Theorems 3.4 and 4.1. The proofs of Theorems 3.4 , 4.1 and 4.2 are similar to the original proofs, but replacing \(\beta \) by \(\alpha \).
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The online version of the original article can be found under doi:10.1007/s13348-014-0131-x.
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Afrouzi, G.A., Mirzapour, M. & Rădulescu, V.D. Erratum to: Nonlocal fourth-order Kirchhoff systems with variable growth: low and high energy solutions. Collect. Math. 67, 225–226 (2016). https://doi.org/10.1007/s13348-016-0165-3
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DOI: https://doi.org/10.1007/s13348-016-0165-3