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:

  • \((M_1)\) 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

$$\begin{aligned} m_1\theta _1(p^-)^{\alpha -1}>\alpha m_2,\quad m_1\theta _2(q^-)^{\alpha -1}>\alpha m_2. \end{aligned}$$
(0.1)

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

$$\begin{aligned} \begin{aligned} c_7&\ge \displaystyle J(u_n,v_n)\ge \displaystyle \dfrac{m_1}{\alpha }\left( \int _{\Omega }\frac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx\right) ^\alpha -\int _{\Omega }\frac{u_n}{\theta _1}\frac{\partial F}{\partial u}(x,u_n,v_n)\,dx\\&\quad +\dfrac{m_1}{\alpha }\left( \int _{\Omega }\frac{1}{q(x)}|\Delta v_n|^{q(x)}\,dx\right) ^\alpha -\int _{\Omega }\frac{v_n}{\theta _2}\frac{\partial F}{\partial v}(x,u_n,v_n)\,dx-c_8,\\ \end{aligned} \end{aligned}$$

where \(c_8\) is a positive constant. Therefore

$$\begin{aligned} \begin{aligned} c_7&\ge \displaystyle J(u_n,v_n)\\&\ge \displaystyle \frac{m_1}{\alpha }\left( \int _\Omega \dfrac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx\right) ^\alpha -\dfrac{m_2}{\theta _1}\left( \int _\Omega \dfrac{1}{p(x)}|\Delta u_n|^{p(x)}\,dx\right) ^{\alpha -1}\int _\Omega |\Delta u_n|^{p(x)}\,dx\\&\quad +\frac{1}{\theta _1}D_1J(u_n,v_n)(u_n)\\&\quad \displaystyle +\frac{m_1}{\alpha }\left( \int _{\Omega }\dfrac{1}{q(x)}|\Delta v_n|^{q(x)}dx\right) ^\alpha -\dfrac{m_2}{\theta _2}\left( \int _\Omega \dfrac{1}{q(x)}|\Delta v_n|^{p(x)}\,dx\right) ^{\alpha -1}\int _\Omega |\Delta v_n|^{p(x)}\,dx\\&\quad +\frac{1}{\theta _2}D_2J(u_n,v_n)(v_n)-c_8\\&\ge \displaystyle \left( \frac{m_1}{\alpha }-\frac{m_2}{\theta _1(p^-)^{\alpha -1}}\right) \left( \int _\Omega |\Delta u_n|^{p(x)}\,dx\right) ^\alpha +\displaystyle \left( \frac{m_1}{\alpha }-\frac{m_2}{\theta _2(q^-)^{\alpha -1}}\right) \left( \int _{\Omega }|\Delta v_n|^{q(x)}dx\right) ^\alpha \\&\quad \displaystyle -\frac{1}{\theta _1}\Vert D_1J(u_n,v_n)\Vert _{*,p(x)}\Vert u_n\Vert -\frac{1}{\theta _2}\Vert D_2J(u_n,v_n)\Vert _{*,q(x)}\Vert v_n\Vert -c_8. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} c_7\ge & {} \displaystyle \left( \frac{m_1}{\alpha }-\frac{m_2}{\theta _1(p^-)^{\alpha -1}}\right) \Vert u_n\Vert _{p(x)}^{\alpha p^{-}}\nonumber \\&-\left( \frac{1}{\theta _1}\Vert D_1J(u_n,v_n)\Vert _{*,p} +\frac{1}{\theta _2}\Vert D_2J(u_n,v_n)\Vert _{*,q}\right) \Vert u_n\Vert _{p(x)}. \end{aligned}$$

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 \(\alpha \).