Correction to: Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity

1 Correction to: Calc. Var. (2019) 58:57 https://doi.org/10.1007/s00526-019-1499-y

In [1], we studied the existence of ground state solutions for a fractional Kirchhoff problem with exponential growth. The correct statement of Lemma 3.3 in [1] is the following.

Lemma 3.3

Assume that $(M_2)$, $(f_2)$ and $(f_6)$ hold. Then

$$\begin{aligned} c_*<\frac{s}{N}\mathscr {M}\left( \left( \frac{\alpha _{N,s}}{\alpha _0}\right) ^{(N-s)/s}\right) . \end{aligned}$$

The proof of this lemma is the same as in [1] and it reduces to replacing $\frac{\alpha _{N,s}}{\alpha _0}$ by $(\frac{\alpha _{N,s}}{\alpha _0})^{(N-s)/s}$.

Accordingly, the related parts which used the result of Lemma 3.3 should be corrected. Hence, in the proof of Lemma 4.1,

$$\begin{aligned} \frac{s}{N}\mathscr {M}(\Vert u_n\Vert ^{N/s}) \rightarrow c_*<\frac{s}{N}\mathscr {M} \left( \frac{\alpha _{N,s}}{\alpha _0}\right) \end{aligned}$$

should be corrected as follows

$$\begin{aligned} \frac{s}{N}\mathscr {M}(\Vert u_n\Vert ^{N/s}) \rightarrow c_*<\frac{s}{N}\mathscr {M} \left( \left( \frac{\alpha _{N,s}}{\alpha _0}\right) ^{(N-s)/s}\right) . \end{aligned}$$

• On page 1 line 7, in Abstract, $\exp (\alpha t^2)$ should be replaced with $\exp (\alpha |t|^{N/(N-s)})$.

• On page 6 line 5, the correct definition of $\lambda ^*$ in $(f_3)$ is

  $$\begin{aligned} \lambda ^*:=\inf _{u\in W_0^{s,N/s}(\Omega )\setminus \{0\}} \frac{\Vert u\Vert ^{\theta N /s}}{\Vert u\Vert _{L^{\theta N/s}(\Omega )}^{\theta N/s}}>0. \end{aligned}$$

• On page 6 line 12, in assumption $(f_6)$, $\frac{\alpha _{N,s}}{\alpha _0}$ should be replaced with $(\frac{\alpha _{N,s}}{\alpha _0})^{(N-s)/s}$.

• On page 18 lines 7 and 8, $\Vert u_n\Vert ^{N/s}$ should be replaced by $\Vert u_n\Vert ^{N/(N-s)}$.

• On page 18 line-11, the estimate of $\left| \int _{\Omega }f(x,u_n)u_ndx\right|$ should be corrected as follows

  $$\begin{aligned} \left| \int _\Omega f(x,u_n)u_ndx\right|\le & {} C\left( \int _\Omega |u_n|^{\theta N/s}dx +\int _\Omega |u_n|\exp (\alpha |u_n|^{N/(N-s)})dx\right) \\\le & {} C\left( \Vert u_n\Vert _{L^{\frac{\theta N}{s}}(\Omega )}^{\theta N/s} +\Vert u_n\Vert _{L^{\frac{q}{q-1}}(\Omega )}\right. \\&\left( \int _{\Omega }\exp [q\alpha \Vert u_n\Vert ^{N/(N-s)} (u_n/\Vert u_n\Vert )^{N/(N-s)}]dx\right) ^{\frac{1}{q}}\\\le & {} C\left( \Vert u_n\Vert _{L^{\frac{\theta N}{s}}(\Omega )}^{\theta N/s} +\Vert u_n\Vert _{L^{\frac{q}{q-1}}(\Omega )}\right) \rightarrow 0. \end{aligned}$$

• On page 20 line 5, replace $v_0, u_0$ by v, u, respectively.

• On page 20 line 7, estimate (4.8) should be replaced by

  $$\begin{aligned} \sup _{n\in \mathbb {N}}\int _\Omega \exp (\alpha ^{\prime } v_n^{N/(N-s)}) dx<\infty ,\ \ \forall \ \alpha ^{\prime }<\frac{\alpha _{N,s}}{(1-\Vert v\Vert ^{N/s})^{s/(N-s)}}. \end{aligned}$$

• On page 20 line-8, the inequality should be corrected as follows

  $$\begin{aligned} \xi ^{N/s}<\frac{(\alpha _{N,s}/\alpha _0)^{(N-s)/s}}{1-\Vert v\Vert ^{N/s}}. \end{aligned}$$

• On page 20 line-6, the inequality should be corrected as follows

  $$\begin{aligned} \alpha _0\Vert u_n\Vert ^{N/(N-s)}<\alpha ^{\prime \prime } <\frac{\alpha _{N,s}}{(1-\Vert v\Vert ^{N/s})^{s/(N-s)}}. \end{aligned}$$

• On page 20 line-4, the inequality should be corrected as follows

  $$\begin{aligned} \nu \alpha \Vert u_n\Vert ^{N/(N-s)}\le \alpha ^{\prime \prime } <\frac{\alpha _{N,s}}{(1-\Vert v\Vert ^{N/s})^{s/(N-s)}}. \end{aligned}$$

• On page 21 line 3, replace $\Vert u_n\Vert _{L^{\frac{N\theta }{s}}(\Omega )}^{N\theta /s}$ by $\Vert u_n-u\Vert _{L^{\frac{N\theta }{s}}(\Omega )}^{N\theta /s}$.

• On page 21 line 11, replace $\Vert v\Vert$ by $\Vert v\Vert ^{p-1}$.

• On page 21 line-6, replace $\varepsilon$ by $\varphi$.

• On page 22 line-11, in the estimate of $\mathcal {I}_\lambda (u)$, there is missed the factor $\frac{s}{N}$ before $\frac{\mathscr {M}(t_*)}{t_*^{\theta }}$. Similarly, in the definition of g(t), there is missed the factor $\frac{s}{N}$ before $\frac{\mathscr {M}(t_*)}{t_*^{\theta }}$.

• On page 22 line-1, $t_{\max }$ should be corrected as follows

  $$\begin{aligned} t_{\mathrm{max}}=\left( \frac{\mathscr {M}(t_*)\theta }{C_{N,s} t_*^\theta \lambda }\right) ^{\frac{s}{sq-N\theta }}>0, \end{aligned}$$

  and $\Lambda ^*$ should be replaced by

  $$\begin{aligned} \Lambda ^*=\frac{\mathscr {M}(t_*)\theta }{C_{N,s}t_*^{\theta }\widetilde{\rho }_1^{q-\frac{N\theta }{s}}}. \end{aligned}$$

• On page 25 line 12, $\widetilde{\rho }_\lambda$ should be corrected as follows

  $$\begin{aligned} \widetilde{\rho }_\lambda :=\left( \frac{\mathscr {M}(t_*)\theta }{C_{N,s}t_*^\theta \lambda }\right) ^{\frac{s}{sq-N\theta }}. \end{aligned}$$