All results in the paper [Reference Fiscella and Vitillaro1] are stated under the assumption that $\Omega$ is a bounded open subset of $\mathbb R^n$ ($n\geqslant 1$) with $C^{1,1}$ boundary $\Gamma=\Gamma_0\cup\Gamma_1$, where $\Gamma_0$ and $\Gamma_1$ are disjoint subsets of $\Gamma$. The manifold $\Gamma$ is equipped with the natural Lebesgue measure $\sigma$ on the manifold, $\Gamma_0$ and $\Gamma_1$ are assumed to be measurable with respect to $\sigma$ and, finally, $\sigma(\Gamma_0)$ is assumed to be positive. The authors recently pointed out that $\Omega$ must be connected for most of these results to make sense.

The paper [Reference Fiscella and Vitillaro1] was aimed to relax the assumption $m \lt \frac {2(n+1)p-4(n-1)}{n(p-2)+4}$, which appears in [Reference Vitillaro3, Theorem 7], to the assumption $m \lt 1+p/2$, which appears in [Reference Fiscella and Vitillaro1, Theorems 1.1 and 3.2]. But in [Reference Vitillaro3] the open set $\Omega$ was assumed to be connected, while in [Reference Fiscella and Vitillaro1] this assumption is missing. Unfortunately, as we are going to show, the variational setting under which [Reference Fiscella and Vitillaro1, Theorems 1.1 and 3.2] are stated may fail to hold when $\Omega$ is disconnected, depending on the location of $\Gamma_0$.

We recall that, at p. 761 of [Reference Fiscella and Vitillaro1], we introduced the Hilbert space

\begin{equation*}H^1_{\Gamma_0}(\Omega)=\left\{u\in H^1(\Omega): u_{|\Gamma_0}=0\right\},\end{equation*}

where $u_{|\Gamma_0}$ is intended in the trace sense. Moreover, by recalling the setting of [Reference Vitillaro3], the space $H^1_{\Gamma_0}(\Omega)$ was equipped with the norm $\|\nabla (\cdot)\|_2$, which was asserted to be equivalent to the standard norm of $H^1(\Omega)$, that is to the norm defined by

\begin{equation*}\|u\|_{H^1(\Omega)}=\left(\|u\|_2^2+\|\nabla u\|_2^2\right)^{1/2}\quad\text{for all}~u\in H^1(\Omega).\end{equation*}

Now the authors realized that this setting is correct only when $\Omega$ is connected, i.e. in the original setting of [Reference Vitillaro3], since when $\Omega$ is disconnected the bilinear form $(\cdot,\cdot)$ defined on $H^1_{\Gamma_0}(\Omega)$ by

\begin{equation*}(u,v)=\int_\Omega \nabla u\nabla v\quad\text{for all}~u,\,v\in H^1_{\Gamma_0}(\Omega),\end{equation*}

may even fail to be positive definite.

Indeed, let us consider the particular case when $\Omega=\Omega_1\cup\Omega_2$, $\Omega_1$ and $\Omega_2$ being open and nonempty, $\overline{\Omega_1}\cap\overline{\Omega_2}=\emptyset$, and $\Gamma_0= \partial\Omega_2$, $\Gamma_1= \partial\Omega_1$. Let us set the characteristic function $\chi_{\Omega_1}$, defined on $\Omega$, that is

\begin{equation*}\chi_{\Omega_1}(x)= \begin{cases} 0\quad\text{when}~x\in\Omega_2,\\ 1\quad\text{when}~x\in\Omega_1. \end{cases}\end{equation*}

Its equivalence class with respect to a.e. equivalence, still denoted (as usual) as $\chi_{\Omega_1}$, belongs to $H^1(\Omega)$, and one has ${\chi_{\Omega_1}}_{|\Gamma_0}\equiv 0$, while ${\chi_{\Omega_1}}_{|\Gamma_1}\equiv 1$, and hence $\chi_{\Omega_1}\in H^1_{\Gamma_0}(\Omega).$ Since $\nabla \chi_{\Omega_1}\equiv 0$ in $\Omega$, we then have $(\chi_{\Omega_1},\chi_{\Omega_1})=0$, while $\chi_{\Omega_1}\not\equiv 0$.

On the other hand, when $\Omega$ is connected, the setting is correct. Indeed, by the Area Formula, see [Reference Leoni2, Chapter 9, Theorem 9.27, p. 253], $\sigma(\Gamma_0)=\mathcal{H}^{N-1}(\Gamma_0)$, where $\mathcal{H}^{N-1}$ denotes the Hausdorff measure. Hence, by [Reference Ziemer4, Chapter 2, Theorem 2.6.16, p. 75], one derives that the capacity $B_{1,2}(\Gamma_0)$ is positive. One then applies [Reference Ziemer4, Chapter 4, Corollary 4.5.2, p. 195] to conclude that the following Poincaré-type inequality holds: there is a positive constant $c_1=c_1(\Omega,\Gamma_0)$ such that

\begin{equation*} \|u\|_2\le c_1 \|\nabla u\|_2\end{equation*}

for all $u\in H^1(\Omega)$ such that $u_{|\Gamma_0}=0$. Unfortunately, the authors have not noticed, before the publication of [Reference Fiscella and Vitillaro1], that [Reference Ziemer4, Chapter 4, Corollary 4.5.2, p. 195] is obtained as a consequence of [Reference Ziemer4, Chapter 4, Theorem 4.5.1, p. 195], where $\Omega$ is assumed to be connected, so this proof is based on the connectedness assumption.

Most of the results of [Reference Fiscella and Vitillaro1] can be misleading when $\Omega$ is disconnected. For example, in the particular case $\Omega=\Omega_1\cup\Omega_2$ considered above, the potential well depth $d$ defined in [Reference Fiscella and Vitillaro1, (1.4)] vanishes. Indeed, we recall that

(0.1)\begin{equation} d=\inf_{u\in H^1_{\Gamma_0}(\Omega)\setminus\{0\}}\sup_{\lambda \gt 0} J(\lambda u), \end{equation}

where the functional $J$ is defined by

\begin{equation*} J(u)=\tfrac 12 \|\nabla u\|_2^2 -\tfrac 1p \|u\|^p_p\quad\text{for all}~u\in H^1_{\Gamma_0}(\Omega), \end{equation*}

vanishes. An easy calculation shows, as in the proof of [Reference Fiscella and Vitillaro1, Lemma 4.1] that, for any $u\in H^1_{\Gamma_0}(\Omega)\setminus\{0\}$, one has

\begin{equation*}\sup_{\lambda \gt 0}J(\lambda u)=\max_{\lambda \gt 0}J(\lambda u)=\left(\frac 12 -\frac 1p\right)\left(\frac {\|\nabla u\|_2}{\|u\|_p}\right)^{\frac {2p}{p-2}}\geqslant 0.\end{equation*}

Since we have $\sup_{\lambda \gt 0}J(\lambda \chi_{\Omega_1})=0$, we get that the infimum in (0.1) is attained, as a minimum, when $u=\chi_{\Omega_1}$, so $d=0$. Consequently, the so-called ‘bad part of the potential well’ defined in [Reference Fiscella and Vitillaro1, (1.5)] does not exist in this case, and [Reference Fiscella and Vitillaro1, Theorem 1.1] is wrong. In the particular case considered above, also [Reference Fiscella and Vitillaro1, Theorem 3.2] cannot be stated. Indeed, the positive constant $K_0$ defined in formula [Reference Fiscella and Vitillaro1, (2.15)], that is

\begin{equation*} K_0=\sup_{u\in H^1_{\Gamma_0}(\Omega)\setminus\{0\}} \frac {\int_\Omega F(\cdot,u)}{\|\nabla u\|_2^p},\end{equation*}

is clearly ill-defined, since $\chi_{\Omega_1}\not\equiv 0$, so we can take it in the supremum, but $\nabla \chi_{\Omega_1}\equiv 0$, so the fraction does not make sense.

Finally, also the local existence of weak solutions of problem [Reference Fiscella and Vitillaro1, (1.1)], which is asserted in [Reference Fiscella and Vitillaro1, Theorem 2.6], cannot be taken for granted when $\Omega$ is disconnected. Indeed [Reference Fiscella and Vitillaro1, Remark 2.7], when it asserts that in [Reference Vitillaro3] one can consider also disconnected open sets, is wrong. Indeed, when $\|\nabla (\cdot)\|_2$ is not an equivalent norm in the space $H^1_{\Gamma_0}(\Omega)$, as in the case $\Omega=\Omega_1\cup\Omega_2$ considered above, the functional $I$ defined in the last line of [Reference Vitillaro3, p. 383] is not coercive, so there is substantial gap in the proof of [Reference Fiscella and Vitillaro1, Theorem 2.6]. The same phenomenon occurs when dealing with [Reference Fiscella and Vitillaro1, Theorem 2.8].

In conclusion, the assumption that ‘$\Omega$ is a bounded open subset of $\mathbb R^n$’ must be replaced in paper [Reference Fiscella and Vitillaro1] by ‘$\Omega$ is a bounded domain (that is, a connected open subset) of $\mathbb R^n$’. This assumption appears in three instances, in the Abstract and twice at the beginning of Section 1. Under this new assumption, all the results in paper [Reference Fiscella and Vitillaro1] hold. As correctly pointed out in [Reference Fiscella and Vitillaro1, Remark 2.7], the case in which $\Omega$ is disconnected is not of particular interest.