---
abstract: 'This note is devoted to exploring Hölder’s quasicontinuity for the Riesz-Morrey potentials, and its application to the corresponding nature of some nonnegative weak solutions of the quasilinear Lane-Emden equations for the $p$-Laplacian.'
address:
- 'Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027'
- 'Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7, Canada'
author:
- 'David R. Adams'
- Jie Xiao
title: |
    The Hölder Quasicontinuity for\
    Riesz-Morrey Potentials and Lane-Emden Equations
---

[^1]

Introduction {#s1}
============

Let $(\alpha,p,\lambda)\in (0,n)\times [1,\infty)\times (0,n]$ and $\Omega$ be a bounded domain in the $2\le n$-dimensional Euclidean space $\mathbb R^n$. The main ideas in [@AX04; @AX11b; @AX11a; @AX11c; @AX12a; @Ser] suggest us to deal with two basic concepts in the theory of Morrey spaces and their potentials.

The first one is the so-called Riesz-Morrey potential – the $\alpha$-order Riesz singular integral $$I_\alpha f(x)=\int_{\mathbb R^n} f(y)|y-x|^{\alpha-n}\,dy=\int_{\Omega} f(y)|y-x|^{\alpha-n}\,dy$$ of $f$ (whose value on $\mathbb R^n\setminus\Omega$ is defined to be $0$) in the Morrey space $$L^{p,\lambda}(\Omega)=\left\{g\in L^p(\Omega):\ \|g\|_{L^{p,\lambda}(\Omega)}=\sup_{x\in\Omega,\ 0<r<\hbox{diam}(\Omega)}
\Big(r^{\lambda-n}\int_{B(x,r)\cap\Omega}|g|^p\Big)^\frac1p<\infty\right\},$$ where $\hbox{diam}(\Omega)$ is the diameter of $\Omega$, $B(x,r)$ is the open ball with center $x$ and radius $r$, and the integral is taken with respect to the $n$-dimensional Lebesgue measure $dy$.

The second one is the Riesz-Morrey capacity of a set $E\subseteq\Omega$: $$C_\alpha(E;L^{p,\lambda}(\Omega))=\inf_{E\subseteq\ open\ O\subseteq\Omega}
C_\alpha(O;L^{p,\lambda}(\Omega))=\inf_{E\subseteq\ open\ O\subseteq\Omega}\sup_{compact\ K\subseteq\ open\ O}C_\alpha(K;L^{p,\lambda}(\Omega)),$$ where $$C_\alpha(K;L^{p,\lambda}(\Omega))=\inf\Big\{\|h\|_{L^{p,\lambda}(\Omega)}^p:\
0\le h\in L^{p,\lambda}(\Omega)\ \ \&\ \ I_\alpha h\ge 1_K\Big\}$$ for which $1_K$ stands for the characteristic function of the compact $K\subseteq O$.

In this note, through using the Riesz-Morrey capacity we study the quasicontinuous representative and the Hölder quasicontinuity of each Riesz-Morrey potential – see Theorems \[t31\] & \[t33\]. Certainly, the discovered properties show their worth in connection with investigating Hölderian quasicontinuity of some nonnegative weak solutions $u$ of the quasilinear Lane-Emden equations for $p$-Laplacian: $$-\Delta_p u=-\hbox{div}(|\nabla u|^{p-2}\nabla u)=u^{q+1}\quad\hbox{or}\quad e^u
\quad\hbox{in}\quad \Omega,$$ where $(p,q)\in (1,n)\times (0,\infty)$ – see Theorems \[p42\] & \[p43\].

[*Notation*]{}. In what follows, $\Omega$ is always assumed to be a bounded domain in $\mathbb R^n$. For $E\subseteq\Omega$ define $\fint_E$ to be the integral mean over $E$ with respect to the Lebesgue measure $dy$. And, $\mathsf{X}\lesssim\mathsf{Y}$ stands for $\mathsf{X}\le c\mathsf{Y}$ for a constant $c>0$. Moreover, $\mathsf{X}\approx\mathsf{Y}$ means both $\mathsf{X}\lesssim\mathsf{Y}$ and $\mathsf{Y}\lesssim \mathsf{X}$.

Riesz-Morrey potentials {#s3}
=======================

Quasicontinuous representation for $I_\alpha L^{p,\lambda}$ {#s31}
-----------------------------------------------------------

A function $g$ on $\Omega$ is said to be $C_\alpha(\cdot;L^{p,\lambda})$-quasicontinuous provided that for any $\epsilon>0$ there is a continuous function $\tilde{g}$ on $\Omega$ such that $$C_\alpha\big(\{x\in\Omega: \tilde{g}(x)\not=g(x)\};L^{p,\lambda}(\Omega)\big)<\epsilon.$$ Naturally, $\tilde{g}$ is called a $C_\alpha(\cdot;L^{p,\lambda}(\Omega))$-quasicontinuous representative of $g$. The forthcoming Theorem \[t31\] is an extension of [@AH Theorem 6.2.1] from $L^p$ to $L^{p,\lambda}$.

\[l31\] For $1<p<\infty$ and $0<\gamma\le n$, let $L^{p,\gamma}_0(\Omega)$ be the Zorko space (cf. [@Z1986]) of all $f\in L^{p,\gamma}(\Omega)$ that can be approximated by $C^1$ functions with compact support in $\Omega$ under the norm $\|\cdot\|_{L^{p,\gamma}(\Omega)}$. Then $L^{p,\lambda}(\Omega)\subseteq L^{p,\gamma}_0(\Omega)\ \ \forall\ \ \lambda\in (0,\gamma)$.

If $\gamma=n$, then $L^{p,\gamma}_0(\Omega)=L^p(\Omega)$, and hence $$\int_{\Omega\cap B(x,r)}|f|^p\le\|f\|_{L^{p,\lambda}(\Omega)}^p \big(\hbox{diam}(\Omega)\big)^{n-\lambda}\quad\forall\quad (\lambda,x,r)\in (0,n)\times\Omega\times \big(0,\hbox{diam}(\Omega)\big),$$ thereby deriving the desired inclusion.

If $\gamma<n$, then the desired inclusion follows from [@AX12a Lemma 3.4].

\[t31\] Let $g=I_\alpha f$, $f\in L^{p,\lambda}(\Omega)$, and $1<p< \lambda/\alpha<\mu/\alpha\le n/\alpha$. Then there is a set $\Sigma\subset\Omega$ such that:

[(i)]{} $C_\alpha(\Sigma;L^{p,\mu}(\Omega))=0$;

[(ii)]{} $
\lim_{r\to 0}\fint_{B(x,r)}g=\tilde{g}(x)\quad\forall\quad x\in \Omega\setminus\Sigma;
$

[(iii)]{} $\lim_{r\to 0}\fint_{B(x,r)}\big|g-\tilde{g}(x)\big|=0\quad\forall\quad x\in \Omega\setminus\Sigma.$

Moreover, one has:

[(iv)]{} For any $\epsilon>0$ there is an open set $O\subset\Omega$ such that $C_\alpha(O; L^{p,\mu}(\Omega))<\epsilon$ and the convergence in (ii)-(iii) is uniform on $\Omega\setminus O$;

[(v)]{} $\tilde{g}$ is a $C_\alpha(\cdot;L^{p,\mu}(\Omega))$-quasicontinuous representative of $g$;

[(vi)]{} $
\tilde{g}(x)=g(x)\quad\forall\quad x\in\Omega\setminus O.
$

Given $r\in (0,\infty)$, let $$\chi(x)={1_{\mathbb B^n}(x)}{\omega_n}^{-1}\quad\&\quad\chi_r(x)=r^{-n}\chi(x/r),$$ where $\omega_n$ is the volume of the unit ball $\mathbb B^n$ of $\mathbb R^n$. For $f\in L^{p,\lambda}(\Omega)$, $\epsilon>0$ and $\mu\in (\lambda,n]$, we use Lemma \[l31\] to find a Schwartz function $f_0$ on $\mathbb R^n$ such that $f_0=0$ in $\mathbb R^n\setminus\Omega$ and $\|f-f_0\|_{L^{p,\mu}(\Omega)}<\epsilon$. Consequently, $g_0=I_\alpha f_0$ is a Schwartz function and $\chi_r\ast g_0$ converges to $g_0$ on $\Omega$ as $r\to 0$. Note that $$\fint_{B(x,r)}g=\chi_r\ast g(x)\quad\&\quad \fint_{B(x,r)}g_0=\chi_r\ast g_0(x).$$ Thus, for $\delta>0$ letting $$\mathsf{J}_\delta g(x)=\sup_{0<r<\delta}(\chi_r\ast g)(x)-\inf_{0<r<\delta}(\chi_r\ast g)(x),$$ we have $$\mathsf{J}_\delta g(x)\le \mathsf{J}_\delta (g-g_0)(x)+\mathsf{J}_\delta g_0(x).$$ By the previously-stated convergence, for any given $\epsilon>0$ there exists $\delta>0$ so small that $\sup_{x\in\Omega}\mathsf{J}_\delta g_0(x)<\epsilon$. If $\mathcal{M}$ stands for the Hardy-Littlewood maximal operator, then $$|\chi_r\ast(g-g_0)(x)|\le \mathcal{M}(g-g_0)(x)\quad\forall\quad x\in\Omega,$$ and hence $$\mathsf{J}_\delta g(x)\le \mathcal{M}(g-g_0)(x)+\epsilon\quad\forall\quad x\in\Omega.$$ Upon choosing $\omega/2>\epsilon>0$, the last estimate gives $$E_\omega:=\{x\in\Omega: \mathsf{J}_\delta g(x)>\omega\}\subseteq\{x\in\Omega: \mathsf{J}_\delta (g-g_0)(x)>\omega/2\}=:F_\omega.$$ In view of the definition of $C_\alpha(\cdot;L^{p,\mu}(\Omega))$ and the boundedness of $\mathcal{M}$ acting on $L^{p,\mu}(\Omega)$, we find $$C_\alpha(E_\omega;L^{p,\mu}(\Omega))\le C_\alpha(F_\omega;L^{p,\mu}(\Omega))\lesssim \omega^{-p}\|f-f_0\|^p_{L^{p,\mu}(\Omega)}\lesssim({\epsilon}{\omega}^{-1})^p.$$ For each natural number $j$ let $\omega=2^{-j}$, $\epsilon=4^{-j}$, and $\delta_j$ be their induced number. If $$G_j=\{x\in\Omega: \mathsf{J}_{\delta_j}g(x)>2^{-j}\},$$ then $$C_\alpha(G_j; L^{p,\mu}(\Omega))\lesssim 2^{-jp}.$$ Furthermore, $$O_k=\cup_{j=k}^\infty G_j\Rightarrow
C_\alpha(O_k; L^{p,\mu}(\Omega))\lesssim\sum_{j=k}^\infty 2^{-jp}\to 0\quad\hbox{as}\quad k\to\infty.$$ Consequently, under $$\begin{cases}
1<p<\mu/\alpha;\\ 
\mu-\alpha p<d\le n;\\
0<q<dp/(\mu-\alpha p),
\end{cases}$$ one has $$C_\alpha(\cap_{k=1}^\infty O_k; L^{p,\mu}(\Omega))=0.$$ Note that $$x\notin O_k\Rightarrow\mathsf{J}_\delta g(x)\le 2^{-j}\quad\forall\quad\delta\le\delta_j\quad\&\quad j\ge k.$$ So, $$\lim_{r\to 0}\chi_r\ast g(x)=\tilde{g}(x)\ \ \hbox{exists\ for\ any}\ \ x\notin\cap_{k=1}^\infty O_k.$$ Clearly, this convergence is uniform on $\Omega\setminus O_k$ with sufficiently small $C_\alpha(O_k; L^{p,\mu}(\Omega))$. This proves the results of Theorem \[t31\] with $\Sigma=\cap_{k=1}^\infty O_k$ except the part (iii).

However, the demonstration of the part (iii) follows from a slight modification of the above argument plus defining $$\mathsf{J}_\delta(g-\tilde{g})(x)=\sup_{0<r\le\delta}(\chi_r\ast|g-\tilde{g}(x)|)(x)$$ and so establishing $$\mathsf{J}_\delta(g-\tilde{g})(x)\le \mathcal{M}(g-g_0)(x)+|(\tilde{g}-g_0)(x)|+\epsilon$$ under $$\mathsf{J}_\delta(g-{g}_0(x))(x)<\epsilon.$$

Hölderian quasicontinuity for $I_\alpha L^{p,\lambda}$ {#s32}
------------------------------------------------------

Given $\beta\in (0,1]$. We say that $g\in Lip_\beta(\Omega)$ provided that $g$ satisfies $$\sup\left\{\frac{|g(x)-g(y)|}{|x-y|^\beta}:\quad x,y\in\Omega,\ x\not=y\right\}<\infty.$$ In particular, if $\beta\in (0,1)$ or $\beta=1$ then $g$ is called $\beta$-Hölder continuous or Lipschitz continuous. Moreover, a function $g$ defined on $\Omega$ is called Hölder quasicontinuous if for any $\epsilon>0$ there is a set $E\subset\Omega$ of a given capacity smaller than $\epsilon$ such that $g$ is of the Hölder continuity on $\Omega\setminus E$. The forthcoming Theorem \[t33\] shows that any function in $I_\alpha L^{p,\lambda}$ is of Hölder quasicontinuity. To be more precise, let us recall the Sobolev-Morrey type imbedding (cf. [@ADuke; @ALecture]): $$I_\alpha: L^{p,\lambda}(\Omega)\mapsto\begin{cases}
                              L^{\frac{\lambda p}{\lambda-\alpha
p},\lambda}(\Omega)\cap L^{p,\lambda-\alpha p}(\Omega),  &   1<p<\lambda/\alpha; \\
                              BMO(\Omega), &   1<p=\lambda/\alpha, \\
                          \end{cases}$$ where $$f\in BMO(\Omega)\Longleftrightarrow\sup_{x\in\Omega,\ 0<r<\hbox{diam}(\Omega)}\fint_{B(x,r)\cap\Omega}\left|f-\fint_{B(x,r)\cap\Omega}f\right|<\infty.$$ Interestingly, the above imbedding can be extended from $p\le\lambda/\alpha$ to $p>\lambda/\alpha$.

\[l32\] Let $g=I_\alpha f$, $f\in L^{p,\lambda}(\Omega)$, and $(\alpha,p,\lambda)\in (0,n)\times(1,\infty)\times(0,n]$.

[(i)]{} If $0<\delta=\alpha-\lambda/p<1$, then $g\in Lip_\delta(\Omega)$.

[(ii)]{} If $$\begin{cases}
1<p<\lambda/\alpha;\\
1<q<\min\{p,\lambda/\alpha\};\\
\mu=n-(n-\lambda)q/p;\\
0<\beta<\min\left\{1,\alpha(1-q/p),{\lambda(1-q/p)}/\big({\lambda+(1-\alpha)q}\big)\right\},
\end{cases}$$ then for any $r\in (0,1)$ there exist $f_r\in L^{p,\lambda}(\Omega)$ and $g_r=I_\alpha f_r$ such that $$\begin{cases}
\|f-f_r\|_{L^{q,\mu}(\Omega)}\lesssim r^{\beta};\\
|g_r(x)-g_r(y)|\lesssim |x-y|^\beta\quad\forall\quad y\in B(x,r)\subseteq\Omega.
\end{cases}$$

\(i) Since $\alpha=\delta+\lambda/p$, an application of [@ADuke Corollary (iii)] and [@Carl page 91] gives $$I_\alpha L^{p,\lambda}(\Omega)= I_\delta I_{\lambda/p} L^{p,\lambda}(\Omega)\subseteq I_\delta BMO(\Omega)\subseteq Lip_\delta(\Omega),$$ whence implying $g\in Lip_\delta(\Omega)$.

\(ii) Without loss of generality, we may assume $\|f\|_{L^{p,\lambda}(\Omega)}\le 1$ and $f|_{\mathbb R^n\setminus\Omega}=0$. Since $\Omega$ is bounded, there is a big ball $B(x,r)$ with center $x\in\Omega$ and radius $r\le\hbox{diam}(\Omega)$ such that $\Omega\subseteq B(x,r)$, and consequently, $$\|f\|_{L^p(\Omega)}^p=\int_\Omega|f|^p\le\big(\hbox{diam}(\Omega)\big)^{n-\lambda}.$$ For $r\in (0,1)$ let $O_r=\{x\in\Omega: |f(x)|>s_r\}$, $s_r=r^{\beta q/(q-p)}$, and $$f_r=\begin{cases} f\quad\hbox{on}\quad \Omega\setminus O_r;\\
0\quad\hbox{on}\quad O_r.
\end{cases}$$ Evidently, $$\int_{O_r}1_{O_r}\le s_r^{-p}\big(\hbox{diam}(\Omega)\big)^{n-\lambda}$$ and $g_r=I_\alpha f_r$ is bounded. Moreover, by Hölder’s inequality and the definition of $\|\cdot\|_{L^{q,\mu}(\Omega)}$, one gets $$\begin{aligned}
\|f-f_r\|_{L^{q,\mu}(\Omega)}^q&\le&\|f\|_{L^{p,\lambda}(\Omega)}^{q}\left(\int_{O_r}1_{O_r}\right)^\frac{\mu-\lambda}{n-\lambda}\\
&\le&{(\hbox{diam}(\Omega)\big)^{\mu-\lambda}}{s_r^{\frac{p(\lambda-\mu)}{n-\lambda}}}\\
&\le& (\hbox{diam}(\Omega)\big)^{(n-\lambda)(1-q/p)} r^{q\beta}.\end{aligned}$$ Meanwhile, thanks to $f_r\le s_r$, we can use (i) above to get that if $$p<\bar{p}=\frac{\lambda(p-q)-\beta pq}{\alpha(p-q)-\beta p}\quad\&\quad 0<\bar{\beta}=\alpha-\lambda/\bar{p}<1,$$ then $$|g_r(x)-g_r(y)|=|I_\alpha f_r(x)-I_\alpha f_r(y)|\lesssim\|f_r\|_{L^{\bar{p},\lambda}(\Omega)}|x-y|^{\bar{\beta}}\quad\forall\quad y\in B(x,r).$$ Another application of the Hölder inequality gives $$\|f_r\|^{\bar{p}}_{L^{\bar{p},\lambda}(\Omega)}\le s_r^{\bar{p}-p}\|f\|^p_{L^{p,\lambda}(\Omega)}\le s_{r}^{\bar{p}-p}.$$ Thus, $|g_r(x)-g_r(y)|\lesssim r^\beta$ holds for any $y\in B(x,r)$.

Below is the Hölder quasicontinuity for the Riesz-Morrey potentials which actually gives a nontrivial generalization of [@Mal Theorem 7] (see [@HaK] for a further development of [@Mal]).

\[t33\] Let $g=I_\alpha f$, $f\in L^{p,\lambda}(\Omega)$, and $1<p<\lambda/\alpha\le n/\alpha$. If $$\begin{cases}
 1<q<\min\{p,\lambda/\alpha\}=p;\\
 \mu=n-(n-\lambda)q/p;\\
0<\gamma<\min\left\{1,\alpha(1-q/p),{\lambda(1-q/p)}/\big({\lambda+(1-\alpha)q}\big)\right\},
\end{cases}$$ then for any $\epsilon>0$ there exists an open set $O\subseteq\Omega$ and a $\gamma$-Hölder continuous function $h$ on $\Omega$ such that $$C_\alpha(O;L^{q,\mu}(\Omega))<\epsilon\ \ \&\ \ 
g=h\quad\hbox{in}\quad\Omega\setminus O.$$

The notations introduced in Lemma \[l32\] and its proof will be used in the sequel. Given $\gamma\in (0,\beta)$ with $\beta$ as in Lemma \[l32\]. Now, for each natural number $j$ let $r_j$ be chosen so that $$\label{qo}
r_0=1\quad\&\quad \Big({r_{j+1}}/{r_j}\Big)^\gamma\le 1/2.$$ For simplicity, set $h_j=g_{r_j}$ and then $f_j$ be the corresponding $f_{r_j}$ and $$\sum_{j=1}^\infty\|f_{j+1}-f_{j}\|_{L^{p,\lambda}(\Omega)}<\infty.$$ Choosing $$\begin{cases}
w_j=\max\big\{-r_j^\gamma,\min\{r_j^\gamma,h_{j+1}-h_j\}\big\};\\
O_j=\{x\in\Omega: |h_{j+1}(x)-h_j(x)|>r_j^\gamma\},
\end{cases}$$ we use the already-established estimate $$\|f-f_r\|_{L^{q,\mu}(\Omega)}\le(\hbox{diam}(\Omega)\big)^{(n-\lambda)(1/q-1/p)} r^{\beta}$$ and the definition of $C_\alpha(\cdot; L^{q,\mu}(\Omega))$ to obtain $$C_\alpha(O_j; L^{q,\mu}(\Omega))\le r_j^{-\gamma q}\|f_{j+1}-f_j\|_{L^{q,\mu}(\Omega)}^q\lesssim r_j^{(\beta-\gamma) q},$$ Consequently, for any $\epsilon>0$ there is a big integer $J$ such that $$\sum_{j=J}^\infty C_\alpha(O_j; L^{q,\mu}(\Omega))\lesssim \sum_{j=J}^\infty r_j^{q(\beta-\gamma)}<\epsilon.$$ Putting $$O=\cup_{j=J}^\infty E_j\ \ \&\ \ h=h_{J}+\sum_{j=J}^\infty w_j,$$ we find that $O$ is an open subset of $\Omega$ and $$C_\alpha(O; L^{q,\mu}(\Omega))<\epsilon\quad\&\quad h=g\quad\hbox{on}\quad \Omega\setminus O.$$ It remains to check that $h$ is $\beta$-Hölder continuous on $\Omega$. Of course, it is enough to verify $$\label{short3}
|h(x)-h(y)|\lesssim |x-y|^\beta\quad\forall\quad x,y\in\Omega\quad\hbox{with}\quad |x-y|\le r_{J}.$$ Obviously, $h_J$ is $\beta$-Hölder continuous. To show the similar property for $\sum_{j=J}^\infty w_j$, we may assume $$x,y\in\Omega;\quad 0<|x-y|\le r_J;\quad r_{k+1}<|x-y|\le r_k.$$ From (\[qo\]) it follows that $$\label{short4}
k\le\Big(\frac{\gamma}{\ln 2}\Big)\ln\frac1{r_k}\le\Big(\frac{\gamma}{(\beta-\gamma)\ln 2}\Big)r_k^{\gamma-\beta}\le\Big(\frac{\gamma}{(\beta-\gamma)\ln 2}\Big)|x-y|^{\gamma-\beta}$$ When $1\le j\le k$, an application of the last estimate in Lemma \[l32\] gives $$|w_j(x)-w_j(y)|\lesssim|x-y|^{\beta}.$$ When $j>k$, another application of (\[qo\]) yields $$|w_j(x)-w_j(y)|\le 2r_j^\gamma\le 2^{k-j+2}r_{k+1}^\gamma\le 2^{k-j+2}|x-y|^\gamma.$$ This, together with (\[short4\]) and $h=h_{J}+\sum_{j=J}^\infty w_j$, derives $$|h(x)-h(y)|\lesssim |x-y|^\gamma+k|x-y|^{\beta}\lesssim|x-y|^\gamma.$$

Lane-Emden Equations {#s2}
====================

Hölderian quasicontinuity for $-\Delta_p u=u^{q+1}$
---------------------------------------------------

Recall that for $1\le p<\infty$ the Sobolev space $W^{1,p}(\Omega)$ consists of all functions $f$ with $$\|f\|_{W^{1,p}(\Omega)}=\left(\int_{\Omega}|f|^p\right)^\frac1p+\left(\int_{\Omega}|\nabla f|^p\right)^\frac1p<\infty$$ and the Sobolev space $W^{1,p}_0(\Omega)$ is the completeness of $C^1_0(\Omega)$ (all $C^1$ functions $f$ with compact support in $\Omega$) under $\|\cdot\|_{W^{1,p}(\Omega)}$. According to [@GiTr Lemma 7.14], any $f\in W^{1,1}_0(\Omega)$ can be represented via $$\label{e41}
f(x)=\frac{\Gamma\big(\frac{n}{2}\big)}{2\pi^\frac{n}{2}}\int_{\Omega}|x-y|^{-n}(x-y)\cdot\nabla f(y)\,dy\ \ \&\ \ |f(x)|\le \left(\frac{\Gamma\big(\frac{n}{2}\big)}{2\pi^\frac{n}{2}}\right)\big(I_1|\nabla f|\big)(x)\quad\hbox{a.e.}\quad x\in\Omega,$$ where $\Gamma(\cdot)$ is the usual gamma function. As a variant of $C_1(K; L^{p,n}(\Omega))$ the variational $p$-capacity of a compact $K\subseteq\Omega$ is defined by $$C(K;W_0^{1,p}(\Omega))=\inf\left\{\int_{\Omega}|\nabla f|^p:\ f\in
W^{1,p}_0(\Omega)\ \&\  f\ge 1_K\right\}.$$ Clearly, this definition is extendable to an arbitrary set $E\subseteq\Omega$ through (cf. [@HKM p.27]) $$C(E;W_0^{1,p}(\Omega))=\inf_{E\subseteq\ open\ O\subseteq\Omega}
C(O;W_0^{1,p}(\Omega))=\inf_{E\subseteq\ open\ O\subseteq\Omega}\sup_{compact\ K\subseteq\ open\ O}C(K;W_0^{1,p}(\Omega)).$$ Importantly, such a capacity can used to establish the following relatively independent Sobolev embedding whose (v) is indeed a sort of motivation to investigate the quasilinear Lane-Emden equations.

\[p23\] Given $1<p<\min\{n,q\}$ and $0<r<q(1-p^{-1})$, let $\nu$ be a nonnegative Radon measure on $\Omega$. Then the following properties are mutually equivalent:

[(i)]{} $I_1$ is a continuous operator from $L^{p}(\Omega)$ into $L^q(\Omega,\nu)$;

[(ii)]{} $W^{1,p}_0(\Omega)$ continuously embeds into $L^q(\Omega,\nu)$.

[(iii)]{} Isocapacitary inequality $\nu(K)\lesssim {C}(K; W^{1,p}_0(\Omega))^\frac{q}{p}$ holds for all compact sets $K\subset\Omega$;

[(iv)]{} Isocapacitary inequality $\nu(B(x,r))\lesssim r^\frac{q(n-p)}{p}$ holds for all $B(x,r)\subseteq\Omega$;

[(v)]{} Faber-Krahn’s inequality $\nu(O)^{\frac{p}{q}-1}\lesssim\lambda_{p,\nu}(O)$ holds for all bounded open sets $O\subseteq\Omega$, where $$\lambda_{p,\nu}(O)=\inf\left\{\frac{\int_O|\nabla f|^p}{\int_O|f|^p\,d\nu}:\ f\in C_0^1(O)\ \&\ f\not\equiv 0\ \hbox{on}\ O\right\}.$$

(ii)$\Leftrightarrow$(iii)$\Leftrightarrow$(iv)$\Leftrightarrow$(i) is essentially known – see, for example, [@Maz0; @Maz0a] and [@AH Theorem 7.2.2].

So, it remains to prove (ii)$\Leftrightarrow$(v). If (ii) is valid, then the Hölder inequality yields that for any open set $O\subseteq\Omega$ and $f\in C_0^1(O)$, $$\int_O|f|^p\,d\nu\le\left(\int_{O}|f|^q\,d\nu\right)^\frac{p}{q}\nu(O)^{1-\frac{p}{q}}\lesssim\Big(\int_O|\nabla f|^p\Big)\nu(O)^{1-\frac{p}{q}}$$ holds, whence giving (v). For the converse, we use the argument methods in [@Cha pp. 159-161] and [@Coh] to proceed. Suppose (v) is true. Then for any $f\in W^{1,p}_0(\Omega)$ and any $t>0$, $$\begin{aligned}
\int_{\Omega}|f|^p\,d\nu&\le& \int_{\{y\in\Omega:\  |f(y)|>t\}}|f|^p\,d\nu+ t^{p-1}\int_{\{y\in\Omega:\  |f(y)|\le t\}}|f|\,d\nu\\
&\lesssim&\frac{\int_{\{y\in\Omega:\  |f(y)|>t\}}|\nabla f|^p}{\nu(\{y\in\Omega:\  |f(y)|>t\})^{\frac{p}{q}-1}}
+ t^{p-1}\int_{\{y\in\Omega:\  |f(y)|\le t\}}|f|\,d\nu\\
&\lesssim&\left(t^{-1}{\int_{\Omega}|f|\,d\nu}\right)^{1-\frac{p}{q}}\int_{\Omega}|\nabla f|^p+t^{p-1}\int_{\{y\in\Omega:\  |f(y)|\le t\}}|f|\,d\nu.\end{aligned}$$ Choosing $$t=\left(\frac{\int_{\Omega}|\nabla f|^p}{\big(\int_{\Omega}|f|\,d\nu\big)^\frac{p}{q}}\right)^\frac{q}{p(q-1)},$$ we get a constant $c>0$ such that $$\int_{\Omega}|f|^p\,d\nu\le 2c\left(\int_{\Omega}|\nabla f|^p\right)^\frac{q(p-1)}{p(q-1)}\left(\int_{\Omega}|f|\,d\nu\right)^\frac{q-p}{q-1}.$$ Replacing this $f$ by $$f_k=\min\big\{\max\{f-2^k, 0\},2^k\big\},\ k=0,\pm 1,\pm 2,...,$$ we have $$\begin{aligned}
\left(\int_{\Omega}f_k^p\,d\nu\right)^\frac{p(q-1)}{q(p-1)}\le (2c)^\frac{p(q-1)}{q(p-1)}\left(\int_{\Omega}|\nabla f_k|^p\right)\left(\int_{\Omega}f_k\,d\nu\right)^\frac{p(q-p)}{q(p-1)}.\end{aligned}$$ This implies $$\begin{aligned}
&&\left(2^{kp}\nu(\{y\in\Omega:\ f(y)\ge 2^{k+1}\})\right)^\frac{p(q-1)}{q(p-1)}\\
&&\le (2c)^\frac{p(q-1)}{q(p-1)}\left(\int_{\{y\in\Omega:\ \ 2^k\le f(y)<2^{k+1}\}}|\nabla f|^p\right)\left(2^{k}\nu(\{y\in\Omega:\ f(y)\ge 2^{k}\})\right)^\frac{p(q-p)}{q(p-1)}.\end{aligned}$$ Setting $$\left\{\begin{array} {r@{\quad}l}
a_k & =\ 2^{kq}\nu(\{y\in\Omega:\ f(y)\ge 2^{k}\});\\
b_k &=\ \int_{\{y\in\Omega:\ \ 2^k\le f(y)<2^{k+1}\}}|\nabla f|^p;\\
\theta &=\ \frac{q(p-1)}{p(q-1)},
\end{array}
\right.$$ one has $a_{k+1}\le 2^{1+q}c b_k^\theta a_k^{p(1-\theta)}$. This, together with Hölder’s inequality, derives $$\begin{aligned}
\sum_{k}a_k&\le&2^{1+q}c\sum_{k}b_k^\theta a_k^{p(1-\theta)}\\
&\le&2^{1+q}c\Big(\sum_{k}b_k\Big)^\theta\Big(\sum_{k} a_k\Big)^{p(1-\theta)}\\
&\le&2^{1+q}c\Big(\int_{\Omega}|\nabla f(y)|^p\,dy\Big)^\theta\Big(\sum_{k} a_k\Big)^{p(1-\theta)}.\end{aligned}$$ A simplification of these estimates yields (ii).

\[r22\] The part on Faber-Krahn’s inequality under $(p,q,d\nu)=(2,{2n}/{(n-2)},dy)$ of Proposition \[p23\] appeared in [@Car; @Heb; @Xia1; @Xia2; @Xia3]. In particular, if $$d\nu=\omega dy\ \ \&\ \ 1<p<q<{pn}/{(n-p)},$$ then condition (iv) above says that $0\le\omega$ belongs to the Morrey space $L^{1,n-(n-p)q/p}(\Omega)$ – in other words – the Sobolev imbedding under this circumstance is fully controlled by this Morrey space; see [@MazV] for a similar treatment on the Schrödinger operator $-\Delta +\mathcal{V}$. Furthermore, when the last $\omega$ equals identically $1$, there is a nonnegative function $u\in W^{1,p}_0(\Omega)$ such that the Euler-Lagrange (or Lane-Emden type) equation $$-\Delta_p u=\lambda_{p,\nu}(\Omega)u^{p-1}\quad\hbox{in}\quad \Omega$$ holds in the weak sense: $$\int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla\phi=\lambda_{p,\nu}(\Omega)\int_{\Omega}u^{p-1}\phi\quad\forall\quad\phi\in W_0^{1,p}(\Omega);$$ see e.g. [@KP] and its related references.

In view of Proposition \[p23\], Remark \[r22\], and the research of the Lane-Emden equations in [@Pa; @Pa1; @Pa2; @Pa3; @PoV; @APAMS; @SeZ; @Zou; @GS; @CE], we consider the nonnegative weak solutions of the quasilinear Lane-Emden equation with index $(p,q)\in (1,n)\times (0,\infty)$: $$\label{e47}
-\Delta_p u= u^{q+1}\quad\hbox{in}\quad\Omega,$$ and utilize Theorem \[t33\] to get the following result.

\[p42\] Let $$\begin{cases}
(p,q)\in (1,n)\times(0,\infty);\\
\tilde{q}\ge\max\{p,q+2\};\\
n\ge\lambda\ge\max\left\{\frac{n(q+2)}{\tilde{q}},p\big(\frac{n}{\tilde{q}}+1\big)\right\}.
\end{cases}$$ If $u\in L^{\tilde{q}}(\Omega)$ is a nonnegative weak solution of (\[e47\]), then for any $\epsilon>0$ there is an open set $O\subseteq\Omega$ such that $C_1(O;L^{\hat{q},\hat{\mu}}(\Omega))<\epsilon$ and $I_1|\nabla u|$ is $\hat{\gamma}$-Hölder continuous in $\Omega\setminus O$ where $$\begin{cases}
1<\hat{q}<p\le\lambda<\hat{\mu}=n-(n-\lambda)\hat{q}/p;\\
0<\hat{\gamma}<1-\hat{q}/p.
\end{cases}$$

Suppose $u\in L^{\tilde{q}}(\Omega)$ is a nonnegative weak solution of (\[e47\]). Then $$\label{e47a}
\int_\Omega u^{q+1}\phi=\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla\phi\quad\forall\quad \phi\in W_0^{1,p}(\Omega).$$ Given $x_0\in\Omega$ and $0<r<\hbox{diam}(\Omega)$. Upon taking a test function $\phi=u\eta^2$ such that $$\label{e31f}
\begin{cases}
\eta(x)=1\quad\hbox{for}\quad x\in B(x_0,r/3);\\
\eta(x)=0\quad\hbox{for}\quad x\in \mathbb R^n\setminus B(x_0,r/2);\\
|\nabla \eta(x)|\lesssim r^{-1}\quad\hbox{for}\quad x\in B(x_0,r),
\end{cases}$$ we utilize (\[e47a\]) to get $$\int_\Omega u^{q+2}\eta^2=\int_\Omega |\nabla u|^{p-2}|\nabla u|^2\eta^2+2^{-1}\int_\Omega|\nabla u|^{p-2}\big(\nabla (u^2)\big)\cdot\big(\nabla(\eta^2)\big).$$ Through the properties of $\eta$, Young’s inequality $$\label{CS}
ab\le \frac{\epsilon a^{\theta}}{\theta}+\frac{\epsilon^{\frac{1}{1-\theta}}b^{\theta'}}
{\theta'}\quad{\forall}\quad a,\ b,\
\epsilon,\ \theta-1>0\ \ \&\ \ \theta'=\frac{\theta}{\theta-1},$$ (applied to the last integral), and Hölder’s inequality, we find $$\begin{aligned}
\int_{B(x_0,r/3)\cap\Omega}|\nabla u|^p&\lesssim&\int_{B(x_0,r/3)\cap\Omega}u^{2+q}+r^{-p}\int_{B(x_0,r/3)\cap\Omega}u^p\\
&\lesssim& \left(\int_{B(x_0,r/3)\cap\Omega}u^{\tilde{q}}\right)^{\frac{2+q}{\tilde{q}}}r^{n(1-\frac{2+q}{\tilde{q}})}
+\left(\int_{B(x_0,r/3)\cap\Omega}u^{\tilde{q}}\right)^\frac{p}{\tilde{q}}r^{n(1-\frac{p}{\tilde{q}})-p}\\
&\lesssim&\left(\|u\|^{q+2}_{L^{\tilde{q}}(\Omega)}+\|u\|_{L^{\tilde{q}}(\Omega)}^p\right)r^{n-\lambda}(\hbox{diam}(\Omega))^{\tilde{\lambda}},\end{aligned}$$ where the assumption on $p,q,\tilde{q},\lambda$ and the following definition $$\tilde{\lambda}=\lambda-\max\left\{\frac{n(q+2)}{\tilde{q}},\ \frac{p(n+\tilde{q})}{\tilde{q}}\right\}\ge 0$$ have been used. Therefore, $|\nabla u|\in L^{p,\lambda}(\Omega)$ and desired assertion follows from applying Theorem \[t33\] to the Riesz-Morrey potential $I_1|\nabla u|$.

Hölderian quasicontinuity for $-\Delta_p u=e^u$
-----------------------------------------------

The recent works [@W1; @W2; @D; @F], along with Theorem \[p42\], have driven us to consider the nonnegative weak solutions to the quasilinear Lane equation for the $1<p<n$ Laplacian: $$\label{e477}
-\Delta_p u= e^u\quad\hbox{in}\quad\Omega,$$ thereby discovering the following fact.

\[p43\] Let $1<p<n$. If $u$ with $\int_{\Omega}ue^u<\infty$ is a nonnegative weak solution of (\[e477\]), then for any $\epsilon>0$ there is an open set $O\subseteq\Omega$ such that $C_1(O;L^{\hat{q},n}(\Omega))<\epsilon$ and $I_1|\nabla u|$ is $\hat{\gamma}$-Hölder continuous in $\Omega\setminus O$ where $$\begin{cases}
1<\hat{q}<p;\\
0<\hat{\gamma}<1-\hat{q}/p.
\end{cases}$$

Suppose $u\ge 0$ is a weak solution of (\[e477\]) with the integrability $\int_{\Omega}ue^u<\infty$. Then $$\label{e47ab}
\int_\Omega e^u\phi=\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla\phi\quad\forall\quad \phi\in W_0^{1,p}(\Omega).$$ Given $(x_0,r)\in\Omega\times \big(0,\hbox{diam}(\Omega)\big)$. Choosing $\phi=u\eta^2$ with (\[e31f\]) we obtain via (\[e47ab\]): $$\int_\Omega ue^u \eta^2=\int_\Omega |\nabla u|^{p-2}|\nabla u|^2\eta^2+2^{-1}\int_\Omega|\nabla u|^{p-2}\big(\nabla (u^2)\big)\cdot\big(\nabla(\eta^2)\big).$$ An application of the Young inequality (\[CS\]), the Hölder inequality and the assumption $p\in (1,n)$ yields $$\begin{aligned}
\int_{B(x_0,r/3)\cap\Omega}|\nabla u|^p&\lesssim&\int_{B(x_0,r/3)\cap\Omega}u e^u+r^{-p}\int_{B(x_0,r/3)\cap\Omega}u^p\\
&\lesssim& \int_{B(x_0,r/3)\cap\Omega} u e^{u}
+ r^{-p}\int_{B(x_0,r/3)\cap\Omega} u^{1-p/n}u^{p-1+p/n}\\
&\lesssim&\int_{B(x_0,r/3)\cap\Omega} u e^{u}
+ r^{-p}\int_{B(x_0,r/3)\cap\Omega} (ue^u)^{1-p/n}\\
&\lesssim&\int_{\Omega} u e^{u}
+\left(\int_{\Omega} ue^{u}\right)^{1-p/n}.\end{aligned}$$ Thus, $|\nabla u|\in L^{p,n}(\Omega)$. This, together with Theorem \[t33\] for the Riesz-Morrey potential $I_1|\nabla u|$, derives the desired assertion.

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[^1]: JX is in part supported by NSERC of Canada and URP of Memorial University.
