1. Introduction
Let $\Omega \subset \mathbb {R}^{n} \,(n\geq 2)$ be a bounded domain. In this paper, we are interested in the following class of integro-differential equations with general growth
with
where the symbol $\mathrm {P.V.}$ represents ‘in the principal value sense’, $s\in (0,\,1)$ and the function $K(x,\,y):\mathbb {R}^{n}\times \mathbb {R}^{n}\rightarrow (0,\,\infty ]$ is a symmetric measurable kernel such that
Particularly when $\Lambda =1$, Eq. (1.1) is called $s$-fractional $G$-Laplace equation. The function $g:[0,\,\infty )\rightarrow [0,\,\infty )$ is continuous and strictly increasing fulfilling $g(0)=0$, $\lim _{t\rightarrow \infty }g(t)=\infty$ and
where $G(\cdot )$ is an $N$-function possessing the $\Delta _2$ and $\nabla _2$ conditions (see § 2).
In recent years, a great attention has been concentrated on the nonlocal $p$-Laplacian problems, which is the special case that $g(t)=t^{p-1}$. For the regularity theory on this kind of problems, Kassmann [Reference Kassmann28] proved the nonlocal Harnack inequality with tail-term for the fractional Laplacian. Di Castro-Kuusi-Palatucci [Reference Di Castro, Kuusi and Palatucci13] further investigated the local behaviour of weak solutions incorporating boundedness and Hölder continuity in the spirit of De Giorgi-Nash-Moser iteration; see also [Reference Di Castro, Kuusi and Palatucci12] for the nonlocal Harnack inequalities. The Hölder regularity up to the boundary was whereafter showed by Iannizzotto–Mosconi–Squassina [Reference Iannizzotto, Mosconi and Squassina27]. We also refer the readers to [Reference Brasco and Lindgren1] for higher Sobolev regularity, [Reference Kuusi, Mingione and Sire33] for self-improving properties [Reference Korvenpää, Kuusi and Lindgren30, Reference Korvenpää, Kuusi and Palatucci32], for the viscosity and potential theory [Reference Brasco and Parini3, Reference Franzina and Palatucci20], for fractional $p$-eigenvalue problems. When it comes to the parabolic counterpart, several features of solutions have already been studied, such as the local regularity [Reference Brasco, Lindgren and Strömqvist2, Reference Ding, Zhang and Zhou15, Reference Strömqvist45] and the well-posedness [Reference Mazón, Rossi and Toledo37, Reference Vázquez46]. For more results on the nonlocal nonlinear problems of the $p$-Laplacian type, one can see for instance [Reference Caffarelli, Chan and Vasseur7, Reference Felsinger and Kassmann17, Reference Korvenpää, Kuusi and Palatucci31, Reference Kuusi, Mingione and Sire34, Reference Palatucci40, Reference Vázquez47].
When $g(\cdot )$ carries a more general structure, Eq. (1.1) can be viewed naturally as the nonlocal analogue of the $G$-Laplace equation whose classical model is
The so-called $G$-Laplace equations have been extensively studied over the past years. The regularity theory, especially for the scenario that $g(t)\approx t^{p-1}+t^{q-1}$, is initially explored by the celebrated papers of Marcellini [Reference Marcellini35, Reference Marcellini36]. More results on the generalized $p$-Laplace equations can be found in [Reference Buryachenko and Skrypnik4, Reference De Filippis and Mingione10, Reference Diening, Stroffolini and Verde14, Reference Harjulehto, Hästö and Lee26, Reference Salort41, Reference Skrypnik and Voitovych44]. On the other hand, Fernández Bonder–Salort–Vivas [Reference Fernández Bonder, Salort and Vivas18] established the Hölder continuity for weak solutions to the fractional $g$-Laplacian with Dirichlet boundary values; see also [Reference Fernández Bonder, Salort and Vivas19] for the global regularity of eigenfunctions. Chaker–Kim–Weidner [Reference Chaker, Kim and Weidner9] proved, via De Giorgi classes, the interior regularity properties for the nonlocal functionals with $(p,\,q)$-growth and related equations. More recently, the weak solutions to (1.1) were proved to be locally bounded and Hölder continuous in [Reference Byun, Kim and Ok5] under the assumption (1.3). Regarding further studies of the nonlocal problems possessing non-standard growth, including also double phase equations and equations with variable exponents, one can refer to [Reference Byun, Ok and Song6, Reference Chaker and Kim8, Reference De Filippis and Palatucci11, Reference Fang and Zhang16, Reference Giacomoni, Kumar and Sreenadh21, Reference Giacomoni, Kumar and Sreenadh22, Reference Goel, Kumar and Sreenadh25, Reference Ok39, Reference Salort42, Reference Salort and Vivas43] and references therein.
Although pretty abundant research results have been obtained for the nonlocal problems with non-standard growth, to the best of our knowledge, there are few results regarding the pointwise estimates such as the Harnack inequalities. To this end, our aim of this manuscript is to investigate Harnack estimate for Eq. (1.1), which can be regarded as a natural outgrowth of the result in [Reference Di Castro, Kuusi and Palatucci12]. Due to the possibly inhomogeneous growth of the function $G$, we have to explore the suitable conditions on $G$ in order to infer the desired result. Additionally, we require that the function $G$ satisfies the following condition:
for any $t,\,\tau \geq 0$ and $c_0$ being a positive constant. Examples of $G$ satisfying the requirements (1.3) and (1.5) include
• $G(t)=t^{p}$, $t\ge 0$, $p>1$;
• $G(t)=\max \{t^{p},\,t^{q}\}$, $t\ge 0$, $1< p\le q<\infty$;
• $G(t)=t^{p}+a_0t^{q} \text { with } a_0>0$, $t\ge 0$, $1< p\le q<\infty$;
• $G(t)=t^{p}\log (e+t)$, $t\ge 0$, $p>1$.
Before giving our main result, we introduce the so-called ‘tail space’,
The corresponding nonlocal tail of $u$ is given by
Notice that $u\in L^{g}_s(\mathbb {R}^{n})$ if and only if $\mathrm {Tail}(u;x_0,\,R)$ is finite for any $x_0\in \mathbb {R}^{n}$ and $R>0$. The details can be found in [Reference Byun, Kim and Ok5, subsection 2.3].
Now we are in a position to state the main result as follows.
Theorem 1.1 Suppose that $s\in (0,\,1)$ and the assumptions (1.3) and (1.5) are in force. Let $u\in \mathbb {W}^{s,G}(\Omega )\cap L^{g}_s(\mathbb {R}^{n})$ be a weak solution of Eq. (1.1) such that $u\geq 0$ in $B_R:=B_R(x_0)\subset \Omega$. Then, for every $B_r:=B_r(x_0)\subset B_\frac {R}{2}(x_0),$ we have the following nonlocal Harnack inequality
where Tail $(\cdot )$ is defined in (1.6), $u_-:=\max \{-u,\,0\}$, the positive constant $C$ depends on $n,\,p,\,q,\,s,\,\Lambda$ as well as the structural constant $c_0$ given by (1.5), and the absolute constant $\epsilon \in (0,\,1)$, coming from lemma 3.3 below, is a priori determined by $n,\,p,\,q,\,s,\,\Lambda$.
Remark 1.2 Let us point out that the extra hypothesis (1.5) is only exploited in the proof of theorem 1.1 below. The reason why we impose the additional strong condition on $G$ is that we need to split the term $G(u)$ into $G(u^{1-\varepsilon })G(u^{\varepsilon })$ with $\varepsilon$ being an arbitrary number in $(0,\,1)$, and then get the integral of $u^{\varepsilon '}$ $(\varepsilon '\in (0,\,1))$ as the integrand, which enables us to apply lemma 3.3. Observe that, if $g(t)=t^{p-1}$, then $q=p$ and
Hence, our result is reduced to the Harnack inequality obtained in [Reference Di Castro, Kuusi and Palatucci12, theorem 1.1].
Remark 1.3 The result obtained in theorem 1.1 can be extended to the nonhomogeneous equation $\mathcal {L}u=f$ with $f$ being bounded locally. In fact, we just need to consider the additional integral involving $f$, $\int _\Omega f\phi \,{\rm d}x$, in proposition 3.1, lemmas 3.2, 4.1 and 4.2, where $\phi$ is a test function varying in different contexts. For the nonhomogeneous counterpart, we could deduce the following Harnack inequality
The paper is organized as follows. In § 2, we give the definition of weak solutions to Eq. (1.1), and collect some notations and auxiliary inequalities to be used later. Section 3 is devoted to deducing infimum estimates for weak supersolutions by employing the expansion of positivity. Finally, we prove the Harnack inequality in § 4.
2. Preliminaries
In this section, we shall give some basic inequalities, state the notions of some functional spaces and weak solutions, and then provide a covering lemma.
In what follows, we denote by $C$ a generic positive constant which may change from line to line. Relevant dependencies on parameters will be illustrated utilizing parentheses, i.e., $C\equiv C(n,\,p,\,q)$ means that $C$ depends on $n,\,p,\,q$. Let $B_r(x_0):=\{x\in \mathbb {R}^{n}:|x-x_0|< r\}$ stand for the open ball with centre $x_0$ and radius $r>0$. If not important, or clear from the context, we do not denote the centre as follows: $B_r:=B_r(x_0)$. If $f\in L^{1}(A)$ and $A\subset \mathbb {R}^{n}$ is a measurable subset with positive measure $0<|A|<\infty$, we denote its integral average by
The function $G:[0,\,\infty )\rightarrow [0,\,\infty )$ is an $N$-function which means that it is convex and increasing, and satisfies that
The conjugate function of $N$-function $G$ is denoted by
From the relation (1.3), we now give several inequalities to be utilized later:
(a) for $t\in [0,\,\infty )$,
(2.1)\begin{equation} \begin{cases} a^{q}G(t)\leq G(at)\leq a^{p}G(t) & \text{if } a\in(0,1),\\ a^{p}G(t)\leq G(at)\leq a^{q}G(t) & \text{if } a\in(1,\infty) \end{cases} \end{equation}and(2.2)\begin{equation} \begin{cases} a^{p'}G^{*}(t)\leq G^{*}(at)\leq a^{q'}G^{*}(t) & \text{if} a\in(0,1),\\ a^{q'}G^{*}(t)\leq G^{*}(at)\leq a^{p'}G^{*}(t) & \text{if} a\in(1,\infty), \end{cases} \end{equation}where $p',\,q'$ are the Hölder conjugates of $p,\,q$.(b) Young's inequality with $\epsilon \in (0,\,1]$
(2.3)\begin{equation} t\tau\leq \epsilon^{1-q}G(t)+\epsilon G^{*}(\tau), \quad t,\tau\geq 0. \end{equation}(c) for $t,\,\tau \geq 0$,
(2.4)\begin{equation} G^{*}(g(t))\leq (q-1)G(t), \end{equation}and(2.5)\begin{equation} 2^{{-}1}(G(t)+G(\tau))\leq G(t+\tau)\leq 2^{q-1}(G(t)+G(\tau)). \end{equation}
Moreover, the function $G$ fulfills the following $\Delta _2$ and $\nabla _2$ conditions (see [Reference Mihăilescu and Rădulescu38, proposition 2.3]):
(Δ2) there is a constant $\mu >1$ such that $G(2t)\leq \mu G(t)$ for $t\geq 0$;
(∇2) there is a constant $\nu >1$ such that $G(t)\leq \frac {1}{2\nu }G(\nu t)$ for $t\geq 0$,
where $\mu,\,\nu$ depend on $p,\,q$. As a matter of fact, the condition $\nabla _2$ is just $\Delta _2$ applied to $G^{*}$.
We next introduce the notion of Orlicz–Sobolev spaces. For an $N$-function $G$ with the $\Delta _2$ and $\nabla _2$ conditions, the Orlicz space $L^{G}(\Omega )$ is defined as
equipped with the Luxemburg norm
The fractional Orlicz–Sobolev space $W^{s,G}(\Omega )$ ($s\in (0,\,1$)) is given by
endowed with the norm
where $[u]_{s,G,\Omega }$ is the Gagliardo semi-norm defined as
Let $C_{\Omega }\equiv (\Omega \times \mathbb {R}^{n})\cup (\mathbb {R}^{n}\times \Omega )$. For measurable function $u$ in $\mathbb {R}^{n}$, we define
which is the space weak solutions of (1.1) belong to.
Now we give the definition of weak solutions to (1.1).
Definition 2.1 We call $u\in \mathbb {W}^{s,G}(\Omega )$ a weak supersolution of Eq. (1.1) if
for each nonnegative function $\psi \in \mathbb {W}^{s,G}(\Omega )$ with compact support in $\Omega$. For weak subsolution, the above inequality is reversed. $u\in \mathbb {W}^{s,G}(\Omega )$ is a weak solution to (1.1) if and only if it is both a weak supersolution and a weak subsolution.
We conclude this section by presenting the Krylov–Sofonov covering lemma (see for instance [Reference Kinnunen and Shanmugalingam29]) playing an important role in proving lemma 3.3 below.
Lemma 2.2 Let $\overline {\delta }\in (0,\,1)$ and $E\subset B_r(x_0)$ be a measurable set. Denote
Then one of the following must hold:
(i) $|[E]_{\overline {\delta }}|\geq \frac {c(n)}{\overline {\delta }}|E|;$
(ii) $[E]_{\overline {\delta }}=B_r(x_0).$
3. Expansion of positivity
This section is devoted to deriving the infimum estimates on the weak supersolutions of (1.1) by expansion of positivity. The following proposition exhibits the spread of pointwise positivity in space.
Proposition 3.1 Let $k\geq 0$ and $u\in \mathbb {W}^{s,G}(\Omega )$ be a weak supersolution to Eq. (1.1) such that $u\geq 0$ in $B_R(x_0)\subset \Omega$. If
for some $\sigma \in (0,\,1]$ and $r$ fulfilling $0< r<\frac {R}{16}\leq 1$, then there is $\delta \in (0,\,\frac {1}{2})$, which depends on $n,\,p,\,q,\,s,\,\Lambda,\,\sigma$, such that
Before proving this proposition, we first need the propagation of positivity in measure, that is the forthcoming lemma.
Lemma 3.2 Let $k\geq 0$ and $u\in \mathbb {W}^{s,G}(\Omega )$ be a weak supersolution to Eq. (1.1) such that $u\geq 0$ in $B_R(x_0)\subset \Omega$. If there is a $\sigma \in (0,\,1]$ satisfying
with $0< r<\frac {R}{16}\leq 1$, then we infer that, for any $\delta \in (0,\,\frac {1}{2})$,
with the constant $C>0$ depending only on $n,\,p,\,q,\,s,\,\Lambda$.
Proof. Let $v(x):=u(x)+d$ with $d=r^{s}g^{-1}(r^{s}\mathrm {Tail}(u_-;x_0,\,R))$. Now take a cut-off function $\varphi \in C^{\infty }_0(B_{7r})$ such that
We select $\eta :=\varphi ^{q}\frac {v}{G(v/r^{s})}$ as a test function in the weak formulation (2.6), and then slightly modify the expression to have
Following the arguments of steps 1–3 in [Reference Byun, Kim and Ok5, proposition 3.4], we get
For the integral $I_2$,
We first evaluate $I_{21}$. Note that, by (1.3), (2.1) and (2.5),
Then,
When $x\in B_{7r}$ and $y\in \mathbb {R}^{n}\setminus B_{8r}$,
we further get
where we utilized the fact $u(y)\geq 0$ in $B_R\supset B_{8r}$ and the constant $C$ depends on $n,\,p,\,q,\,s,\,\Lambda$. We next estimate $I_{22}$ as
Recalling the definition of $d$, we arrive at
with $C$ depending on $n,\,p,\,q,\,s,\,\Lambda$.
Merging (3.2), (3.3) with (3.1) yields that
For all $\delta \in (0,\,\frac {1}{2})$, set
Owing to $w$ being a truncation of $\log (k+d)-\log v$, there holds that
Observe that
Hence,
In the same way as the computations in [Reference Di Castro, Kuusi and Palatucci12, page 1819], we finally deduce that
We now have finished the proof.
Based on the above lemma, we can conclude the proof of proposition 3.1.
Proof of proposition 3.1. We may suppose, with no loss of generality, that
We now choose a cut-off function $\varphi \in C^{\infty }_0(B_\rho )$ with $4r\leq \rho \leq 6r$ and take the test function $\eta =v_-\varphi ^{q}:=(l-u)_+\varphi ^{q}$ for $l\in (\frac {1}{2}\delta k,\,2\delta k)$ in the weak formulation (2.6). Then we have
We first evaluate $I_2$,
We proceed with treating the integral $I_1$. This procedure is similar to the estimate on $I$ in [Reference Byun, Kim and Ok5, proposition 3.1], but for the sake of readability we give a sketched proof. Assume $u(x)\geq u(y)$. Then
by distinguishing three cases that $l\geq u(x)\geq u(y)$, $u(x)\geq l>u(y)$ and $u(x)\geq u(y)\geq l$. Exchanging the roles of $x$ and $y$, we in general case also have the previous inequality. We next consider two cases:
The case that $v_-(x)\leq v_-(y)$ is symmetric. In case 1, from (1.3) and (2.2)–(2.4), we have
where we take $\epsilon =\frac {p}{2q(q-1)}$. In the other case,
In summary, we derive
Therefore,
Combining the estimates on $I_1$ and $I_2$ with (3.4), we know that
Next, we will perform an iteration process. Set
for $j=0,\,1,\,2,\,\cdots$. We can find that
and
Take cut-off functions $\varphi _j\in C^{\infty }_0(B_{\tilde {\rho }_j}(x_0))$ ($j=0,\,1,\,2,\,\cdots$) such that
With $v_-,\,\varphi,\,l,\,\rho$ being replaced by $v_j,\,\varphi _j,\,l_j,\,\rho _j$ respectively, (3.5) turns into
where we have employed (2.1) and the facts that
and for $x\in \mathrm {supp}\,\varphi _j\subset B_{\tilde {\rho }_j}$ and $y\in \mathbb {R}^{n}\setminus B_j$,
Observe that, by (1.3) and (2.5),
Since $u(y)\geq 0$ in $B_R$, for the integral in $J_2$ there holds that
where in the last inequality we note that
namely,
As for $J_1$, it follows from (2.5) that
Putting together these preceding estimates yields that
According to lemma 4.1 in [Reference Byun, Kim and Ok5], we obtain
with $\theta >1$ depending only on $n,\,s$.
On the other hand, by means of (2.5) and Jensen's inequality, the following display
is valid. Moreover, via (3.6) and $4r\leq \rho _{j+1}\leq 6r$,
It then follows from (3.7)–(3.9) that
that is,
Denote
Then
We can apply the iteration lemma (see, e.g., [Reference Giusti24, lemma 7.1]) to deduce that if
then $A_j\rightarrow 0$ as $j\rightarrow \infty$. Now from lemma 3.2 we examine
As long as we choose such small $\delta$ that
the desired result $\lim _{j\rightarrow \infty }A_j=0$ can be justified. In other words, we draw a conclusion that there exists $\delta$, determined by $n,\,p,\,q,\,s,\,\Lambda$ and $\sigma$, such that
in $B_{4r}$. We now complete the proof.
At the end of this section, as a consequence of proposition 3.1 and the Krylov–Sofonov covering lemma, we derive the following result.
Lemma 3.3 Suppose that $u\in \mathbb {W}^{s,G}(\Omega ),$ satisfying $u\geq 0$ in $B_R(x_0)\subset \Omega,$ is a weak supersolution to Eq. (1.1). Then we can find two constants $\epsilon \in (0,\,1)$ and $C\geq 1,$ both of which depend only upon $n,\,p,\,q,\,s,\,\Lambda,$ such that, when $B_r(x_0)\subset B_R(x_0),$
is valid.
Proof. Define for any $t>0$
where $\delta$ is identical to that of proposition 3.1, and $T$ stands for
Recalling lemma 2.2 and proposition 3.1, we could follow the proof of [Reference Di Castro, Kuusi and Palatucci12, lemma 4.1] verbatim, except substituting $\delta$ in [Reference Di Castro, Kuusi and Palatucci12, lemma 4.1] with $\frac {1}{2}\delta$ here, to arrive at
This directly implies the desired result.
4. Nonlocal Harnack inequality
In this section, we are going to show the nonlocal Harnack inequality by merging the local boundedness on subsolutions (lemma 4.2) along with the infimum estimate of supersolution (lemma 3.3), and taking into account the tail estimate for solutions (lemma 4.1) in a suitable way.
Lemma 4.1 Assume that $u\in \mathbb {W}^{s,G}(\Omega )\cap L^{g}_s(\mathbb {R}^{n})$ is a weak solution to Eq. (1.1) such that $u\geq 0$ in $B_R(x_0)\subset \Omega$. Then the tail estimate
holds true for all $0< r< R,$ where $C>0$ depends only on $n,\,p,\,q,\,s,\,\Lambda$.
Proof. Let $l=\sup _{B_r}u$. We take the test function
in the weak formulation (2.6), where $\varphi \in C^{\infty }_0(B_r)$ satisfies that
to derive
For $I_2$, we can see that
We know from (1.3) and (2.5) that
Thereby,
where we have used (1.3) and the facts that, for $x\in B_\frac {r}{2}$ and $y\in \mathbb {R}^{n}\setminus B_r$
and for $x\in B_\frac {3r}{4}$ and $y\in \mathbb {R}^{n}\setminus B_r$
On the other hand, with the help of (1.3) and (2.5), we get
As a result,
Next it remains to deal with the integral $I_1$. Set $v:=u-2\,l$. Suppose, without loss of generality, that $\varphi (x)\geq \varphi (y)$. Then $\varphi ^{q}(x)-\varphi ^{q}(y)\leq q\varphi ^{q-1}(x)(\varphi (x)-\varphi (y))$. For $(x,\,y)\in B_r\times B_r$, we in turn employ the inequalities (1.3), (2.2)–(2.4) to arrive at
Here we need note $\varphi ^{q-1}(x)\leq 1$ and take $\epsilon =\frac {p}{2q}$. From this, we find that
Consequently, it holds, by combing (4.2), (4.3) with (4.1), that
Finally, observe that for $a,\,b\geq 0$ and $c\geq 1$,
and
Otherwise, by (1.3), (2.1) and the strictly increasing property of $g$,
which is a contradiction. Then we have
and
which means the desired result.
In order to infer Harnack inequality for Eq. (1.1), we need the following local boundedness result on weak subsolutions that is a slightly modified version of [Reference Byun, Kim and Ok5, theorem 4.4].
Lemma 4.2 Let $B_r(x_0)\subset \subset \Omega$. Assume that $u\in \mathbb {W}^{s,G}(\Omega )\cap L^{g}_s(\mathbb {R}^{n})$ is a weak subsolution to Eq. (1.1). Then there holds that
where $C$ depends on $n,\,p,\,q,\,s,\,\Lambda$.
Proof. The process is the same as that of [Reference Byun, Kim and Ok5, theorem 4.4]. Let us point out that the notations below adopt identically those in [Reference Byun, Kim and Ok5, theorem 4.4]. We just need to notice that, after the inequality (4.14) in [Reference Byun, Kim and Ok5], the parameter $k$ is first chosen so large that
with $\delta \in (0,\,1]$, instead of the value of $k$ there. Then the inequality (4.14) in [Reference Byun, Kim and Ok5] becomes
Let $C_0=2^{\theta } C$ and $B=2^{(n+sq+2q)\theta }$. Then
By the iteration lemma (see, e.g. [Reference Giusti24, lemma 7.1]), we need
namely,
so that $a_j\rightarrow 0$ as $j\rightarrow \infty$. Now we pick
Terminally, the limit $\lim _{j\rightarrow \infty }a_j=0$ leads to (4.5).
Finally, we implement the proof of the nonlocal Harnack inequality stated in theorem 1.1. From this procedure, one can apparently understand the reason why we impose the condition (1.5).
Proof of theorem 1.1. For simplicity, let $\lambda =\frac {\theta }{1-\theta }$. Putting together the local boundedness estimate (lemma 4.2) and the tail estimate (lemma 4.1), we derive that, for $B_\rho \subset \subset \Omega$,
Here we have utilized
which can be justified in a similar way to (4.4).
We next would like to apply the iteration [Reference Giaquinta and Giusti23, lemma 1] (see also [Reference Di Castro, Kuusi and Palatucci12, lemma 2.7]). Denote $\rho =(\gamma -\gamma ')r$ with $\frac {1}{2}\leq \gamma '<\gamma \leq 1$. By a covering argument, we obtain
where we note the positivity of $u$ in $B_R(x_0)$. Observe that from (1.3) we can get an important inequality $\min \{t^{p},\,t^{q}\}\leq G(t)\leq c\max \{t^{p},\,t^{q}\}$. Now making use of this inequality and the assumption (1.5), we evaluate, for any $\varepsilon \in (0,\,1)$,
Via selecting $\delta =(\frac {1}{4C})^{q-1}$, merging the displays (4.6), (4.7) and an application of Young's inequality, we have
We can apply [Reference Giaquinta and Giusti23, lemma 1] to infer that
In order to make use of lemma 3.3, we need to invoke the relation that $G(t)\leq c\max \{t^{p},\,t^{q}\}$.
We proceed by considering the integral term in the above display,
Combining the last two displays and choosing $\varepsilon =\frac {\epsilon }{q}$ with $\epsilon$ given by lemma 3.3, we finally arrive at
The proof is complete now.
Acknowledgments
The authors wish to thank the anonymous reviewer for valuable comments and suggestions to improve the manuscript. This work was supported by the National Natural Science Foundation of China (No. 12071098) and the National Postdoctoral Program for Innovative Talents of China (No. BX20220381).