A note on gradient estimates for p-Laplacian equations

TL;DR

This paper improves gradient estimate methods for p-Laplacian equations by relaxing assumptions, utilizing nonlinear potential theory and compactness results to analyze weak solutions.

Contribution

It demonstrates that certain assumptions can be dropped when seeking weak solutions, enhancing the theoretical framework for p-Laplacian equations.

Findings

Relaxation of assumptions in gradient estimates

Use of nonlinear potential theory for $L^ty$ estimates

Application of Riesz-Fre9chet-Kolmogorov theorem for compactness

Abstract

The aim of this short paper is to show that some assumptions in [10] can be relaxed and even dropped when looking for weak solutions instead of strong ones. This improvement is a consequence of two results concerning gradient terms: an $L^\infty$ estimate, which exploits nonlinear potential theory, and a compactness result, based on the classical Riesz-Fr\'echet-Kolmogorov theorem.

Equations

$$\{\begin{array}{rlrl}-{\Delta }_{p}u& =f(x,u,v,\nabla u,\nabla v)& & \text{mbox}in{\mathbb{R}}^N,\\ -{\Delta }_{q}v& =g(x,u,v,\nabla u,\nabla v)& & \text{mbox}in{\mathbb{R}}^N,\\ u,v& >0& & \text{mbox}in{\mathbb{R}}^N,\end{array}$$

$$m_1{a}_1(x)s_1^{{\alpha }_1}{s}_2^{{\beta }_1}\le f(x,s_1,s_2,{\mathbf{t}}_1,{\mathbf{t}}_2)\le {\hat{m}}_1{a}_1(x)\left(s_1^{{\alpha }_1}{s}_2^{{\beta }_1}+|{\mathbf{t}}_1{|}^{{\gamma }_1}+|{\mathbf{t}}_2{|}^{{\delta }_1}\right)$$

$$m_2{a}_2(x)s_1^{{\alpha }_2}{s}_2^{{\beta }_2}\le g(x,s_1,s_2,{\mathbf{t}}_1,{\mathbf{t}}_2)\le {\hat{m}}_2{a}_2(x)\left(s_1^{{\alpha }_2}{s}_2^{{\beta }_2}+|{\mathbf{t}}_1{|}^{{\gamma }_2}+|{\mathbf{t}}_2{|}^{{\delta }_2}\right)$$

$$\frac{1}{{\zeta }_1}<1-\frac{p}{p^{\ast }}-{\theta }_1,\frac{1}{{\zeta }_2}<1-\frac{q}{q^{\ast }}-{\theta }_2,$$

$$\begin{array}{r}{\theta }_1:=\max \left\{\frac{{\beta }_1}{q^{\ast }},\frac{{\gamma }_1}{p},\frac{{\delta }_1}{q}\right\}<1-\frac{p}{p^{\ast }},{\theta }_2:=\max \left\{\frac{{\alpha }_2}{p^{\ast }},\frac{{\gamma }_2}{p},\frac{{\delta }_2}{q}\right\}<1-\frac{q}{q^{\ast }}.\end{array}$$

$${\eta }_1{\eta }_2<(p-1-{\gamma }_1)(q-1-{\delta }_2).$$

$$\frac{1}{s_p}+\max \left\{\frac{{\gamma }_1}{p},\frac{{\delta }_1}{q}\right\}<\frac{1}{N},\frac{1}{s_q}+\max \left\{\frac{{\gamma }_2}{p},\frac{{\delta }_2}{q}\right\}<\frac{1}{N}.$$

$$-{\Delta }_{p}u=f(x)\text{mbox}in{\mathbb{R}}^N$$

$${\int }_{\Omega }|\nabla u{|}^{p-2}\nabla u\cdot \nabla \phi \mathrm{d}x={\int }_{\Omega }f\phi \mathrm{d}x\forall \phi \in C_c^{\infty }({\mathbb{R}}^N).$$

$$\begin{array}{rl}& P_f(x,R):={\int }_0^R{\left(\frac{|f{|}^2(B_{\rho }(x))}{{\rho }^{N-2}}\right)}^{\frac{1}{2}}\frac{\mathrm{d}\rho }{\rho },\text{mbox}with|f{|}^2(B_{\rho }(x)):=\Vert f{\Vert }_{L^2(B_{\rho }(x))}^2.\end{array}$$

$$-{\Delta }_{p}u=f(x)\text{mbox}in\Omega ,$$

$$\begin{array}{rl}\Vert \nabla u{\Vert }_{L^{\infty }(B_R)}& \le C\left[{\left(\frac{1}{|B_{2R}|}{\int }_{B_{2R}}|\nabla u{|}^p\mathrm{d}x\right)}^{\frac{1}{p}}+\Vert P_f(\cdot ,2R){\Vert }_{L^{\infty }(B_{2R})}^{\frac{1}{p-1}}\right]\end{array}$$

$$-{\Delta }_{p}u=f(x)\text{mbox}in{\mathbb{R}}^N,$$

$$\Vert \nabla u{\Vert }_{L^{\infty }({\mathbb{R}}^N)}^{p-1}\le C\left(\Vert \nabla u{\Vert }_{L^p({\mathbb{R}}^N)}^{p-1}+\Vert f{\Vert }_{L^r({\mathbb{R}}^N)}\right).$$

$$\begin{array}{rl}|\nabla u(x){|}^{p-1}& \le \Vert \nabla u{\Vert }_{L^{\infty }(B_1(x))}^{p-1}\\ & \le C\left[{\left(\frac{1}{|B_2(x)|}{\int }_{B_2(x)}|\nabla u{|}^p\mathrm{d}x\right)}^{\frac{1}{p^{\prime }}}+\Vert P_f(\cdot ,2){\Vert }_{L^{\infty }(B_2(x))}\right]\\ & \le C\left[\Vert \nabla u{\Vert }_{L^p({\mathbb{R}}^N)}^{p-1}+\underset{y\in B_2(x)}{\sup }{\int }_0^2{\rho }^{-\frac{N}{2}}\Vert f{\Vert }_{L^2(B_{\rho }(y))}\mathrm{d}\rho \right]\\ & \le C\left[\Vert \nabla u{\Vert }_{L^p({\mathbb{R}}^N)}^{p-1}+\Vert f{\Vert }_{L^r({\mathbb{R}}^N)}{\int }_0^2{\rho }^{-\frac{N}{r}}\mathrm{d}\rho \right]\\ & \le C\left(\Vert \nabla u{\Vert }_{L^p({\mathbb{R}}^N)}^{p-1}+\Vert f{\Vert }_{L^r({\mathbb{R}}^N)}\right).\end{array}$$

$$u_h(x):=u(x+h),{\delta }_{h}u:=u_h-u.$$

$$-{\Delta }_p{u}_n=f_n(x)\text{mbox}\mathrm{in}\Omega$$

$${\int }_{B_R}|\nabla u_n{|}^{p-2}\nabla u_n\cdot \nabla \phi \mathrm{d}x={\int }_{B_R}{f}_n\phi \mathrm{d}x$$

$${\int }_{B_R}{\eta }^2{V}_n\cdot {\delta }_h(\nabla u_n)\mathrm{d}x+2{\int }_{B_R}\eta {\delta }_h{u}_n{V}_n\cdot \nabla \eta \mathrm{d}x={\int }_{B_R}{f}_n\phi \mathrm{d}x.$$

$${\int }_{B_R}{\eta }^2(V_n{)}_h\cdot {\delta }_h(\nabla u_n)\mathrm{d}x+2{\int }_{B_R}\eta {\delta }_h{u}_n(V_n{)}_h\cdot \nabla \eta \mathrm{d}x={\int }_{B_R}{f}_n{\phi }_{-h}\mathrm{d}x.$$

$${\int }_{B_R}{\eta }^2{\delta }_h{V}_n\cdot {\delta }_h(\nabla u_n)\mathrm{d}x+2{\int }_{B_R}\eta {\delta }_h{u}_n{\delta }_h{V}_n\cdot \nabla \eta \mathrm{d}x={\int }_{B_R}{f}_n{\delta }_{-h}\phi \mathrm{d}x.$$

$$\begin{array}{rl}{\int }_{B_t}{\delta }_h{V}_n\cdot {\delta }_h(\nabla u_n)\mathrm{d}x& \le {\int }_{B_R}{\eta }^2{\delta }_h{V}_n\cdot {\delta }_h(\nabla u_n)\mathrm{d}x\\ & \le 2{\int }_{B_R}|{\delta }_h{u}_n||{\delta }_h{V}_n||\nabla \eta |\mathrm{d}x+{\int }_{B_R}|f_n||{\delta }_{-h}\phi |\mathrm{d}x\\ & \le \frac{C}{s-t}\Vert {\delta }_h{u}_n{\Vert }_{L^p(B_s)}\Vert {\delta }_h{V}_n{\Vert }_{L^{p^{\prime }}(B_s)}+\Vert f_n{\Vert }_{L^{r^{\prime }}(B_s)}\Vert {\delta }_{-h}\phi {\Vert }_{L^r(B_s)}\\ & \le \frac{C}{s-t}\Vert {\delta }_h{u}_n{\Vert }_{L^p(B_R)}\left(\Vert (V_n{)}_h{\Vert }_{L^{p^{\prime }}(B_s)}+\Vert V_n{\Vert }_{L^{p^{\prime }}(B_s)}\right)\\ & +\Vert f_n{\Vert }_{L^{r^{\prime }}(B_R)}\left(\Vert {\phi }_{-h}{\Vert }_{L^r(B_s)}+\Vert \phi {\Vert }_{L^r(B_s)}\right)\\ & \le \frac{2C}{s-t}\Vert {\delta }_h{u}_n{\Vert }_{L^p(B_R)}\Vert V_n{\Vert }_{L^{p^{\prime }}(B_R)}+2\Vert f_n{\Vert }_{L^{r^{\prime }}(B_R)}\Vert {\delta }_h{u}_n{\Vert }_{L^r(B_R)}\\ & \le C\left(\Vert {\delta }_h{u}_n{\Vert }_{L^p(B_R)}\Vert \nabla u_n{\Vert }_{L^p(B_R)}^{p-1}+\Vert f_n{\Vert }_{L^{r^{\prime }}(B_R)}\Vert {\delta }_h{u}_n{\Vert }_{L^r(B_R)}\right),\end{array}$$

$$\begin{array}{rl}{\int }_{B_t}{\delta }_h{V}_n\cdot {\delta }_h(\nabla u_n)\mathrm{d}x& ={\int }_{B_t}(|\nabla (u_n{)}_h{|}^{p-2}\nabla (u_n{)}_h-|\nabla u_n{|}^{p-2}\nabla u_n)\cdot (\nabla (u_n{)}_h-\nabla u_n)\mathrm{d}x\\ & \ge c\Vert (\nabla u_n{)}_h-\nabla u_n{\Vert }_{L^p(B_t)}^p=c\Vert {\delta }_h(\nabla u_n){\Vert }_{L^p(B_t)}^p,\end{array}$$

$$\begin{array}{rl}{\int }_{B_t}{\delta }_h{V}_n\cdot {\delta }_h(\nabla u_n)\mathrm{d}x& ={\int }_{B_t}(|\nabla (u_n{)}_h{|}^{p-2}\nabla (u_n{)}_h-|\nabla u_n{|}^{p-2}\nabla u_n)\cdot (\nabla (u_n{)}_h-\nabla u_n)\mathrm{d}x\\ & \ge c{\int }_{B_t}(1+|\nabla (u_n{)}_h{|}^2+|\nabla u_n{|}^2{)}^{\frac{p-2}{2}}|\nabla (u_n{)}_h-\nabla u_n{|}^2\mathrm{d}x\\ & =c{\int }_{B_t}{W}_{nh}|{\delta }_h(\nabla u_n){|}^2\mathrm{d}x,\end{array}$$

$$\begin{array}{rl}\Vert {\delta }_h(\nabla u_n){\Vert }_{L^p(B_t)}^p& ={\int }_{B_t}{W}_{nh}^{\frac{p}{2}}|{\delta }_h(\nabla u_n){|}^p{W}_{nh}^{-\frac{p}{2}}\mathrm{d}x\\ & \le {\left({\int }_{B_t}{W}_{nh}|{\delta }_h(\nabla u_n){|}^2\mathrm{d}x\right)}^{\frac{p}{2}}{\left({\int }_{B_t}{W}_{nh}^{\frac{p}{p-2}}\mathrm{d}x\right)}^{\frac{2-p}{2}}\\ & \le {\left({\int }_{B_t}{W}_{nh}|{\delta }_h(\nabla u_n){|}^2\mathrm{d}x\right)}^{\frac{p}{2}}{\left(|B_t|+2\Vert \nabla u_n{\Vert }_{L^p(B_R)}^p\right)}^{\frac{2-p}{2}}\\ & \le C{\left({\int }_{B_t}{W}_{nh}|{\delta }_h(\nabla u_n){|}^2\mathrm{d}x\right)}^{\frac{p}{2}}.\end{array}$$

$$\nabla u_n\to \nabla u\text{mbox}in{L}_{\mathrm{loc}}^q(\Omega )\forall q\in (1,p).$$

$$\frac{1}{{\zeta }_1}+{\theta }_1<\frac{1}{r^{\prime }}<1-\frac{p}{p^{\ast }},\frac{1}{{\zeta }_2}+{\theta }_2<\frac{1}{s^{\prime }}<1-\frac{q}{q^{\ast }},$$

$$\{\begin{array}{rlrl}-{\Delta }_{p}u& =f(x,u+{\epsilon }_n,v,\nabla u,\nabla v)& & \text{mbox}in{\mathbb{R}}^N,\\ -{\Delta }_{q}v& =g(x,u,v+{\epsilon }_n,\nabla u,\nabla v)& & \text{mbox}in{\mathbb{R}}^N,\\ u,v& >0& & \text{mbox}in{\mathbb{R}}^N,\end{array}$$

$$(u_n,v_n)$$

Full text

††MSC 2020: 35J15, 35J47, 35D30, 35D35.††Keywords: a priori estimates, compactness, convection terms, strong solutions.††✉ Corresponding author: Umberto Guarnotta (umberto.guarnotta@studium.unict.it).

1. Introduction

In this brief note, whose starting point is [10], we consider the problem

[TABLE]

where $N\geq 2$, $1<p,q<N$, $\Delta_{r}z:={\rm div}(|\nabla z|^{r-2}\nabla z)$ denotes the $r$-Laplacian of $z$ for $1<r<+\infty$, while $f,g:\mathbb{R}^{N}\times(0,+\infty)^{2}\times\mathbb{R}^{2N}\to(0,+\infty)$ are Carathéodory functions satisfying the following hypotheses.

1. $\underline{\rm H_{1}(f)}$

There exist $\alpha_{1}\in(-1,0]$, $\beta_{1},\delta_{1}\in[0,q-1)$, $\gamma_{1}\in[0,p-1)$, $m_{1},\hat{m}_{1}>0$, and a measurable $a_{1}:\mathbb{R}^{N}\to(0,+\infty)$ such that

[TABLE]

in $\mathbb{R}^{N}\times(0,+\infty)^{2}\times\mathbb{R}^{2N}$. Moreover, $\displaystyle{\operatorname*{ess\,inf}_{B_{\rho}}a_{1}>0}$ for all $\rho>0$.

1. $\underline{\rm H_{1}(g)}$

There exist $\beta_{2}\in(-1,0]$, $\alpha_{2},\gamma_{2}\in[0,p-1)$, $\delta_{2}\in[0,q-1)$, $m_{2},\hat{m}_{2}>0$, and a measurable $a_{2}:\mathbb{R}^{N}\to(0,+\infty)$ such that

[TABLE]

in $\mathbb{R}^{N}\times(0,+\infty)^{2}\times\mathbb{R}^{2N}$. Moreover, $\displaystyle{\operatorname*{ess\,inf}_{B_{\rho}}a_{2}>0}$ for all $\rho>0$.

1. $\underline{\rm H_{1}(a)}$

There exist $\zeta_{1},\zeta_{2}\in(N,+\infty]$ such that $a_{i}\in L^{1}(\mathbb{R}^{N})\cap L^{\zeta_{i}}(\mathbb{R}^{N})$, $i=1,2$, where

[TABLE]

with

[TABLE]

1. $\underline{\rm H_{2}}$

If $\eta_{1}:=\max\{\beta_{1},\delta_{1}\}$ and $\eta_{2}:=\max\{\alpha_{2},\gamma_{2}\}$ then

[TABLE]

In the sequel, by ${\rm H_{1}}$ we mean the set of hypotheses $\rm H_{1}(f)$, $\rm H_{1}(g)$, and $\rm H_{1}(a)$.

Unlike [10], we restrict our attention to weak solutions instead of strong ones. This allows us to weaken several assumptions, in particular:

• •

$p,q>2-\frac{1}{N}$ is relaxed to $p,q>1$;

• •

hypothesis ${\rm H_{3}}$ (cf. [10, p. 743]), ensuring a high local summability of reactions, is dropped;

• •

no local summability for $a_{1},a_{2}$ is required (cf. $\rm H_{1}(f)$–$\rm H_{1}(g)$).

Let us briefly comment these improvements, focusing our attention on the first equation of (${\rm P}$), since our arguments are scalar (i.e., they do not depend on the system structure). The lower bound on $p$ was used to prove [10, Lemma 2.1] and, jointly with ${\rm H_{3}}$, to guarantee the strong convergence of $\{|\nabla u_{n}|^{p-2}\nabla u_{n}\}$ in $L^{p^{\prime}}_{\rm loc}(\mathbb{R}^{N})$, being $\{u_{n}\}$ a sequence of solutions (precisely, their first components) to problems approximating (${\rm P}$) (see [10, formula (4.5)]). On the other hand, in the assumption $a_{1}\in L^{s_{p}}_{\rm loc}(\mathbb{R}^{N})$, the number $s_{p}$ was supposed to be greater than $p^{\prime}N$, which ensures the local $C^{1,\alpha}$ regularity in [10, Lemma 3.1] by [5]. Actually, the same result can be obtained by requiring merely $s_{p}>N$, according to [12], so that one can take $s_{p}:=\zeta_{1}$ with no additional assumptions, where $\zeta_{1}$ stems from $\rm H_{1}(a)$. Another consequence of using [12] instead of [5] is that ${\rm H_{3}^{\prime}}$ in [10, Remark 4.4] can be relaxed to

[TABLE]

Convergence of gradient terms comes into play whenever a second-order differential problem needs to be approximated: this can occur because of lack of ellipticity (or uniform ellipticity) of the principal part and/or presence of non-smooth reaction terms; see, e.g., [11, Theorem 3.3]. An approximation procedure is necessary also in the context of singular problems, that is, problems whose reaction term blows up when the solution approaches to zero, as (${\rm P}$); for an account on this topic, vide [8, 9].

Here, we proceed as follows. Lemma 2.1 of [10] is restated in a new, general fashion and its proof is given patterned after the one in [10]; see Lemma 2.3. Next, we prove a compactness result (Lemma 2.4) for gradient terms, which is self-contained (unlike the alternative proofs mentioned in Remark 2.5) and relies on the basic Riesz-Fréchet-Kolmogorov $L^{p}$-compactness criterion. Finally, it is shown (in Theorem 2.6) how to modify the proof of [10, Lemma 4.1] to get a weak solution under ${\rm H_{1}}$–$\rm H_{2}$, besides commenting the unavailability of [10, Lemma 4.3], pertaining strong solutions, in this context (see Remark 2.7).

Notations

Hereafter $\Omega$ is a bounded domain of $\mathbb{R}^{N}$, $N\geq 2$, and $p\in(1,+\infty)$. We set $p^{\prime}:=\frac{p}{p-1}$ and, provided $p<N$, $p^{*}:=\frac{Np}{N-p}$. If $p\geq N$ then $p^{*}:=\infty$ and $(p^{*})^{\prime}:=1$. We write $\operatorname*{dist}(A,B)$ for the distance between the sets $A,B\subseteq\mathbb{R}^{N}$. The symbol $B_{R}(x)$ indicates the (open) ball of center $x\in\mathbb{R}^{N}$ and radius $R>0$, while $\overline{B}_{R}(x)$ stands for the closure of $B_{R}(x)$. By $B_{R}(x)\Subset\Omega$ we mean $\overline{B}_{R}(x)\subseteq\Omega$. The center of any ball will be omitted when it is irrelevant.

Given $f\in L^{1}_{\rm loc}(\mathbb{R}^{N})$, a distributional solution to

[TABLE]

is a function $u\in W^{1,p}_{\rm loc}(\mathbb{R}^{N})$ such that

[TABLE]

If $f\in L^{(p^{*})^{\prime}}(\mathbb{R}^{N})$, by weak solution to (1.1) we mean a function $u\in\mathcal{D}^{1,p}_{0}(\mathbb{R}^{N})$ satisfying (1.2) for all $\varphi\in\mathcal{D}^{1,p}_{0}(\mathbb{R}^{N})$. Analogous definitions hold when $\Omega$ replaces $\mathbb{R}^{N}$ or $f$ depends on $u,\nabla u$.

The number $C>0$ represents a suitable constant, which may change its value at each passage. For further details, we address the reader to [10, Section 2].

2. Main results

For any $f\in L^{2}_{\rm loc}(\Omega)$ we define the nonlinear potential

[TABLE]

We recall the following result, provided in [6].

Proposition 2.1**.**

Let $u\in W^{1,p}_{\rm loc}(\Omega)$ be a distributional solution to

[TABLE]

with $f\in L^{r}_{\rm loc}(\Omega)$, $r:=\max\{2,(p^{*})^{\prime}\}$. Then there exists $C=C(N,p)>0$ such that

[TABLE]

for any $B_{2R}\Subset\Omega$.

Remark 2.2**.**

As observed in [6, p. 1363], the condition $r\geq(p^{*})^{\prime}$ is not used to prove the result, but it guarantees that $u$ is a weak solution, and not merely a very weak solution; in the latter case, an approximation procedure yields the existence of a very weak solution $u\in W^{1,p-1}(\Omega)$ of (2.1). For a thorough treatment on approximable solutions, see [3].

Lemma 2.3**.**

Let $u\in\mathcal{D}^{1,p}_{0}(\mathbb{R}^{N})$ be a distributional solution to

[TABLE]

with $f\in L^{r}(\mathbb{R}^{N})$, $r>N$. Then $\nabla u\in L^{\infty}(\mathbb{R}^{N})$. More precisely, there exists $C=C(N,p)>0$ such that

[TABLE]

Proof.

Pick any $x\in\mathbb{R}^{N}$. By Proposition 2.1 and Hölder’s inequality (with exponents $\frac{r}{2}$ and $\frac{r}{r-2}$), after observing that $r>N\geq\max\{2,(p^{*})^{\prime}\}$, we get

[TABLE]

Taking the supremum in $x\in\mathbb{R}^{N}$ on the left yields the conclusion. ∎

For every $u\in W^{1,p}_{\rm loc}(\Omega)$, $x\in B_{R}\Subset\Omega$, and $h\in\mathbb{R}^{N}$ such that $|h|<\operatorname*{dist}(B_{R},\partial\Omega)$, we set

[TABLE]

Analogous definitions hold for vector-valued functions.

Lemma 2.4**.**

Let $\{u_{n}\}\subseteq W^{1,p}_{\rm loc}(\Omega)$ and $\{f_{n}\}\subseteq L^{r^{\prime}}_{\rm loc}(\Omega)$, $r\in(1,p^{*})$, be such that $u_{n}$ is a distributional solution to

[TABLE]

for all $n\in\mathbb{N}$. Suppose that:

1. ${\rm(K_{1})}$

$\{\nabla u_{n}\}\;\;\mbox{is bounded in}\;\;L^{p}_{\rm loc}(\Omega);$** 2. ${\rm(K_{2})}$

$\{f_{n}\}\;\;\mbox{is bounded in}\;\;L^{r^{\prime}}_{\rm loc}(\Omega);$** 3. ${\rm(K_{3})}$

$u_{n}\to u\;\;\mbox{in}\;\;L^{p}_{\rm loc}(\Omega)\cap L^{r}_{\rm loc}(\Omega).$**

Then $\{\nabla u_{n}\}$ admits a strongly convergent subsequence in $L^{p}_{\rm loc}(\Omega)$.

Proof.

Fix $R>0$ such that $B_{R}\Subset\Omega$. A density argument produces

[TABLE]

for any $n\in\mathbb{N}$ and $\varphi\in W^{1,p}_{0}(B_{R})$. Now pick $t,s>0$ such that $B_{t}\Subset B_{s}\Subset B_{R}$ and $\eta\in C^{\infty}_{c}(B_{s})$ such that $0\leq\eta\leq 1$, $\eta\equiv 1$ on $B_{t}$, and $|\nabla\eta|\leq\frac{C}{s-t}$ for some $C>0$. If $V_{n}:=|\nabla u_{n}|^{p-2}\nabla u_{n}$ then using (2.2) with $\varphi:=\eta^{2}\delta_{h}u_{n}$, where $|h|<R-s$, gives

[TABLE]

Next, exploit (2.2) with $\varphi_{-h}$, perform the change of variable $x\mapsto x+h$ on the left-hand side, and recall that $B_{s+|h|}\subseteq B_{R}$, to achieve

[TABLE]

So, subtracting (2.3) from (2.4) yields

[TABLE]

Since $\operatorname*{supp}\eta\subseteq B_{s}$, this entails

[TABLE]

where Hölder’s inequality has been used twice while $C=C(N,t,s)>0$. Notice that, thanks to ${\rm(K_{1})}$–${\rm(K_{3})}$ and [2, Exercise 4.34], the last term of (2.5) vanishes as $h\to 0^{+}$ uniformly in $n$. Let us now distinguish two cases, namely $p\geq 2$ and $p\in(1,2)$.

Case 1. If $p\geq 2$ then

[TABLE]

with $c>0$ small enough; cf. [13, Chapter 12, inequality (I)]. By (2.5)–(2.6) we thus obtain $\delta_{h}(\nabla u_{n})\to 0$ in $L^{p}(B_{t})$ as $h\to 0^{+}$ uniformly in $n$, and the Riesz-Fréchet-Kolmogorov yields the conclusion, because $t>0$ was arbitrary.

Case 2. For $p\in(1,2)$ one has (see [13, Chapter 12, inequality (VII)])

[TABLE]

where $c>0$ is sufficiently small while $W_{nh}:=(1+|\nabla(u_{n})_{h}|^{2}+|\nabla u_{n}|^{2})^{\frac{p-2}{2}}$. Hölder’s inequality with exponents $\frac{2}{p}$ and $\frac{2}{2-p}$, besides ${\rm(K_{1})}$, produce

[TABLE]

Reasoning as in the above case, the conclusion directly follows from (2.5), (2.7), and (2.8). ∎

Remark 2.5**.**

Lemma 2.4 can be proved also (in a less direct way) through a result by Boccardo and Murat [1] which, under the hypotheses of Lemma 2.4, ensures that

[TABLE]

In particular, (2.9) implies $\nabla u_{n}\to\nabla u$ a.e. in $\Omega$. A development of this approach, allowing $q=p$, is contained in [7, Lemma 2.5 and Remark 3]. Another way [4, 11] to get convergence of gradient terms is using a differentiability result for the stress field, i.e., the field whose divergence represents the elliptic operator (as $|\nabla u|^{p-2}\nabla u$ for the $p$-Laplacian). In fact, by Rellich-Kondrachov’s theorem [2, Theorem 9.16], such a differentiability allows to gain compactness.

Theorem 2.6**.**

Let ${\rm H_{1}}$–$\rm H_{2}$ be satisfied. Then problem (${\rm P}$) possesses a weak solution $(u,v)\in\mathcal{D}^{1,p}_{0}(\mathbb{R}^{N})\times\mathcal{D}^{1,q}_{0}(\mathbb{R}^{N})$.

Proof.

The reasoning is patterned after that of [10, Lemma 4.1], so here we only sketch it. Pick $r,s>1$ such that

[TABLE]

which is possible thanks to $\rm H_{1}(a)$. Fix $\rho>0$ and define $\varepsilon_{n}:=\frac{1}{n}$, $n\in\mathbb{N}$. By [10, Lemmas 3.5–3.8], for all $n\in\mathbb{N}$ there exists $(u_{n},v_{n})\in(\mathcal{D}^{1,p}_{0}(\mathbb{R}^{N})\times\mathcal{D}^{1,q}_{0}(\mathbb{R}^{N}))\cap C^{1,\alpha}_{\rm loc}(\mathbb{R}^{N})^{2}$ solution to

[TABLE]

such that the following properties hold true, with appropriate $(u,v)\in\mathcal{D}^{1,p}_{0}(\mathbb{R}^{N})\times\mathcal{D}^{1,q}_{0}(\mathbb{R}^{N})$ and $M,\sigma_{2\rho}>0$:

[TABLE]

Hence, $\rm H_{1}(f)$ and (2.11) yield, for almost every $x\in B_{2\rho}$,

[TABLE]

By (2.11) the sequence $\{(\nabla u_{n},\nabla v_{n})\}$ is bounded in $L^{p}(B_{2\rho})\times L^{q}(B_{2\rho})$. Exploiting $\rm H_{1}(a)$, (2.10), and (2.12) we thus see that

[TABLE]

Accordingly, Lemma 2.4, with $\Omega:=B_{2\rho}$, besides (2.11), produces $\nabla u_{n}\to\nabla u$ in $L^{p}(B_{\rho})$. Now the proof goes on exactly as in [10, Lemma 4.1], ensuring that $(u,v)$ is a distributional solution to (${\rm P}$). The conclusion is achieved by invoking [10, Lemma 4.2], which shows that any distributional solution to (${\rm P}$) is actually a weak one. ∎

Remark 2.7**.**

An advantage of using differentiability results for the stress field (see Remark 2.5) in this context is the possibility to obtain strong solutions to (${\rm P}$), as done in [10, Lemma 4.3]: indeed, otherwise we do not know how to give a pointwise (a.e.) sense to the $p$-Laplacian operator, seen as the divergence of the stress field $|\nabla u|^{p-2}\nabla u$. This issue is linked to a well-known conjecture for (2.1), which can be stated as

[TABLE]

For a discussion about this conjecture, see [11, Section 1].

Acknowledgments

The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

They were supported by the following research projects: 1) PRIN 2017 ‘Nonlinear Differential Problems via Variational, Topological and Set-valued Methods’ (Grant no. 2017AYM8XW) of MIUR; 2) ‘MO.S.A.I.C.’ PRA 2020–2022 ‘PIACERI’ Linea 2 (S.A. Marano) and Linea 3 (U. Guarnotta) of the University of Catania. U. Guarnotta also acknowledges the support of GNAMPA-INdAM Project CUP_E55F22000270001.

Conflict of interest statement. On behalf of all authors, the corresponding author states that there is no conflict of interest.