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# Local boundedness results for very weak solutions of double obstacle problems

Journal of Inequalities and Applications20122012:43

https://doi.org/10.1186/1029-242X-2012-43

• Received: 17 May 2011
• Accepted: 27 February 2012
• Published:

## Abstract

This article mainly concerns double obstacle problems for second order divergence type elliptic equation divA(x, u, u) = divf(x). We give local boundedness for very weak solutions of double obstacle problems.

## Keywords

• double obstacle problems
• local boundedness
• elliptic equation

## 1 Introduction

Let Ω be a bounded open set of R n , n 2. We consider the second order divergence type elliptic equation (also called A-harmonic equation or Leray-Lions equation)
$\mathsf{\text{div}}A\left(x,\phantom{\rule{2.77695pt}{0ex}}u\left(x\right),\phantom{\rule{2.77695pt}{0ex}}\nabla u\left(x\right)\right)=\mathsf{\text{div}}f\left(x\right).$
(1.1)
where A : Ω × R × R n R n is a Carathéodory function satisfying the coercivity and growth conditions: for almost all x Ω, all u R, and ξ R n ,
1. (i)

A(x, u, ξ), ξ〉 ≥ α |ξ| p ,

2. (ii)

|A(x, u, ξ)| ≤ β1 |ξ|p-1+ β2 |u| m + h(x),

where α > 0, β1 and β2 are some nonnegative constants, 1 < p < n, $p-1\le m\le \frac{n\left(p-1\right)}{n-r}$ and $h\left(x\right)\in {L}_{\mathsf{\text{loc}}}^{s/\left(p-1\right)}\left(\mathrm{\Omega }\right)$, $f\left(x\right)\in {\left({L}_{\mathsf{\text{loc}}}^{s/\left(p-1\right)}\left(\mathrm{\Omega }\right)\right)}^{n}$ for some s > r.

Suppose that ψ1, ψ2 are any functions in Ω with values in R {±∞}, and that θ W1,r(Ω) with max{1, p - 1} < r ≤ p. Let
${K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}\left(\mathrm{\Omega }\right)=\left\{v\in {W}^{1,r}\left(\mathrm{\Omega }\right):{\psi }_{1}\le v\le {\psi }_{2},\phantom{\rule{2.77695pt}{0ex}}a.e\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{0.3em}{0ex}}v-\theta \in {W}_{0}^{1,r}\left(\mathrm{\Omega }\right)\right\}.$

The functions ψ1, ψ2 are two obstacles and θ determines the boundary values.

For any $u,\phantom{\rule{0.3em}{0ex}}v\in {K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}\left(\mathrm{\Omega }\right)$, we introduce the Hodge decomposition for $|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)\in {L}^{\frac{r}{r-p+1}}$, see :
$|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)=\nabla {\varphi }_{v,u}+{h}_{v,u}$
(1.2)
where ${\varphi }_{v,u}\in {W}_{0}^{1,\frac{r}{r-p+1}}\left(\mathrm{\Omega }\right)$, ${h}_{v,u}\in {L}^{\frac{r}{r-p+1}}\left(\mathrm{\Omega }\right)$ is a divergence-free vector field and the following estimates hold:
${∥\nabla {\varphi }_{v,u}∥}_{\frac{r}{r-p+1}}\le c{∥\nabla \left(v-u\right)∥}_{r}^{r-p+1},$
(1.3)
${∥{h}_{v,u}∥}_{\frac{r}{r-p+1}}\le c\left(p-r\right){∥\nabla \left(v-u\right)∥}_{r}^{r-p+1},$
(1.4)

where c = c(n, p) is a constant depending only on n and p.

Definition 1.1. A very weak solution to the ${K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}\left(\mathrm{\Omega }\right)$-double obstacle problem for the A-harmonic Equation (1.1) is a function $u\in {K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}\left(\mathrm{\Omega }\right)$ such that
$\underset{\mathrm{\Omega }}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)-h⟩dx\ge \underset{\mathrm{\Omega }}{\int }⟨f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)-h⟩dx,$
(1.5)

whenever $v\in {K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}\left(\mathrm{\Omega }\right)$.

The obstacle problem has a strong background, and has many applications in physics and engineering. The local boundedness for solutions of obstacle problems plays a central role in many aspects. Based on the local boundedness, we can further study the regularity of the solutions. In , Gao et al. first considered the local boundedness for very weak solutions of obstacle problems to the A-harmonic equation in 2010. Precisely, the authors considered the local boundedness for very weak solutions of Kψ,θ(Ω)-obstacle problems to the A-harmonic equation div A(x, u(x)) = 0 with the obstacle function ψ 0, where operator A satisfies conditions 〈A(x, ξ), ξ α|ξ| p and |A(x, ξ)| ≤ β|ξ|p-1with A(x, 0) = 0. For the property of weak solutions of nonlinear elliptic equations, we refer the reader to .

In this article, we continue to consider the local boundedness property. Under some general conditions (i) and (ii) given above on the operator A, we obtain a local boundedness result for very weak solutions of ${K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}$-double obstacle problems to the A-harmonic Equation (1.1).

Theorem. Let operator A satisfies conditions (i) and (ii). Suppose that ${\psi }_{1},{\psi }_{2}\in {W}_{\mathsf{\text{lo}}\phantom{\rule{1em}{0ex}}\mathsf{\text{c}}}^{1,\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. Then a very weak solution u to the ${K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}\left(\mathrm{\Omega }\right)$-obstacle problem of (1.1) is locally bounded.

Remark. Since we have assumed that operator A satisfies the conditions (ii), in the proof of the theorem, we have to estimate the integral of some power of |u| by means of |u|. To deal with this difficulty, we will make use of the Sobolev inequality that was used in .

## 2 Preliminary knowledge and lemmas

We give some symbols and preliminary lemmas used in the proof. If x0 Ω and t > 0, then B t denotes the ball of radius t centered at x0. For a function u(x) and k > 0, let
$\begin{array}{c}{A}_{k}=\left\{x\in \mathrm{\Omega }:|u\left(x\right)|\phantom{\rule{2.77695pt}{0ex}}>k\right\},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}{A}_{k}^{+}=\left\{x\in \mathrm{\Omega }:u\left(x\right)>k\right\},\\ {A}_{k,t}={A}_{k}\cap {B}_{t},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}{A}_{k,t}^{+}={A}_{k}^{+}\cap {B}_{t}.\end{array}$

Moreover, if s < n, s* is always the real number satisfying 1/s* = 1/s - 1/n. Let t k (u) = min{u, k}.

Lemma 2.1.  Let f(τ) be a nonnegative bounded function defined for 0 R0 ≤ t ≤ R1. Suppose that for R0 ≤ τ < t ≤ R1 one has
$f\left(\tau \right)\le A{\left(t\phantom{\rule{2.77695pt}{0ex}}-\tau \right)}^{-\alpha }+B\phantom{\rule{2.77695pt}{0ex}}+\theta f\left(t\right),$
where A, B, α, θ are nonnegative constants and θ < 1. Then there exists a constant c = c(α, θ), depending only on α and θ, such that for every ρ, R, R0 ≤ ρ < R ≤ R1, one has
$f\left(\rho \right)\le c\left[A{\left(R\phantom{\rule{2.77695pt}{0ex}}-\rho \right)}^{-\alpha }+B\right].$
Definition 2.2.  A function $u\in {W}_{\mathsf{\text{lo}}\phantom{\rule{1em}{0ex}}\mathsf{\text{c}}}^{1,m}\left(\mathrm{\Omega }\right)$ belongs to the class B(Ω, γ, m, k0), if for all k > k0, k0 > 0 and all B ρ = B ρ (x0), B ρ-ρσ = B ρ-ρσ (x0), B R = B R (x0), one has
$\underset{{A}_{k,\rho -\rho \sigma }^{+}}{\int }|\nabla u{|}^{m}dx\phantom{\rule{2.77695pt}{0ex}}\le \gamma \left\{{\sigma }^{-m}{\rho }^{-m}\underset{{A}_{k,\rho }^{+}}{\int }{\left(u-k\right)}^{m}dx\phantom{\rule{2.77695pt}{0ex}}+|{A}_{k,\rho }^{+}|\right\},$

for R/ 2 ≤ ρ - ρσ < ρ < R, m < n, where $\left|{A}_{k,\rho }^{+}\right|$ is the n-dimensional Lebesgue measure of the set ${A}_{k,\rho }^{+}$.

Lemma 2.3.  Suppose that u(x) is an arbitrary function belonging to the class B(Ω, γ, m, k0) and B R Ω. Then one has
$\underset{{B}_{R/2}}{\text{max}}u\left(x\right)\le c,$

in which the constant c is determined only by γ, m, k0, R, ||u|| m .

## 3 Proof of theorem

Proof. Let u be a very weak solution to the ${K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}\left(\mathrm{\Omega }\right)$-obstacle problem for the A-harmonic Equation (1.1). Let ${B}_{{R}_{1}}\subset \subset \mathrm{\Omega }$ and 0 < R1/ 2 τ < t R1 be arbitrarily fixed. Fix a cutoff function $\varphi \in {C}_{0}^{\mathrm{\infty }}\left({B}_{{R}_{1}}\right)$, such that
$\mathsf{\text{supp}}\varphi \subset {B}_{t},\phantom{\rule{2.77695pt}{0ex}}0\le \varphi \le 1,\phantom{\rule{2.77695pt}{0ex}}\varphi \equiv 1\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}{B}_{\tau },\phantom{\rule{2.77695pt}{0ex}}|\nabla \varphi |\phantom{\rule{0.3em}{0ex}}\le 2{\left(t-\tau \right)}^{-1}.$
(3.1)
If ψ2 is an arbitrary function in Ω with values in R {+∞}, consider the function
$v=u-{\varphi }^{r}\left(u-{\psi }_{k}\right),$
(3.2)
where
${\psi }_{k}=\text{min}\left\{\text{max}\left\{{\psi }_{1},\phantom{\rule{2.77695pt}{0ex}}{t}_{k}\left(u\right)\right\},\phantom{\rule{2.77695pt}{0ex}}{\psi }_{2}\right\},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}{t}_{k}\left(u\right)=\text{min}\left\{u,\phantom{\rule{2.77695pt}{0ex}}k\right\},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}k\ge 0.$
It is easy to see ψ1ψ k ψ2. Now, $v\in {K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}\left(\mathrm{\Omega }\right)$; indeed, since $u\in {K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}\left(\mathrm{\Omega }\right)$ and $\varphi \in {C}_{0}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$, then
$\begin{array}{c}v-\theta =u-\theta -{\varphi }^{r}\left(u-{\psi }_{k}\right)\in {W}_{0}^{1,r}\left(\mathrm{\Omega }\right),\\ v-{\psi }_{1}=u-{\psi }_{1}-{\varphi }^{r}\left(u-{\psi }_{k}\right)\ge \left(1-{\varphi }^{r}\right)\left(u-{\psi }_{1}\right)\ge 0\phantom{\rule{0.3em}{0ex}}\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.in}}\phantom{\rule{0.3em}{0ex}}\mathrm{\Omega },\\ v-{\psi }_{2}=u-{\psi }_{2}-{\varphi }^{r}\left(u-{\psi }_{k}\right)\le \left(1-{\varphi }^{r}\right)\left(u-{\psi }_{2}\right)\le 0\phantom{\rule{0.3em}{0ex}}\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.in}}\phantom{\rule{0.3em}{0ex}}\mathrm{\Omega }.\end{array}$
(3.3)
For any fixed k > 0, let
${v}_{0}=\left\{\begin{array}{cc}\hfill u,\hfill & \hfill \mathsf{\text{if}}u\phantom{\rule{2.77695pt}{0ex}}\le k,\hfill \\ \hfill v,\hfill & \hfill \mathsf{\text{if}}u\phantom{\rule{2.77695pt}{0ex}}>k.\hfill \end{array}\right\$
It is easy to see that ${v}_{0}\in {K}_{{\psi }_{1},{\psi }_{2}}^{\theta ,r}\left(\mathrm{\Omega }\right)$. Then by Definition 1.1 we have
$\underset{\mathrm{\Omega }}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left({v}_{0}-u\right){|}^{r-p}\nabla \left({v}_{0}-u\right)-{\stackrel{̃}{h}}_{v,u}⟩dx\ge \underset{\mathrm{\Omega }}{\int }⟨f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left({v}_{0}-u\right){|}^{r-p}\nabla \left({v}_{0}-u\right)-{\stackrel{̃}{h}}_{v,u}⟩dx.$
(3.4)
If uk, then ${\stackrel{̃}{h}}_{v,u}=0,\phantom{\rule{2.77695pt}{0ex}}\nabla {\stackrel{̃}{\varphi }}_{v,u}=0$; If u > k, then ${\stackrel{̃}{h}}_{v,u}={h}_{v,u}$, ${\stackrel{̃}{\varphi }}_{v,u}={\varphi }_{v,u}$. It's derived from the uniqueness of Hodge decomposition. This means that
$\begin{array}{c}\phantom{\rule{1em}{0ex}}\underset{{A}_{k,t}^{+}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}{h}_{v,u}⟩dx+\underset{{A}_{k,t}^{+}}{\int }⟨f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left({v}_{0}-u\right){|}^{r-p}\nabla \left({v}_{0}-u\right)-{\stackrel{̃}{h}}_{v,u}⟩dx\\ \le \underset{\mathrm{\Omega }}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}{h}_{v,u}⟩dx+\underset{\mathrm{\Omega }}{\int }⟨f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left({v}_{0}-u\right){|}^{r-p}\nabla \left({v}_{0}-u\right)-{\stackrel{̃}{h}}_{v,u}⟩dx\\ \le \underset{\mathrm{\Omega }}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left({v}_{0}-u\right){|}^{r-p}\nabla \left({v}_{0}-u\right)⟩dx\\ =\left(\underset{\mathrm{\Omega }\cap \left\{u\le k\right\}}{\int }+\underset{\mathrm{\Omega }\cap \left\{u>k\right\}}{\int }\right)⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left({v}_{0}-u\right){|}^{r-p}\nabla \left({v}_{0}-u\right)⟩dx\\ =\underset{\mathrm{\Omega }\cap \left\{u>k\right\}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left({v}_{0}-u\right){|}^{r-p}\nabla \left({v}_{0}-u\right)⟩dx\\ =\underset{{A}_{k,t}^{+}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)⟩dx.\end{array}$
(3.5)
Let
$E\left(v,\phantom{\rule{2.77695pt}{0ex}}u\right)=|{\varphi }^{r}\nabla u{|}^{r-p}{\varphi }^{r}\nabla u+|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right).$
(3.6)
By an elementary inequality [, P. 271, (4.1)],
$||X{|}^{-\epsilon }X-|Y{|}^{-\epsilon }Y|\phantom{\rule{0.3em}{0ex}}\le {2}^{\epsilon }\frac{1+\epsilon }{1-\epsilon }|X-Y{|}^{1-\epsilon },\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}X,\phantom{\rule{2.77695pt}{0ex}}Y\in {R}^{n},\phantom{\rule{2.77695pt}{0ex}}0\le \epsilon <1,$
(3.7)
$\nabla v=\nabla u-{\varphi }^{r}\left(\nabla u-\nabla {\psi }_{k}\right)-r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right),$
(3.8)
one can derive that
$|E\left(v,\phantom{\rule{2.77695pt}{0ex}}u\right)|\le {2}^{p-r}\frac{p-r+1}{r-p+1}|{\varphi }^{r}\nabla {\psi }_{k}-r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right){|}^{r-p+1}.$
(3.9)
We get from the definition of E(v, u) and (3.5) that
$\begin{array}{c}\phantom{\rule{1em}{0ex}}\underset{{A}_{k,t}^{+}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}|{\varphi }^{r}\nabla u{|}^{r-p}{\varphi }^{r}\nabla u⟩dx\\ =\underset{{A}_{k,t}^{+}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}E\left(v,\phantom{\rule{2.77695pt}{0ex}}u\right)⟩dx-\underset{{A}_{k,t}^{+}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)⟩dx\\ \le \underset{{A}_{k,t}^{+}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}E\left(v,\phantom{\rule{2.77695pt}{0ex}}u\right)⟩dx-\underset{{A}_{k,t}^{+}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}{h}_{v,u}⟩dx-\underset{{A}_{k,t}^{+}}{\int }⟨f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}E\left(v,\phantom{\rule{2.77695pt}{0ex}}u\right)⟩dx\\ \phantom{\rule{1em}{0ex}}+\underset{{A}_{k,t}^{+}}{\int }⟨f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}|{\varphi }^{r}\nabla u{|}^{r-p}{\varphi }^{r}\nabla u⟩dx+\underset{{A}_{k,t}^{+}}{\int }⟨f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}{h}_{v,u}⟩dx\\ ={I}_{1}+{I}_{2}+{I}_{3}+{I}_{4}+{I}_{5}.\end{array}$
(3.10)
We now estimate the left-hand side and the right-hand side of (3.10), respectively. Firstly,
$\begin{array}{ll}\hfill \underset{{A}_{k,t}^{+}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}|{\varphi }^{r}\nabla u{|}^{r-p}{\varphi }^{r}\nabla u⟩dx& \ge \underset{{A}_{k,\tau }^{+}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}|\nabla u{|}^{r-p}\nabla u⟩dx\phantom{\rule{2em}{0ex}}\\ \ge \alpha \underset{{A}_{k,\tau }^{+}}{\int }|\nabla u{|}^{r}dx,\phantom{\rule{2em}{0ex}}\end{array}$
(3.11)
here we have used condition (i). Secondly, by condition (ii) and (3.9),
$\begin{array}{c}|{I}_{1}|\phantom{\rule{2.77695pt}{0ex}}=\left|\underset{{A}_{k,t}}{\int }⟨A\left(x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}\nabla u\right),\phantom{\rule{2.77695pt}{0ex}}E\left(v,\phantom{\rule{2.77695pt}{0ex}}u\right)⟩dx\right|\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le \underset{{A}_{k,t}^{+}}{\int }\left[{\beta }_{1}|\nabla u{|}^{p-1}+{\beta }_{2}|u{|}^{m}+{h}_{1}\right]|E\left(v,\phantom{\rule{2.77695pt}{0ex}}u\right)|dx\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le {2}^{p-r}\frac{p-r+1}{r-p+1}\underset{{A}_{k,t}^{+}}{\int }\left[{\beta }_{1}|\nabla u{|}^{p-1}+{\beta }_{2}|u{|}^{m}+{h}_{1}\right]|{\varphi }^{r}\nabla {\psi }_{k}-r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right){|}^{r-p+1}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}{C}_{1}{\beta }_{1}\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{p-1}|{\varphi }^{r}\nabla {\psi }_{k}{|}^{r-p+1}+{C}_{1}{\beta }_{1}\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{p-1}|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right){|}^{r-p+1}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+{C}_{1}{\beta }_{2}\underset{{A}_{k,t}^{+}}{\int }|u{|}^{m}|{\varphi }^{r}\nabla {\psi }_{k}{|}^{r-p+1}+{C}_{1}{\beta }_{2}\underset{{A}_{k,t}^{+}}{\int }|u{|}^{m}|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right){|}^{r-p+1}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+{C}_{1}\underset{{A}_{k,t}^{+}}{\int }|h|{\left|{\varphi }^{r}\nabla {\psi }_{k}\right|}^{r-p+1}+{C}_{1}\underset{{A}_{k,t}^{+}}{\int }|h|{\left|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right)\right|}^{r-p+1}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}{I}_{11}+{I}_{12}+{I}_{13}+{I}_{14}+{I}_{15}+{I}_{16},\end{array}$
(3.12)
where ${C}_{1}={2}^{p-r}\frac{p-r+1}{r-p+1}$. By Young's inequality
and $\frac{p-1}{r}+\frac{r-p+1}{r}=1$, we have the estimates
$|{I}_{11}|\le {C}_{1}{\beta }_{1}\left[\epsilon \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}dx\right],$
(3.13)
$|{I}_{12}|\le {C}_{1}{\beta }_{2}\left[\epsilon \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)\underset{{A}_{k,t}^{+}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right){|}^{r}dx\right],$
(3.14)
$|{I}_{13}|\le {C}_{1}\left[\epsilon \underset{{A}_{k,t}^{+}}{\int }|u{|}^{\frac{mr}{p-1}}dx+C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}dx\right],$
(3.15)

where we have used |ψ k | ≤ |ψ1| + |ψ2| in ${A}_{k,t}^{+}$.

We observe now that, if w W1,p(B t ) and | suppw| ≤ 1/ 2|B t |, then we have the Sobolev inequality (see also ),
${\left(\underset{{B}_{t}}{\int }|w{|}^{p*}dx\right)}^{p/p*}\le {c}_{1}\left(n,p\right)\underset{{B}_{t}}{\int }|\nabla w{|}^{p}dx.$
(3.16)
Set
${g}_{k}\left(u\right)=\left\{\begin{array}{cc}\hfill u,\hfill & \hfill \mathsf{\text{if}}u\phantom{\rule{2.77695pt}{0ex}}\le k,\hfill \\ \hfill 0,\hfill & \hfill \mathsf{\text{if}}u\phantom{\rule{2.77695pt}{0ex}}>k.\hfill \end{array}\right\$
Since $p-1\le m\le \frac{n\left(p-1\right)}{n-r}$ by assumption, then $r\le \frac{mr}{p-1}\le {r}^{*}$. (3.16) implies
$\begin{array}{c}\phantom{\rule{1em}{0ex}}\underset{{A}_{k,t}^{+}}{\int }|u{|}^{\frac{mr}{p-1}}dx=\underset{{B}_{t}}{\int }|u-{g}_{k}\left(u\right){|}^{\frac{mr}{p-1}}dx\\ \le \phantom{\rule{2.77695pt}{0ex}}||u-{g}_{k}\left(u\right)|{|}_{{r}^{*}}^{\frac{mr}{p-1}-r}|{B}_{t}{|}^{1-\frac{mr}{\left(p-1\right)r*}}{\left(\underset{{B}_{t}}{\int }|u-{g}_{k}\left(u\right){|}^{r*}dx\right)}^{r\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}r*}\\ \le \phantom{\rule{2.77695pt}{0ex}}{c}_{1}\phantom{\rule{2.77695pt}{0ex}}\left(n,\phantom{\rule{2.77695pt}{0ex}}p\right)||u-{g}_{k}\left(u\right)|{|}_{{r}^{*}}^{\frac{mr}{p-1}-r}|{B}_{t}{|}^{1-\frac{mr}{\left(p-1\right)r*}}\underset{{B}_{t}}{\int }|\nabla \left(u-{g}_{k}\left(u\right)\right){|}^{r}dx\\ =\phantom{\rule{2.77695pt}{0ex}}{c}_{1}\phantom{\rule{2.77695pt}{0ex}}\left(n,\phantom{\rule{2.77695pt}{0ex}}p\right)||u-{g}_{k}\left(u\right)|{|}_{{r}^{*}}^{\frac{mr}{p-1}-r}|{B}_{t}{|}^{1-\frac{mr}{\left(p-1\right){r}^{*}}}\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx,\end{array}$
(3.17)
provided that $|\text{supp(}u–{g}_{k}\left(u\right)\text{)}{|}_{B}{}_{t}|\le \text{1}/\text{2}|{B}_{t}|$. Since $\text{sup}\mathsf{\text{p}}\left(u-{g}_{k}\left(u\right)\right){{|}_{B}}_{{}_{t}}\subset {A}_{k,t}^{+}$, then |supp$|\text{sup}\mathsf{\text{p}}\left(u-{g}_{k}\left(u\right)\right){{|}_{B}}_{{}_{t}}|\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}|{A}_{k,t}^{+}|$. On the other hand, we have
$||u|{|}_{{r}^{*},{B}_{t}}^{{r}^{*}}=\underset{{B}_{t}}{\int }{u}^{{r}^{*}}dx\ge \underset{{A}_{k,t}^{+}}{\int }|u{|}^{{r}^{*}}dx\ge {k}^{{r}^{*}}|{A}_{k,t}^{+}|.$
Thus, there exists a constant k0 > 0, such that for all k ≥ k0, we have $\left|{A}_{k,t}^{+}\right|\le 1/2|{B}_{t}|$. We can also suppose that k0 such that
$\underset{{A}_{{k}_{0},t}}{\int }{u}^{{r}^{*}}dx\le 1.$
For such values of k we then have inequality
$\begin{array}{ll}\hfill \underset{{A}_{k,t}^{+}}{\int }|u{|}^{m\frac{r}{p-1}}dx& \le {C}_{2}\left(n,p\right)||u-{g}_{k}\left(u\right)|{|}_{r*}^{\frac{mr}{p-1}-r}|{B}_{t}{|}^{1-\frac{mr}{\left(p-1\right){r}^{*}}}\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\phantom{\rule{2em}{0ex}}\\ \le {C}_{2}\left(n,p\right)||u-{g}_{k}\left(u\right)|{|}_{r*}^{\frac{mr}{p-1}-r}|\mathrm{\Omega }{|}^{1-\frac{mr}{\left(p-1\right){r}^{*}}}\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\phantom{\rule{2em}{0ex}}\\ \le {C}_{2}\left(n,p\right)|\mathrm{\Omega }{|}^{1-\frac{mr}{\left(p-1\right){r}^{*}}}\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\phantom{\rule{2em}{0ex}}\\ ={C}_{3}\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx,\phantom{\rule{2em}{0ex}}\end{array}$
(3.18)
here C3 = C3(n, m, p, r, k0, | Ω|). We derive from (3.15) and (3.18) that
$|{I}_{13}|\phantom{\rule{2.77695pt}{0ex}}\le {C}_{1}{C}_{3}\epsilon \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+{C}_{1}C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}dx.$
(3.19)
$|{I}_{14}|\phantom{\rule{2.77695pt}{0ex}}\le {C}_{1}{C}_{3}\epsilon \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+{C}_{1}C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right){|}^{r}dx.$
(3.20)
I15 and I16 can be estimated as follows:
$|{I}_{15}|\phantom{\rule{2.77695pt}{0ex}}\le {C}_{1}\epsilon \underset{{A}_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx+{C}_{1}C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}dx.$
(3.21)
$|{I}_{16}|\phantom{\rule{2.77695pt}{0ex}}\le {C}_{1}\epsilon \underset{{A}_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx+{C}_{1}C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)\underset{{A}_{k,t}^{+}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right){|}^{r}dx.$
(3.22)
In conclusion, we derive from (3.12)-(3.14), (3.19)-(3.22) that
$\begin{array}{c}|{I}_{1}|\phantom{\rule{2.77695pt}{0ex}}\le \left({C}_{1}{\beta }_{1}\epsilon +{C}_{1}{\beta }_{2}\epsilon +2{C}_{1}{C}_{3}\epsilon \right)\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+2{C}_{1}\epsilon \underset{{A}_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+\left({C}_{1}{\beta }_{1}+2{C}_{1}\right)C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}dx\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+\left({C}_{1}{\beta }_{2}+2{C}_{1}\right)C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)\underset{{A}_{k,t}^{+}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right){|}^{r}dx.\end{array}$
(3.23)
By |ϕ| ≤ 2(t - τ)-1 and |u -ψ k | ≤ |u - k| a.e. in ${A}_{k,t}^{+}$, we have
$\begin{array}{c}|{I}_{1}|\phantom{\rule{2.77695pt}{0ex}}\le {C}_{4}\epsilon \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+2{C}_{1}\epsilon \underset{{A}_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx+{C}_{5}C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}dx\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+{C}_{6}C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)\frac{{2}^{r}r}{{\left(t-\tau \right)}^{r}}\underset{{A}_{k,t}^{+}}{\int }|u-k{|}^{r}dx.\end{array}$
(3.24)
We now estimate |I2|. By condition (ii),
$\begin{array}{c}|{I}_{2}|\phantom{\rule{2.77695pt}{0ex}}=\left|\underset{{A}_{k,t}^{+}}{\int }⟨A\left(x,u,\nabla u\right),{h}_{v,u}⟩dx\right|\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le \underset{{A}_{k,t}^{+}}{\int }\left[{\beta }_{1}|\nabla u{|}^{p-1}+{\beta }_{2}|u{|}^{m}+h\right]|{h}_{v,u}|dx\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le {\beta }_{1}\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{p-1}|{h}_{v,u}|dx+{\beta }_{2}\underset{{A}_{k,t}^{+}}{\int }|u{|}^{m}|{h}_{v,u}|dx+\underset{{A}_{k,t}^{+}}{\int }|h||{h}_{v,u}|dx\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}={I}_{21}+{I}_{22}+{I}_{23}.\end{array}$
(3.25)
By Young's inequality, Hölder's inequality and (1.4), I21 and I23 can be estimated as
$\begin{array}{c}|{I}_{21}|\phantom{\rule{2.77695pt}{0ex}}\le {\beta }_{1}{\left(\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\right)}^{\frac{p\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1}{r}}{\left(\underset{{A}_{k,t}^{+}}{\int }|{h}_{v,u}{|}^{\frac{r}{r\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}p+1}}dx\right)}^{{}^{\frac{r\phantom{\rule{0.3em}{0ex}}-p\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1}{r}}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}{\beta }_{1}{C}_{7}\left(p-r\right){\left(\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\right)}^{\frac{p\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1}{r}}{\left(\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\right)}^{{}^{\frac{r\phantom{\rule{0.3em}{0ex}}-p\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1}{r}}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}{\beta }_{1}{C}_{7}\left(p-r\right)\epsilon \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+{\beta }_{1}{C}_{7}\left(p-r\right)C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx.\end{array}$
(3.26)
$\begin{array}{ll}\hfill |{I}_{23}|\phantom{\rule{2.77695pt}{0ex}}& \le {\left(\underset{{A}_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{{A}_{k,t}^{+}}{\int }|{h}_{v,u}{|}^{\frac{r}{r-p+1}}dx\right)}^{\frac{r-p+1}{r}}\phantom{\rule{2em}{0ex}}\\ \le {C}_{7}\left(p-r\right){\left(\underset{{A}_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\right)}^{\frac{r-p+1}{r}}\phantom{\rule{2em}{0ex}}\\ \le {C}_{7}\left(p-r\right)\epsilon \underset{{A}_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx+{C}_{7}\left(p-r\right)C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx.\phantom{\rule{2em}{0ex}}\end{array}$
(3.27)
By (3.18), we know that if k ≥ k0, then
$\begin{array}{ll}\hfill |{I}_{22}|\phantom{\rule{2.77695pt}{0ex}}& \le {\beta }_{2}{\left(\underset{{A}_{k,t}^{+}}{\int }|u{|}^{\frac{mr}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{{A}_{k,t}^{+}}{\int }|{h}_{v,u}{|}^{\frac{r}{r-p+1}}dx\right)}^{\frac{r-p+1}{r}}\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}{\beta }_{2}{C}_{4}\left(p-r\right){\left(\underset{{A}_{k,t}^{+}}{\int }|u{|}^{\frac{mr}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\right)}^{\frac{r-p+1}{r}}\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}{\beta }_{2}{C}_{7}\left(p-r\right)\epsilon {\int }_{{A}_{k,t}^{+}}|u{|}^{\frac{mr}{p-1}}dx+{\beta }_{2}{C}_{7}\left(p-r\right)C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}{\beta }_{2}{C}_{7}\left(p-r\right)\epsilon {C}_{3}\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+{\beta }_{2}{C}_{7}\left(p-r\right)C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx.\phantom{\rule{2em}{0ex}}\end{array}$
(3.28)
Combining (3.25) with (3.26), (3.27), and (3.28), we obtain
$\begin{array}{ll}\hfill |{I}_{2}|\phantom{\rule{2.77695pt}{0ex}}& \le \left({\beta }_{1}{C}_{4}+{\beta }_{2}{C}_{4}{C}_{3}\right)\left(p-r\right)\epsilon \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+{C}_{4}\left(p-r\right)\epsilon \underset{{A}_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\left({\beta }_{1}{C}_{4}+{\beta }_{2}{C}_{4}+{C}_{4}\right)\left(p-r\right)C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\phantom{\rule{2em}{0ex}}\\ =\phantom{\rule{2.77695pt}{0ex}}{C}_{8}\epsilon \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+{C}_{9}\underset{{A}_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx+{C}_{10}\left(p-r\right)C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx.\phantom{\rule{2em}{0ex}}\end{array}$
(3.29)
We now estimate |I3|, |I4|, and |I5|.
$\begin{array}{ll}\hfill |{I}_{3}|\phantom{\rule{2.77695pt}{0ex}}& =\left|\underset{{A}_{k,t}^{+}}{\int }⟨f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}{E}_{v,u}⟩dx\right|\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}{2}^{p-r}\frac{p-r+1}{r-p+1}\underset{{A}_{k,t}^{+}}{\int }|f\left(x\right)|{\left|{\varphi }^{r}\nabla {\psi }_{k}-r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right)\right|}^{r-p+1}dx\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}{C}_{1}\underset{{A}_{k,t}^{+}}{\int }|f\left(x\right)||{\varphi }^{r}\nabla {\psi }_{k}{|}^{r-p+1}dx+{C}_{1}\underset{{A}_{k,t}^{+}}{\int }|f\left(x\right)|{\left|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right)\right|}^{r-p+1}dx\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}{C}_{1}\epsilon \underset{{A}_{k,t}^{+}}{\int }|f\left(x\right){|}^{\frac{r}{p-1}}dx+{C}_{1}C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}dx\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{C}_{1}C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_{k}\right){|}^{r}dx\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}{C}_{1}\epsilon \underset{{A}_{k,t}^{+}}{\int }|f\left(x\right){|}^{\frac{r}{p-1}}dx+{C}_{1}C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}dx\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{C}_{1}C\left(\epsilon ,p\right)\frac{{2}^{r}r}{{\left(t-\tau \right)}^{r}}\underset{{A}_{k,t}^{+}}{\int }|u-k{|}^{r}dx.\phantom{\rule{2em}{0ex}}\end{array}$
(3.30)
$\begin{array}{c}|{I}_{4}|\phantom{\rule{2.77695pt}{0ex}}=\left|\underset{{A}_{k,t}^{+}}{\int }⟨f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}|{\varphi }^{r}\nabla u{|}^{r-p}{\varphi }^{r}\nabla u⟩dx\right|\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le \underset{{A}_{k,t}^{+}}{\int }|f\left(x\right)|{\left|\nabla u\right|}^{r-p+1}dx\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le \epsilon \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|f\left(x\right){|}^{\frac{r}{p-1}}dx.\end{array}$
(3.31)
$\begin{array}{c}|{I}_{5}|\phantom{\rule{2.77695pt}{0ex}}=\left|\underset{{A}_{k,t}^{+}}{\int }⟨f\left(x\right),\phantom{\rule{2.77695pt}{0ex}}{h}_{v,u}⟩dx\right|\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}{\left(\underset{{A}_{k,t}^{+}}{\int }|f{|}^{\frac{r}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{{A}_{k,t}^{+}}{\int }|{h}_{v,u}{|}^{\frac{r}{r-p+1}}dx\right)}^{\frac{r-p+1}{r}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}{C}_{7}\left(p-r\right){\left(\underset{{A}_{k,t}^{+}}{\int }|f{|}^{\frac{r}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\right)}^{\frac{r-p+1}{r}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le {C}_{7}\left(p-r\right)\epsilon \underset{{A}_{k,t}^{+}}{\int }|f{|}^{\frac{r}{p-1}}dx+{C}_{7}\left(p-r\right)C\left(\epsilon ,p\right)\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx.\end{array}$
(3.32)
By (3.8), we have
$\underset{{A}_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\le \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+\underset{{A}_{k,t}^{+}}{\int }|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}dx+\frac{{2}^{r}r}{{\left(t-\tau \right)}^{r}}\underset{{A}_{k,t}^{+}}{\int }|u-k{|}^{r}dx.$
(3.33)
Thus, the inequalities (3.10), (3.11), (3.24), and (3.29)-(3.33) imply that
$\begin{array}{c}\phantom{\rule{1em}{0ex}}\underset{{A}_{k,\tau }^{+}}{\int }|\nabla u{|}^{r}dx\\ \le \frac{1}{\alpha }\left\{{C}_{4}\epsilon +{C}_{8}\epsilon +\epsilon +\left({C}_{10}+{C}_{7}\right)\left(p-r\right)C\left(\epsilon ,p\right)\right\}\underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\\ \phantom{\rule{1em}{0ex}}+\frac{1}{\alpha }\left\{2{C}_{1}\epsilon +{C}_{9}\right\}\underset{{A}_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx\\ \phantom{\rule{1em}{0ex}}+\frac{1}{\alpha }\left\{{C}_{1}\epsilon +C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)+{C}_{7}\left(p-r\right)\epsilon \right\}\underset{{A}_{k,t}^{+}}{\int }|f{|}^{\frac{r}{p-1}}dx\\ \phantom{\rule{1em}{0ex}}+\frac{1}{\alpha }\left\{{C}_{5}C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)+{C}_{1}C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)+\left({C}_{10}+{C}_{7}\right)\left(p-r\right)C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)\right\}\underset{{A}_{k,t}^{+}}{\int }|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}dx\\ \phantom{\rule{1em}{0ex}}+\frac{1}{\alpha }\left\{{C}_{6}C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)+{C}_{1}C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)+\left({C}_{10}+{C}_{7}\right)\left(p-r\right)C\left(\epsilon ,\phantom{\rule{2.77695pt}{0ex}}p\right)\right\}\frac{{2}^{r}r}{{\left(t-\tau \right)}^{r}}\underset{{A}_{k,t}^{+}}{\int }|u-k{|}^{r}dx.\end{array}$
(3.34)
Choosing ε and p-r small enough such that, the summation θ of the coefficients of the first term in the right-handside of (3.34) is smaller than 1. Let ρ, R be arbitrarily fixed with R1/ 2 ≤ ρ < R ≤ R1. Thus, from (3.34), we deduce that for every t and τ, such that R1/ 2 ≤ τ < t ≤ R1, we have
$\begin{array}{c}\underset{{A}_{k,\tau }^{+}}{\int }|\nabla u{|}^{r}dx\le \frac{{C}_{11}}{\alpha }\underset{{A}_{k,R}^{+}}{\int }\left(|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}+|h{|}^{\frac{r}{p-1}}+|f{|}^{\frac{r}{p-1}}\right)\phantom{\rule{2.77695pt}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}+\frac{{C}_{12}}{\alpha {\left(t-\tau \right)}^{r}}\underset{{A}_{k,R}^{+}}{\int }|u-k{|}^{r}dx+\theta \underset{{A}_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx,\end{array}$
(3.35)
where C11,C12 are some constants depending only on n, p, r, m, k0, | Ω|, α, β1 and β2. Applying Lemma 2.1, we conclude that
$\begin{array}{c}\underset{{A}_{k,\rho }^{+}}{\int }|\nabla u{|}^{r}dx\le \frac{c{C}_{11}}{\alpha {\left(R-\rho \right)}^{r}}\underset{{A}_{k,R}^{+}}{\int }|u-k{|}^{r}dx\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\frac{c{C}_{12}}{\alpha }\underset{{A}_{k,R}^{+}}{\int }\left(|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}+|{h}_{1}{|}^{\frac{r}{p-1}}+|{h}_{2}{|}^{\frac{r}{p-1}}\right)\phantom{\rule{2.77695pt}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}\frac{c{C}_{11}}{\alpha {\left(R-\rho \right)}^{r}}\underset{{A}_{k,R}^{+}}{\int }|u-k{|}^{r}dx+\frac{c{C}_{12}{C}_{13}}{\alpha }|{A}_{k,R}^{+}|,\end{array}$
(3.36)
where c is the constant given by Lemma 2.1 and ${C}_{13}={∥|\nabla {\psi }_{1}{|}^{r}+|\nabla {\psi }_{2}{|}^{r}+|h{|}^{\frac{r}{p-1}}+|f{|}^{\frac{r}{p-1}}∥}_{{L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)}$. Thus u belongs to the class B with γ = max{c2c7/α, c2c6c8/α} and m = r. Lemma 2.3 yields
$\underset{{B}_{R/2}}{\text{max}}u\left(x\right)\le c.$

If ψ1 is an arbitrary function in Ω with values in R {-∞}, noticing 2 ≤ -u ≤ -ψ1, we only use -u in place of u above.

These results together with the assumptions ψ1 ≤ u ≤ ψ2 and ψ1, ${\psi }_{2}\in {W}_{1\mathsf{\text{oc}}}^{1,\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ yield the desired result.

## Declarations

### Acknowledgements

The research was supported by the Natural Science Foundation of Hebei Province (A2010000910), Scientific Research Fund of Zhejiang Provincial Education Department(Y201016044) and Ningbo Natural Science Foundation(2011A610170).

## Authors’ Affiliations

(1)
College of Science, Hebei United University, Tangshan, 063009, China
(2)
Department of Mathematics, Ningbo University, Ningbo, 315211, China

## References

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