Local boundedness results for very weak solutions of double obstacle problems

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.

1 Introduction

Let Ω be a bounded open set of Rⁿ, n ≥ 2. We consider the second order divergence type elliptic equation (also called A-harmonic equation or Leray-Lions equation)

$$\mathsf{\text{div}}A\left(x,u\left(x\right),\nabla u\left(x\right)\right)=\mathsf{\text{div}}f\left(x\right).$$

(1.1)

where A : Ω × R × Rⁿ → Rⁿ is a Carathéodory function satisfying the coercivity and growth conditions: for almost all x ∈ Ω, all u ∈ R, and ξ ∈ Rⁿ,

• (i) 〈A(x, u, ξ), ξ〉 ≥ α |ξ|^p,

• (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 ∈ W^1,r(Ω) with max{1, p - 1} < r ≤ p. Let

$$K_{{\psi }_1,{\psi }_2}^{\theta ,r}\left(\mathrm{\Omega }\right)=\left\{v\in W^{1,r}\left(\mathrm{\Omega }\right):{\psi }_1\le v\le {\psi }_2,a.e.\mathsf{\text{and}}v-\theta \in W_0^{1,r}\left(\mathrm{\Omega }\right)\right\}.$$

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

For any $u,v\in K_{{\psi }_1,{\psi }_2}^{\theta ,r}\left(\mathrm{\Omega }\right)$, 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 [1]:

$$|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)=\nabla {\varphi }_{v,u}+h_{v,u}$$

(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

$$\underset{\mathrm{\Omega }}{\int }⟨A\left(x,u,\nabla u\right),|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)-h⟩dx\ge \underset{\mathrm{\Omega }}{\int }⟨f\left(x\right),|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)-h⟩dx,$$

(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 [2], 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 K_ψ,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 [3–6].

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 ${\psi }_1,{\psi }_2\in W_{\mathsf{\text{lo}}\mathsf{\text{c}}}^{1,\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. 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 [4].

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

$$\begin{array}{c}{A}_k=\left\{x\in \mathrm{\Omega }:\left|u\left(x\right)\right|>k\right\},A_k^{+}=\left\{x\in \mathrm{\Omega }:u\left(x\right)>k\right\},\\ A_{k,t}=A_k\cap B_t,A_{k,t}^{+}=A_k^{+}\cap B_t.\end{array}$$

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. [7] Let f(τ) be a nonnegative bounded function defined for 0 ≤ R₀ ≤ t ≤ R₁. Suppose that for R₀ ≤ τ < t ≤ R₁ one has

$$f\left(\tau \right)\le A{\left(t-\tau \right)}^{-\alpha }+B+\theta f\left(t\right),$$

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

$$f\left(\rho \right)\le c\left[A{\left(R-\rho \right)}^{-\alpha }+B\right].$$

Definition 2.2. [8] A function $u\in W_{\mathsf{\text{lo}}\mathsf{\text{c}}}^{1,m}\left(\mathrm{\Omega }\right)$ 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

$$\underset{A_{k,\rho -\rho \sigma }^{+}}{\int }|\nabla u{|}^{m}dx\le \gamma \left\{{\sigma }^{-m}{\rho }^{-m}\underset{A_{k,\rho }^{+}}{\int }{\left(u-k\right)}^{m}dx+\left|A_{k,\rho }^{+}\right|\right\},$$

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. [8] 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

$$\mathsf{\text{supp}}\varphi \subset B_t,0\le \varphi \le 1,\varphi \equiv 1\mathsf{\text{in}}{B}_{\tau },|\nabla \varphi |\le 2{\left(t-\tau \right)}^{-1}.$$

(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

$${\psi }_k=\text{min}\left\{\text{max}\left\{{\psi }_1,t_k\left(u\right)\right\},{\psi }_2\right\},t_k\left(u\right)=\text{min}\left\{u,k\right\},k\ge 0.$$

It is easy to see ψ₁ ≤ ψ _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

$$\begin{array}{c}v-\theta =u-\theta -{\varphi }^r\left(u-{\psi }_k\right)\in W_0^{1,r}\left(\mathrm{\Omega }\right),\\ v-{\psi }_1=u-{\psi }_1-{\varphi }^r\left(u-{\psi }_k\right)\ge \left(1-{\varphi }^r\right)\left(u-{\psi }_1\right)\ge 0\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.in}}\mathrm{\Omega },\\ v-{\psi }_2=u-{\psi }_2-{\varphi }^r\left(u-{\psi }_k\right)\le \left(1-{\varphi }^r\right)\left(u-{\psi }_2\right)\le 0\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.in}}\mathrm{\Omega }.\end{array}$$

(3.3)

For any fixed k > 0, let

$$v_0=\left\{\begin{array}{cc}\hfill u,\hfill & \hfill \mathsf{\text{if}}u\le k,\hfill \\ \hfill v,\hfill & \hfill \mathsf{\text{if}}u>k.\hfill \end{array}\right.$$

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

$$\underset{\mathrm{\Omega }}{\int }⟨A\left(x,u,\nabla u\right),|\nabla \left(v_0-u\right){|}^{r-p}\nabla \left(v_0-u\right)-{\tilde{h}}_{v,u}⟩dx\ge \underset{\mathrm{\Omega }}{\int }⟨f\left(x\right),|\nabla \left(v_0-u\right){|}^{r-p}\nabla \left(v_0-u\right)-{\tilde{h}}_{v,u}⟩dx.$$

(3.4)

If u ≤ k, then ${\tilde{h}}_{v,u}=0,\nabla {\tilde{\varphi }}_{v,u}=0$; If u > k, then ${\tilde{h}}_{v,u}=h_{v,u}$, ${\tilde{\varphi }}_{v,u}={\varphi }_{v,u}$. It's derived from the uniqueness of Hodge decomposition. This means that

$$\begin{array}{c}\underset{A_{k,t}^{+}}{\int }⟨A\left(x,u,\nabla u\right),h_{v,u}⟩dx+\underset{A_{k,t}^{+}}{\int }⟨f\left(x\right),|\nabla \left(v_0-u\right){|}^{r-p}\nabla \left(v_0-u\right)-{\tilde{h}}_{v,u}⟩dx\\ \le \underset{\mathrm{\Omega }}{\int }⟨A\left(x,u,\nabla u\right),h_{v,u}⟩dx+\underset{\mathrm{\Omega }}{\int }⟨f\left(x\right),|\nabla \left(v_0-u\right){|}^{r-p}\nabla \left(v_0-u\right)-{\tilde{h}}_{v,u}⟩dx\\ \le \underset{\mathrm{\Omega }}{\int }⟨A\left(x,u,\nabla u\right),|\nabla \left(v_0-u\right){|}^{r-p}\nabla \left(v_0-u\right)⟩dx\\ =\left(\underset{\mathrm{\Omega }\cap \left\{u\le k\right\}}{\int }+\underset{\mathrm{\Omega }\cap \left\{u>k\right\}}{\int }\right)⟨A\left(x,u,\nabla u\right),|\nabla \left(v_0-u\right){|}^{r-p}\nabla \left(v_0-u\right)⟩dx\\ =\underset{\mathrm{\Omega }\cap \left\{u>k\right\}}{\int }⟨A\left(x,u,\nabla u\right),|\nabla \left(v_0-u\right){|}^{r-p}\nabla \left(v_0-u\right)⟩dx\\ =\underset{A_{k,t}^{+}}{\int }⟨A\left(x,u,\nabla u\right),|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)⟩dx.\end{array}$$

(3.5)

Let

$$E\left(v,u\right)=|{\varphi }^r\nabla u{|}^{r-p}{\varphi }^r\nabla u+|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right).$$

(3.6)

By an elementary inequality [[9], P. 271, (4.1)],

$$\left|\right|X{|}^{-\epsilon }X-\left|Y{|}^{-\epsilon }Y\right|\le 2^{\epsilon }\frac{1+\epsilon }{1-\epsilon }|X-Y{|}^{1-\epsilon },X,Y\in R^n,0\le \epsilon <1,$$

(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

$$\left|E\left(v,u\right)\right|\le 2^{p-r}\frac{p-r+1}{r-p+1}|{\varphi }^r\nabla {\psi }_k-r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right){|}^{r-p+1}.$$

(3.9)

We get from the definition of E(v, u) and (3.5) that

$$\begin{array}{c}\underset{A_{k,t}^{+}}{\int }⟨A\left(x,u,\nabla u\right),|{\varphi }^r\nabla u{|}^{r-p}{\varphi }^r\nabla u⟩dx\\ =\underset{A_{k,t}^{+}}{\int }⟨A\left(x,u,\nabla u\right),E\left(v,u\right)⟩dx-\underset{A_{k,t}^{+}}{\int }⟨A\left(x,u,\nabla u\right),|\nabla \left(v-u\right){|}^{r-p}\nabla \left(v-u\right)⟩dx\\ \le \underset{A_{k,t}^{+}}{\int }⟨A\left(x,u,\nabla u\right),E\left(v,u\right)⟩dx-\underset{A_{k,t}^{+}}{\int }⟨A\left(x,u,\nabla u\right),h_{v,u}⟩dx-\underset{A_{k,t}^{+}}{\int }⟨f\left(x\right),E\left(v,u\right)⟩dx\\ +\underset{A_{k,t}^{+}}{\int }⟨f\left(x\right),|{\varphi }^r\nabla u{|}^{r-p}{\varphi }^r\nabla u⟩dx+\underset{A_{k,t}^{+}}{\int }⟨f\left(x\right),h_{v,u}⟩dx\\ =I_1+I_2+I_3+I_4+I_5.\end{array}$$

(3.10)

We now estimate the left-hand side and the right-hand side of (3.10), respectively. Firstly,

$$\begin{array}{ll}\hfill \underset{A_{k,t}^{+}}{\int }⟨A\left(x,u,\nabla u\right),|{\varphi }^r\nabla u{|}^{r-p}{\varphi }^r\nabla u⟩dx& \ge \underset{A_{k,\tau }^{+}}{\int }⟨A\left(x,u,\nabla u\right),|\nabla u{|}^{r-p}\nabla u⟩dx\\ \ge \alpha \underset{A_{k,\tau }^{+}}{\int }|\nabla u{|}^{r}dx,\end{array}$$

(3.11)

here we have used condition (i). Secondly, by condition (ii) and (3.9),

$$\begin{array}{c}\left|I_1\right|=\left|\underset{A_{k,t}}{\int }⟨A\left(x,u,\nabla u\right),E\left(v,u\right)⟩dx\right|\\ \le \underset{A_{k,t}^{+}}{\int }\left[{\beta }_1|\nabla u{|}^{p-1}+{\beta }_2|u{|}^m+h_1\right]\left|E\left(v,u\right)\right|dx\\ \le 2^{p-r}\frac{p-r+1}{r-p+1}\underset{A_{k,t}^{+}}{\int }\left[{\beta }_1|\nabla u{|}^{p-1}+{\beta }_2|u{|}^m+h_1\right]|{\varphi }^r\nabla {\psi }_k-r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right){|}^{r-p+1}\\ =C_1{\beta }_1\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{p-1}|{\varphi }^r\nabla {\psi }_k{|}^{r-p+1}+C_1{\beta }_1\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{p-1}|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right){|}^{r-p+1}\\ +C_1{\beta }_2\underset{A_{k,t}^{+}}{\int }\left|u{|}^m\right|{\varphi }^r\nabla {\psi }_k{|}^{r-p+1}+C_1{\beta }_2\underset{A_{k,t}^{+}}{\int }\left|u{|}^m\right|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right){|}^{r-p+1}\\ +C_1\underset{A_{k,t}^{+}}{\int }\left|h\right|{\left|{\varphi }^r\nabla {\psi }_k\right|}^{r-p+1}+C_1\underset{A_{k,t}^{+}}{\int }\left|h\right|{\left|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right)\right|}^{r-p+1}\\ =I_{11}+I_{12}+I_{13}+I_{14}+I_{15}+I_{16},\end{array}$$

(3.12)

where $C_1=2^{p-r}\frac{p-r+1}{r-p+1}$. By Young's inequality

$$ab\le \epsilon a^{p^{\prime }}+C\left(\epsilon ,p\right)b^p\mathsf{\text{valid for }}a,b\ge 0,\epsilon >0\mathsf{\text{and }}p>1,$$

and $\frac{p-1}{r}+\frac{r-p+1}{r}=1$, we have the estimates

$$\left|I_{11}\right|\le C_1{\beta }_1\left[\epsilon \underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^{r}dx\right],$$

(3.13)

$$\left|I_{12}\right|\le C_1{\beta }_2\left[\epsilon \underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right){|}^{r}dx\right],$$

(3.14)

$$\left|I_{13}\right|\le C_1\left[\epsilon \underset{A_{k,t}^{+}}{\int }|u{|}^{\frac{mr}{p-1}}dx+C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^{r}dx\right],$$

(3.15)

where we have used |▽ψ _k | ≤ |▽ψ₁| + |▽ψ₂| in $A_{k,t}^{+}$.

We observe now that, if w ∈ W^1,p(B _t ) and | suppw| ≤ 1/ 2|B _t |, then we have the Sobolev inequality (see also [10]),

$${\left(\underset{B_t}{\int }|w{|}^{p*}dx\right)}^{p/p*}\le c_1\left(n,p\right)\underset{B_t}{\int }|\nabla w{|}^{p}dx.$$

(3.16)

Set

$$g_k\left(u\right)=\left\{\begin{array}{cc}\hfill u,\hfill & \hfill \mathsf{\text{if}}u\le k,\hfill \\ \hfill 0,\hfill & \hfill \mathsf{\text{if}}u>k.\hfill \end{array}\right.$$

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

$$\begin{array}{c}\underset{A_{k,t}^{+}}{\int }|u{|}^{\frac{mr}{p-1}}dx=\underset{B_t}{\int }|u-g_k\left(u\right){|}^{\frac{mr}{p-1}}dx\\ \le \left|\right|u-g_k\left(u\right)\left|{|}_{r^{*}}^{\frac{mr}{p-1}-r}\right|B_t{|}^{1-\frac{mr}{\left(p-1\right)r*}}{\left(\underset{B_t}{\int }|u-g_k\left(u\right){|}^{r*}dx\right)}^{r/r*}\\ \le c_1\left(n,p\right)\left|\right|u-g_k\left(u\right)\left|{|}_{r^{*}}^{\frac{mr}{p-1}-r}\right|B_t{|}^{1-\frac{mr}{\left(p-1\right)r*}}\underset{B_t}{\int }|\nabla \left(u-g_k\left(u\right)\right){|}^{r}dx\\ =c_1\left(n,p\right)\left|\right|u-g_k\left(u\right)\left|{|}_{r^{*}}^{\frac{mr}{p-1}-r}\right|B_t{|}^{1-\frac{mr}{\left(p-1\right)r^{*}}}\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx,\end{array}$$

(3.17)

provided that $|\text{supp(}u–g_k(u)\text{)}{|}_B{}_t|\le \text{1}/\text{2}\left|B_t\right|$. Since $\text{sup}\mathsf{\text{p}}\left(u-g_k\left(u\right)\right){{|}_B}_{{}_t}\subset A_{k,t}^{+}$, then |supp$\left|\text{sup}\mathsf{\text{p}}\left(u-g_k\left(u\right)\right){{|}_B}_{{}_t}\right|\le \left|A_{k,t}^{+}\right|$. On the other hand, we have

$$\left|\right|u|{|}_{r^{*},B_t}^{r^{*}}=\underset{B_t}{\int }{u}^{r^{*}}dx\ge \underset{A_{k,t}^{+}}{\int }|u{|}^{r^{*}}dx\ge k^{r^{*}}\left|A_{k,t}^{+}\right|.$$

Thus, there exists a constant k₀ > 0, such that for all k ≥ k₀, we have $\left|A_{k,t}^{+}\right|\le 1/2\left|B_t\right|$. 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

$$\begin{array}{ll}\hfill \underset{A_{k,t}^{+}}{\int }|u{|}^{m\frac{r}{p-1}}dx& \le C_2\left(n,p\right)\left|\right|u-g_k\left(u\right)\left|{|}_{r*}^{\frac{mr}{p-1}-r}\right|B_t{|}^{1-\frac{mr}{\left(p-1\right)r^{*}}}\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\\ \le C_2\left(n,p\right)\left|\right|u-g_k\left(u\right)\left|{|}_{r*}^{\frac{mr}{p-1}-r}\right|\mathrm{\Omega }{|}^{1-\frac{mr}{\left(p-1\right)r^{*}}}\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\\ \le C_2\left(n,p\right)\left|\mathrm{\Omega }{|}^{1-\frac{mr}{\left(p-1\right)r^{*}}}\underset{A_{k,t}^{+}}{\int }\right|\nabla u{|}^{r}dx\\ =C_3\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx,\end{array}$$

(3.18)

here C₃ = C₃(n, m, p, r, k₀, | Ω|). We derive from (3.15) and (3.18) that

$$\left|I_{13}\right|\le C_1{C}_3\epsilon \underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+C_{1}C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^{r}dx.$$

(3.19)

$$\left|I_{14}\right|\le C_1{C}_3\epsilon \underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+C_{1}C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right){|}^{r}dx.$$

(3.20)

I₁₅ and I₁₆ can be estimated as follows:

$$\left|I_{15}\right|\le C_1\epsilon \underset{A_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx+C_{1}C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^{r}dx.$$

(3.21)

$$\left|I_{16}\right|\le C_1\epsilon \underset{A_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx+C_{1}C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right){|}^{r}dx.$$

(3.22)

In conclusion, we derive from (3.12)-(3.14), (3.19)-(3.22) that

$$\begin{array}{c}\left|I_1\right|\le \left(C_1{\beta }_1\epsilon +C_1{\beta }_2\epsilon +2C_1{C}_3\epsilon \right)\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+2C_1\epsilon \underset{A_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx\\ +\left(C_1{\beta }_1+2C_1\right)C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^{r}dx\\ +\left(C_1{\beta }_2+2C_1\right)C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right){|}^{r}dx.\end{array}$$

(3.23)

By |▽ϕ| ≤ 2(t - τ)^-1 and |u -ψ _k | ≤ |u - k| a.e. in $A_{k,t}^{+}$, we have

$$\begin{array}{c}\left|I_1\right|\le C_4\epsilon \underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+2C_1\epsilon \underset{A_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx+C_{5}C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^{r}dx\\ +C_{6}C\left(\epsilon ,p\right)\frac{2^{r}r}{{\left(t-\tau \right)}^r}\underset{A_{k,t}^{+}}{\int }|u-k{|}^{r}dx.\end{array}$$

(3.24)

We now estimate |I₂|. By condition (ii),

$$\begin{array}{c}\left|I_2\right|=\left|\underset{A_{k,t}^{+}}{\int }⟨A\left(x,u,\nabla u\right),h_{v,u}⟩dx\right|\\ \le \underset{A_{k,t}^{+}}{\int }\left[{\beta }_1|\nabla u{|}^{p-1}+{\beta }_2|u{|}^m+h\right]\left|h_{v,u}\right|dx\\ \le {\beta }_1\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{p-1}|h_{v,u}|dx+{\beta }_2\underset{A_{k,t}^{+}}{\int }|u{|}^m\left|h_{v,u}\right|dx+\underset{A_{k,t}^{+}}{\int }\left|h\right|\left|h_{v,u}\right|dx\\ =I_{21}+I_{22}+I_{23}.\end{array}$$

(3.25)

By Young's inequality, Hölder's inequality and (1.4), I₂₁ and I₂₃ can be estimated as

$$\begin{array}{c}\left|I_{21}\right|\le {\beta }_1{\left(\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\right)}^{\frac{p-1}{r}}{\left(\underset{A_{k,t}^{+}}{\int }|h_{v,u}{|}^{\frac{r}{r-p+1}}dx\right)}^{{}^{\frac{r-p+1}{r}}}\\ \le {\beta }_1{C}_7\left(p-r\right){\left(\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\right)}^{\frac{p-1}{r}}{\left(\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\right)}^{{}^{\frac{r-p+1}{r}}}\\ \le {\beta }_1{C}_7\left(p-r\right)\epsilon \underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+{\beta }_1{C}_7\left(p-r\right)C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx.\end{array}$$

(3.26)

$$\begin{array}{ll}\hfill \left|I_{23}\right|& \le {\left(\underset{A_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{A_{k,t}^{+}}{\int }|h_{v,u}{|}^{\frac{r}{r-p+1}}dx\right)}^{\frac{r-p+1}{r}}\\ \le C_7\left(p-r\right){\left(\underset{A_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\right)}^{\frac{r-p+1}{r}}\\ \le C_7\left(p-r\right)\epsilon \underset{A_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx+C_7\left(p-r\right)C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx.\end{array}$$

(3.27)

By (3.18), we know that if k ≥ k₀, then

$$\begin{array}{ll}\hfill \left|I_{22}\right|& \le {\beta }_2{\left(\underset{A_{k,t}^{+}}{\int }|u{|}^{\frac{mr}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{A_{k,t}^{+}}{\int }|h_{v,u}{|}^{\frac{r}{r-p+1}}dx\right)}^{\frac{r-p+1}{r}}\\ \le {\beta }_2{C}_4\left(p-r\right){\left(\underset{A_{k,t}^{+}}{\int }|u{|}^{\frac{mr}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\right)}^{\frac{r-p+1}{r}}\\ \le {\beta }_2{C}_7\left(p-r\right)\epsilon {\int }_{A_{k,t}^{+}}|u{|}^{\frac{mr}{p-1}}dx+{\beta }_2{C}_7\left(p-r\right)C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\\ \le {\beta }_2{C}_7\left(p-r\right)\epsilon C_3\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+{\beta }_2{C}_7\left(p-r\right)C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx.\end{array}$$

(3.28)

Combining (3.25) with (3.26), (3.27), and (3.28), we obtain

$$\begin{array}{ll}\hfill \left|I_2\right|& \le \left({\beta }_1{C}_4+{\beta }_2{C}_4{C}_3\right)\left(p-r\right)\epsilon \underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+C_4\left(p-r\right)\epsilon \underset{A_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx\\ +\left({\beta }_1{C}_4+{\beta }_2{C}_4+C_4\right)\left(p-r\right)C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\\ =C_8\epsilon \underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+C_9\underset{A_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx+C_{10}\left(p-r\right)C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx.\end{array}$$

(3.29)

We now estimate |I₃|, |I₄|, and |I₅|.

$$\begin{array}{ll}\hfill \left|I_3\right|& =\left|\underset{A_{k,t}^{+}}{\int }⟨f\left(x\right),E_{v,u}⟩dx\right|\\ \le 2^{p-r}\frac{p-r+1}{r-p+1}\underset{A_{k,t}^{+}}{\int }\left|f\left(x\right)\right|{\left|{\varphi }^r\nabla {\psi }_k-r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right)\right|}^{r-p+1}dx\\ \le C_1\underset{A_{k,t}^{+}}{\int }\left|f\left(x\right)\right||{\varphi }^r\nabla {\psi }_k{|}^{r-p+1}dx+C_1\underset{A_{k,t}^{+}}{\int }|f\left(x\right)|{\left|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right)\right|}^{r-p+1}dx\\ \le C_1\epsilon \underset{A_{k,t}^{+}}{\int }|f\left(x\right){|}^{\frac{r}{p-1}}dx+C_{1}C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^{r}dx\\ +C_{1}C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|r{\varphi }^{r-1}\nabla \varphi \left(u-{\psi }_k\right){|}^{r}dx\\ \le C_1\epsilon \underset{A_{k,t}^{+}}{\int }|f\left(x\right){|}^{\frac{r}{p-1}}dx+C_{1}C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^{r}dx\\ +C_{1}C\left(\epsilon ,p\right)\frac{2^{r}r}{{\left(t-\tau \right)}^r}\underset{A_{k,t}^{+}}{\int }|u-k{|}^{r}dx.\end{array}$$

(3.30)

$$\begin{array}{c}\left|I_4\right|=\left|\underset{A_{k,t}^{+}}{\int }⟨f\left(x\right),|{\varphi }^r\nabla u{|}^{r-p}{\varphi }^r\nabla u⟩dx\right|\\ \le \underset{A_{k,t}^{+}}{\int }\left|f\left(x\right)\right|{\left|\nabla u\right|}^{r-p+1}dx\\ \le \epsilon \underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx+C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|f\left(x\right){|}^{\frac{r}{p-1}}dx.\end{array}$$

(3.31)

$$\begin{array}{c}\left|I_5\right|=\left|\underset{A_{k,t}^{+}}{\int }⟨f\left(x\right),h_{v,u}⟩dx\right|\\ \le {\left(\underset{A_{k,t}^{+}}{\int }|f{|}^{\frac{r}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{A_{k,t}^{+}}{\int }|h_{v,u}{|}^{\frac{r}{r-p+1}}dx\right)}^{\frac{r-p+1}{r}}\\ \le C_7\left(p-r\right){\left(\underset{A_{k,t}^{+}}{\int }|f{|}^{\frac{r}{p-1}}dx\right)}^{\frac{p-1}{r}}{\left(\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx\right)}^{\frac{r-p+1}{r}}\\ \le C_7\left(p-r\right)\epsilon \underset{A_{k,t}^{+}}{\int }|f{|}^{\frac{r}{p-1}}dx+C_7\left(p-r\right)C\left(\epsilon ,p\right)\underset{A_{k,t}^{+}}{\int }|\nabla \left(v-u\right){|}^{r}dx.\end{array}$$

(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

$$\begin{array}{c}\underset{A_{k,\tau }^{+}}{\int }|\nabla u{|}^{r}dx\\ \le \frac{1}{\alpha }\left\{C_4\epsilon +C_8\epsilon +\epsilon +\left(C_{10}+C_7\right)\left(p-r\right)C\left(\epsilon ,p\right)\right\}\underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx\\ +\frac{1}{\alpha }\left\{2C_1\epsilon +C_9\right\}\underset{A_{k,t}^{+}}{\int }|h{|}^{\frac{r}{p-1}}dx\\ +\frac{1}{\alpha }\left\{C_1\epsilon +C\left(\epsilon ,p\right)+C_7\left(p-r\right)\epsilon \right\}\underset{A_{k,t}^{+}}{\int }|f{|}^{\frac{r}{p-1}}dx\\ +\frac{1}{\alpha }\left\{C_{5}C\left(\epsilon ,p\right)+C_{1}C\left(\epsilon ,p\right)+\left(C_{10}+C_7\right)\left(p-r\right)C\left(\epsilon ,p\right)\right\}\underset{A_{k,t}^{+}}{\int }|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^{r}dx\\ +\frac{1}{\alpha }\left\{C_{6}C\left(\epsilon ,p\right)+C_{1}C\left(\epsilon ,p\right)+\left(C_{10}+C_7\right)\left(p-r\right)C\left(\epsilon ,p\right)\right\}\frac{2^{r}r}{{\left(t-\tau \right)}^r}\underset{A_{k,t}^{+}}{\int }|u-k{|}^{r}dx.\end{array}$$

(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

$$\begin{array}{c}\underset{A_{k,\tau }^{+}}{\int }|\nabla u{|}^{r}dx\le \frac{C_{11}}{\alpha }\underset{A_{k,R}^{+}}{\int }\left(|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^r+|h{|}^{\frac{r}{p-1}}+|f{|}^{\frac{r}{p-1}}\right)dx\\ +\frac{C_{12}}{\alpha {\left(t-\tau \right)}^r}\underset{A_{k,R}^{+}}{\int }|u-k{|}^{r}dx+\theta \underset{A_{k,t}^{+}}{\int }|\nabla u{|}^{r}dx,\end{array}$$

(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

$$\begin{array}{c}\underset{A_{k,\rho }^{+}}{\int }|\nabla u{|}^{r}dx\le \frac{c{C}_{11}}{\alpha {\left(R-\rho \right)}^r}\underset{A_{k,R}^{+}}{\int }|u-k{|}^{r}dx\\ +\frac{c{C}_{12}}{\alpha }\underset{A_{k,R}^{+}}{\int }\left(|\nabla {\psi }_1{|}^r+|\nabla {\psi }_2{|}^r+|h_1{|}^{\frac{r}{p-1}}+|h_2{|}^{\frac{r}{p-1}}\right)dx\\ \le \frac{c{C}_{11}}{\alpha {\left(R-\rho \right)}^r}\underset{A_{k,R}^{+}}{\int }|u-k{|}^{r}dx+\frac{c{C}_{12}{C}_{13}}{\alpha }|A_{k,R}^{+}|,\end{array}$$

(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 -ψ₂ ≤ -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.