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The obstacle problem for non-coercive equations with lower order term and \(L^{1}\)-data

Abstract

The aim of this paper is to study the obstacle problem associated with an elliptic operator having degenerate coercivity, a low order term, and \(L^{1}\)-data. We prove the existence of an entropy solution to the obstacle problem and show its continuous dependence on the \(L^{1}\)-data in \(W^{1,q}(\varOmega )\) with some \(q>1\).

Introduction

Problem setting and main result

Let Ω be a bounded domain in \(\mathbb{R}^{N}\) (\(N\geq 2\)), \(1< p<+\infty \), and \(\theta \geq 0\). Given functions \(g, \psi \in W ^{1,p}(\varOmega )\cap L^{\infty }(\varOmega )\) and data \(f\in L^{1}(\varOmega )\), the aim of this paper is to study the obstacle problem for nonlinear non-coercive elliptic equations with lower order term, governed by the operator

$$\begin{aligned} Au=-\operatorname{div} \frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}+b \vert u \vert ^{r-2}u, \end{aligned}$$
(1)

where \(b>0\) is a constant, and \(a:\varOmega \times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}\) is a Carathéodory function, satisfying the following conditions:

$$\begin{aligned} &a(x,\xi )\cdot \xi \geq \alpha \vert \xi \vert ^{p}, \end{aligned}$$
(2)
$$\begin{aligned} & \bigl\vert a(x,\xi ) \bigr\vert \leq \beta \bigl(j(x)+ \vert \xi \vert ^{p-1}\bigr), \end{aligned}$$
(3)
$$\begin{aligned} &\bigl(a(x, \xi )-a(x, \eta )\bigr) (\xi -\eta )>0, \end{aligned}$$
(4)
$$\begin{aligned} & \bigl\vert a(x, \xi )-a(x, \zeta ) \bigr\vert \leq \gamma \textstyle\begin{cases} \vert \xi -\zeta \vert ^{p-1}, & \text{if } 1< p< 2, \\ (1+ \vert \xi \vert + \vert \zeta \vert )^{p-2} \vert \xi -\zeta \vert , & \text{if } p\geq 2 \end{cases}\displaystyle \end{aligned}$$
(5)

for almost every x in Ω and for every ξ, η, ζ in \(\mathbb{R}^{N}\) with \(\xi \neq \eta \), where \(\alpha ,\beta ,\gamma >0\) are constants, and j is a nonnegative function in \(L^{p'}( \varOmega )\).

If f has a fine regularity, e.g., \(f\in W^{-1,p'}(\varOmega )\), the obstacle problem corresponding to \((f,\psi ,g) \) can be formulated in terms of the inequality

$$\begin{aligned} & \int _{\varOmega }\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\cdot \nabla (u-v)\, \mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}u(u-v)\, \mathrm{d}x \\ & \quad \leq \int _{\varOmega }f(u-v)\,\mathrm{d}x, \quad \forall v\in K_{g,\psi }\cap L^{\infty }(\varOmega ), \end{aligned}$$
(6)

whenever \(1\leq r< p\) and the convex subset

$$\begin{aligned} K_{g,\psi }= \bigl\{ v\in W^{1,p}(\varOmega ); v-g \in W^{1,p}_{0}(\varOmega ), v\geq \psi , \text{a.e. in } \varOmega \bigr\} \end{aligned}$$

is nonempty. However, if \(f\in L^{1}(\varOmega )\), (6) is not well-defined. Following [1, 3, 5, 19] etc., we are led to the more general definition of a solution to the obstacle problem, using the truncation function

$$\begin{aligned} T_{s}(t)=\max \bigl\{ -s,\min \{s,t\}\bigr\} , \quad s,t\in \mathbb{R}. \end{aligned}$$

Definition 1

An entropy solution of the obstacle problem associated with operator A and functions \((f,\psi ,g)\) with \(f\in L^{1}(\varOmega )\) is a measurable function u such that \(u\geq \psi \) a.e. in Ω, \(\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\in (L^{1}( \varOmega ))^{N}\), \(\vert u \vert ^{r-1}\in L^{1}(\varOmega )\), and, for every \(s>0\), \(T_{s}(u)-T_{s}(g)\in W_{0}^{1,p}(\varOmega )\) and

$$\begin{aligned} & \int _{\varOmega }\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\cdot \nabla \bigl(T_{s}(u-v)\bigr)\,\mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}uT_{s}(u-v) \,\mathrm{d}x \\ & \quad \leq \int _{\varOmega }fT_{s}(u-v)\,\mathrm{d}x, \quad \forall v\in K_{g,\psi }\cap L^{\infty }(\varOmega ). \end{aligned}$$
(7)

Observe that no global integrability condition is required on u nor on its gradient in the definition. As pointed out in [3, 8], if \(T_{s} (u) \in W^{1,p}(\varOmega )\) for all \(s > 0\), then there exists a unique measurable vector field \(U:\varOmega \rightarrow \mathbb{R}^{N}\) such that \(\nabla (T_{s} (u)) = \chi _{\{ \vert u \vert < s\}}U\) a.e. in Ω, \(s > 0\), which, in fact, coincides with the standard distributional gradient of u whenever \(u \in W^{1,1}(\varOmega )\).

Before stating the main result, we make some basic assumptions throughout this paper, i.e., without special statements, we always assume that

$$\begin{aligned} 2-\frac{1}{N}< p< N, \quad\quad 1\leq r< p, \quad\quad 0\leq \theta < \min \biggl\{ \frac{N}{N-1}-\frac{1}{p-1},\frac{p-r}{p-1} \biggr\} ,\quad\quad b>0, \end{aligned}$$

and \(\psi ,g\in W^{1,p}(\varOmega )\cap L^{\infty }(\varOmega )\) with \((\psi -g)^{+}\in W^{1,p}_{0}(\varOmega )\) such that \(K_{g,\psi }\neq \emptyset \). The following theorem is the main result obtained in this paper.

Theorem 1

Let \(f\in L^{1}(\varOmega )\). Then there exists at least one entropy solution u of the obstacle problem associated with \((f,\psi ,g) \). In addition, u depends continuously on f, i.e., if \(f_{n}\rightarrow f\) in \(L^{1}(\varOmega )\) and \(u_{n}\) is a solution to the obstacle problem associated with \((f_{n},\psi ,g)\), then

$$\begin{aligned} u_{n}\rightarrow u \quad \textit{in } W^{1,q}(\varOmega ),\forall q \in \textstyle\begin{cases} ( \frac{N(r-1)}{N+r-1}, \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} ), & \textit{if }\frac{2N-1}{N-1}\leq r< p, \\ (1,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} ), & \textit{if } 1\leq r< \min \{\frac{2N-1}{N-1},p\}. \end{cases}\displaystyle \end{aligned}$$
(8)

Some comments and remarks

Consider the Dirichlet boundary value problem having a form

$$\begin{aligned} \textstyle\begin{cases} -\operatorname{div} \frac{ \vert \nabla u \vert ^{p-2}\nabla u}{(1+ \vert u \vert )^{\theta (p-1)}}+bu=f, & \text{in } \varOmega , \\ u=0, & \text{on } \partial \varOmega , \end{cases}\displaystyle \end{aligned}$$
(9)

with \(p>1\), \(\theta \in (0,1]\), \(b\geq 0\), \(f\in L^{1}(\varOmega )\). The item \(-\operatorname{div}\frac{ \vert \nabla u \vert ^{p-2}\nabla u}{(1+ \vert u \vert )^{\theta (p-1)}}\) may not be coercive when u tends to infinity. Due to this fact, the classical methods used to prove the existence of a solution for elliptic equations, e.g., [14], cannot be applied even if \(b=0\) and the data f is regular. Moreover, \(\frac{ \vert \nabla u \vert ^{p-2}\nabla u}{(1+ \vert u \vert )^{\theta (p-1)}}\), u and f are only in \(L^{1}(\varOmega )\), not in \(W^{-1,p'}(\varOmega )\). All these characteristics prevent us from employing the duality argument [17] or nonlinear monotone operator theory [18] directly.

To overcome these difficulties, “cutting” the non-coercivity term and using the technique of approximation, a pseudomonotone and coercive differential operator on \(W^{1,p}_{0}(\varOmega )\) can be applied to establish a priori estimates on approximating solutions. As a result, existence of solutions, or entropy solutions, can be obtained by taking limitation for \(f\in L^{m}(\varOmega )\), \(m\geq 1\), and \(b> 0\) due to the almost everywhere convergence of gradients of the approximating solutions, see, e.g., [4, 6, 9,10,11, 15] (see also [1, 2, 7, 12, 13, 16] for \(b=0\)). However, there is little literature that considers regularities for entropy solutions of obstacle problems governed by (1) and functions \((f,\psi ,g) \) with \(f\in L^{1}(\varOmega )\), except [19], in which the authors considered the obstacle problem (7) with \(b=0\) and \(L^{1}\)-data.

Motivated by the study on the non-coercive elliptic equations (9) and the problem considered in [19], in this paper, we consider the obstacle problem governed by (1) and functions \((f,\psi ,g) \) with \(f\in L^{1}(\varOmega )\). By the truncation method used in [8] and [19], we prove the existence of an entropy solution and show its continuous dependence on the \(L^{1}\)-data in \(W^{1,q}(\varOmega )\) with some \(q\in (1,p)\).

In the following, we give some remarks on our main result and inequalities that will be needed in the proofs. Some notations are provided at the end of this subsection.

Remark 1

  1. (i)

    \(0\leq \theta < \min \{\frac{N}{N-1}-\frac{1}{p-1}, \frac{p-r}{p-1} \}\Rightarrow r-1<(1-\theta )(p-1)< \frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)}\). Therefore Theorem 1 guarantees \(\vert u \vert ^{r-1}\in L^{1}(\varOmega )\), and the second integration in (7) makes sense.

  2. (ii)

    We will show that \(\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\in (L^{1}(\varOmega ))^{N} \) in Proposition 4. Therefore, the first integration in (7) makes sense.

  3. (iii)

    \(( \frac{N(r-1)}{N+r-1}, \frac{N(p-1)(1-\theta )}{N-1- \theta (p-1)} )\subset (1,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\) if \(\frac{2N-1}{N-1}\leq r< p\). Indeed, \(\theta < \frac{p-r}{p-1}+\frac{p(r-1)}{N(p-1)}\Leftrightarrow \frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)}>\frac{N(r-1)}{N+r-1}\), while \(\frac{2N-1}{N-1}\leq r \) gives \(\frac{N(r-1)}{N+r-1}\geq 1 \). Thus \(u_{n}\rightarrow u\) in \(W^{1,q}(\varOmega )\) for all \(q\in (1,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\).

  4. (iv)

    \(r-1<\frac{Nq}{N-q}\). Indeed, by \(1\leq r<\frac{2N-1}{N-1}\), there holds \(r-1<\frac{N}{N-1}< \frac{Nq}{N-q}\) for any \(q>1\), particularly, for \(q\in (1, \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\). For \(r\geq \frac{2N-1}{N-1}\), it suffices to note that \(q> \frac{N(r-1)}{N+r-1} \Leftrightarrow r-1< \frac{Nq}{N-q}\).

  5. (v)

    \(q< p\). Indeed, \(0\leq \theta <\frac{N}{N-1}-\frac{1}{p-1}< \frac{N-1}{p-1} \Rightarrow \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)}<p \).

Remark 2

Checking proofs in this paper (e.g., setting \(r=1\)), one may find that, for \(b=0\), (8) holds with

$$\begin{aligned} u_{n}\rightarrow u \quad \text{in } W^{1,q}(\varOmega ),\forall q \in \biggl(1,\frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)} \biggr), \end{aligned}$$
(10)

which is the same as [19, Theorem 1]. Thus, Theorem 1 can be seen as an extension of [19, Theorem 1].

Notations

\(\Vert u \Vert _{p}:= \Vert u \Vert _{L^{p}(\varOmega )}\) is the norm of u in \(L^{p}(\varOmega )\), where \(1\leq p\leq \infty \). \(\Vert u \Vert _{1,p}:= \Vert u \Vert _{W^{1,p}(\varOmega )}\) is the norm of u in \(W^{1,p}(\varOmega )\), where \(1< p<\infty \). \(p':=\frac{p}{p-1}\) with \(1< p<\infty \). \(\{u>s\}:=\{x\in \varOmega ;u(x)>s \}\). \(\{u\leq s\}:=\varOmega \setminus \{u>s\}\). \(\{u< s\}:=\{x\in \varOmega ;u(x)< s\}\). \(\{u\geq s\}:=\varOmega \setminus \{u< s\}\). \(\{u=s\}:=\{x \in \varOmega ;u(x)=s\}\). \(\{t\leq u < s\}:=\{u\geq t\}\cap \{u<s\} \). For a measurable set E in \(\mathbb{R}^{N}\), \(\vert E \vert :=\mathcal{L}^{N}(E)\), where \(\mathcal{L}^{N}\) is the Lebesgue measure of \(\mathbb{R}^{N}\). For a real-valued function u, \(u^{+}=\max \{u,0\}\), \(u^{-}=(-u)^{+}\). Without special statements, positive integers are denoted by n, h, k, \(k_{0}\)K. C is a positive constant, which may be different from each other.

Lemmas on entropy solutions

It is worthy to note that, for any smooth function \(f_{n}\), there exists at least one solution to the obstacle problem (6). Indeed, one can proceed exactly as in [1, 11] to obtain \(W^{1,p}\)-solutions due to assumptions (2)–(4) on a and \(r-1< p\). These solutions, in particular, are also entropy solutions. In this section, using the method of [8] and [19], we establish several auxiliary results on convergence of sequences of entropy solutions when \(f_{n}\rightarrow f\) in \(L^{1}(\varOmega )\).

Lemma 2

Let \(v_{0}\in K_{g,\psi }\cap L^{\infty }(\varOmega )\), and let u be an entropy solution of the obstacle problem associated with \((f,\psi ,g)\). Then we have

$$\begin{aligned} \int _{\{ \vert u \vert < t\}}\frac{ \vert \nabla u \vert ^{p}}{(1+ \vert u \vert )^{\theta (p-1)}}\,\mathrm{d}x \leq C \bigl(1+t^{r}\bigr), \quad \forall t>0, \end{aligned}$$

where C is a positive constant depending only on α, β, p, r, b, \(\Vert j \Vert _{p'}\), \(\Vert \nabla v_{0} \Vert _{p}\), \(\Vert v_{0} \Vert _{\infty }\), and \(\Vert f \Vert _{1}\).

Proof

Take \(v_{0}\) as a test function in (7). For t large enough such that \(t- \Vert v_{0} \Vert _{\infty }>0\), we get

$$\begin{aligned} \int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla u}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x\leq{}& \int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{\theta (p-1)}}\,\mathrm{d}x \\ & {} + \int _{\varOmega }\bigl(f-b \vert u \vert ^{r-2}u\bigr) T_{t}(u-v_{0})\,\mathrm{d}x. \end{aligned}$$
(11)

We estimate each integration in the right-hand side of (11). It follows from (3) and Young’s inequality with \(\varepsilon >0\) that

$$\begin{aligned}& \int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x\leq \int _{\{ \vert v_{0}-u \vert < t\}}\frac{\beta ( \vert j \vert + \vert \nabla u \vert ^{p-1})\cdot \vert \nabla v_{0} \vert }{(1+ \vert u \vert )^{\theta (p-1)}}\,\mathrm{d}x \\& \hphantom{\int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x}\leq \int _{\{ \vert v_{0}-u \vert < t\}}\frac{\beta \varepsilon ( \vert j \vert ^{p'}+ \vert \nabla u \vert ^{p})}{(1+ \vert u \vert )^{\theta (p-1)}}\,\mathrm{d}x \\& \hphantom{\int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x}\quad {} + \int _{\{ \vert v_{0}-u \vert < t\}}\frac{\beta C(\varepsilon ) \vert \nabla v_{0} \vert ^{p}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x \\& \hphantom{\int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x} \leq \varepsilon \int _{\{ \vert v_{0}-u \vert < t\}}\frac{ \vert \nabla u \vert ^{p}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x \\& \hphantom{\int _{\{ \vert v_{0}-u \vert < t\}}\frac{a(x, \nabla u)\cdot \nabla v_{0}}{(1+ \vert u \vert )^{ \theta (p-1)}}\,\mathrm{d}x}\quad {} +C\bigl( \Vert j \Vert _{p'}^{p'}+ \Vert \nabla v_{0} \Vert _{p}^{p}\bigr), \end{aligned}$$
(12)
$$\begin{aligned}& \begin{aligned}[b] - \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \,\mathrm{d}x={}&- \int _{\{ \vert u-v_{0} \vert \leq t\}} b \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \,\mathrm{d}x \\ & {} - \int _{\{ \vert u-v_{0} \vert > t\}}b \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \,\mathrm{d}x. \end{aligned} \end{aligned}$$
(13)

Note that on the set \(\{ \vert u-v_{0} \vert \leq t\}\),

$$\begin{aligned} \bigl\vert \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \bigr\vert \leq t \bigl\vert t+ \Vert v_{0} \Vert _{\infty } \bigr\vert ^{r-1}\leq C\bigl(1+t ^{r} \bigr), \end{aligned}$$
(14)

where C is a constant depending only on r, \(\Vert v_{0} \Vert _{\infty }\).

On the set \(\{ \vert u-v_{0} \vert > t\}\), we have \(\vert u \vert \geq t- \Vert v_{0} \Vert _{\infty } >0\), thus u and \(T_{t}(u-v_{0})\) have the same sign. It follows

$$\begin{aligned} - \int _{\{ \vert u-v_{0} \vert > t\}} b \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \,\mathrm{d}x\leq 0. \end{aligned}$$
(15)

Combining (13)–(15), we get

$$\begin{aligned}& - \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v_{0}) \,\mathrm{d}x\leq C\bigl(1+t^{r}\bigr), \end{aligned}$$
(16)
$$\begin{aligned}& \begin{aligned}[b] \int _{\{ \vert v_{0}-u \vert < t\}}\frac{ \vert \nabla u \vert ^{p}}{(1+ \vert u \vert )^{\theta (p-1)}} \,\mathrm{d}x\leq{}& C\bigl( \Vert j \Vert _{p'}^{p'}+ \Vert \nabla v_{0} \Vert _{p}^{p}+t \Vert f \Vert _{1}+1+t ^{r}\bigr) \\ \leq{}& C\bigl(1+t^{r}\bigr). \end{aligned} \end{aligned}$$
(17)

Replacing t with \(t+ \Vert v_{0} \Vert _{\infty }\) in (17) and noting that \(\{ \vert u \vert < t\}\subset \{ \vert v_{0}-u \vert < t+ \Vert v_{0} \Vert _{\infty }\}\), one may obtain the desired result. □

In the rest of this section, let \(\{u_{n}\}\) be a sequence of entropy solutions of the obstacle problem associated with \((f_{n},\psi ,g)\) and assume that

$$\begin{aligned} f_{n}\rightarrow f \quad \text{in } L^{1}(\varOmega ) \quad \text{and} \quad \Vert f_{n} \Vert _{1}\leq \Vert f \Vert _{1}+1. \end{aligned}$$

Lemma 3

There exists a measurable function u such that \(u_{n}\rightarrow u \) in measure, and \(T_{k}(u_{n})\rightharpoonup T_{k}(u)\) weakly in \(W^{1,p}(\varOmega )\) for any \(k>0\). Thus \(T_{k}(u_{n})\rightarrow T _{k}(u) \) strongly in \(L^{p}(\varOmega )\) and a.e. in Ω.

Proof

Let s, t, and ε be positive numbers. One may verify that, for every \(m,n\geq 1\),

$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n}-u_{m} \vert >s\bigr\} \bigr)\leq{}& \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n} \vert >t \bigr\} \bigr)+ \mathcal{L}^{N}\bigl( \bigl\{ \vert u_{m} \vert >t\bigr\} \bigr) \\ & {} +\mathcal{L}^{N}\bigl(\bigl\{ \bigl\vert T_{k}(u_{n})-T_{k}(u_{m}) \bigr\vert >s\bigr\} \bigr), \end{aligned}$$
(18)

and

$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n} \vert >t \bigr\} \bigr)=\frac{1}{t^{p}} \int _{\{ \vert u_{n} \vert >t\}}t ^{p}\,\mathrm{d}x\leq \frac{1}{t^{p}} \int _{\varOmega } \bigl\vert T_{t}(u_{n}) \bigr\vert ^{p} \,\mathrm{d}x. \end{aligned}$$
(19)

Due to \(v_{0}=g+(\psi -g)^{+}\in K_{g,\psi }\cap L^{\infty }(\varOmega )\), by Lemma 2, one has

$$\begin{aligned} \int _{\varOmega } \bigl\vert \nabla T_{t}(u_{n}) \bigr\vert ^{p}\,\mathrm{d}x= \int _{\{ \vert u_{n} \vert < t\}} \vert \nabla u_{n} \vert ^{p}\,\mathrm{d}x \leq C(1+t)^{\theta (p-1)}\bigl(1+t^{r} \bigr). \end{aligned}$$
(20)

Note that \(T_{t}(u_{n})-T_{t}(g)\in W^{1,p}_{0}(\varOmega )\). By (19), (20), and Poincaré’s inequality, for every \(t> \Vert g \Vert _{\infty }\) and for some positive constant C independent of n and t, there holds

$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n} \vert >t \bigr\} \bigr)\leq{}& \frac{1}{t^{p}} \int _{\varOmega } \bigl\vert T _{t}(u_{n}) \bigr\vert ^{p}\,\mathrm{d}x \\ \leq{}& \frac{2^{p-1}}{t^{p}} \int _{\varOmega } \bigl\vert T_{t}(u_{n})-T_{t}(g) \bigr\vert ^{p} \,\mathrm{d}x+\frac{2^{p-1}}{t^{p}} \Vert g \Vert _{p}^{p} \\ \leq{}& \frac{C}{t^{p}} \int _{\varOmega } \bigl\vert \nabla T_{t}(u_{n})- \nabla T _{t}(g) \bigr\vert ^{p}\,\mathrm{d}x+ \frac{2^{p-1}}{t^{p}} \Vert g \Vert _{p}^{p} \\ \leq{}&\frac{C}{t^{p}} \int _{\varOmega } \bigl\vert \nabla T_{t}(u_{n}) \bigr\vert ^{p}\,\mathrm{d}x+\frac{C}{t ^{p}} \Vert g \Vert _{1,p}^{p} \\ \leq{}& \frac{C(1+t^{r+\theta (p-1)})}{t^{p}}. \end{aligned}$$

Since \(0\leq \theta <\frac{p-r}{p-1}\), there exists \(t_{\varepsilon }>0\) such that

$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n} \vert >t \bigr\} \bigr)< \frac{\varepsilon }{3}, \quad \forall t\geq t_{\varepsilon }, \forall n\geq 1. \end{aligned}$$
(21)

Now we have as in (19)

$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \bigl\vert T_{t_{\varepsilon }}(u_{n})-T_{t_{\varepsilon }}(u _{m}) \bigr\vert >s\bigr\} \bigr) &=\frac{1}{s^{p}} \int _{\{ \vert T_{t_{\varepsilon }} (u_{n})-T_{t_{\varepsilon }}(u_{m}) \vert >s \}} s^{p}\,\mathrm{d}x \\ &\leq \frac{1}{s^{p}} \int _{\varOmega } \bigl\vert T_{t_{\varepsilon }}(u_{n})-T _{t_{\varepsilon }}(u_{m}) \bigr\vert ^{p}\,\mathrm{d}x. \end{aligned}$$
(22)

Using (20) and the fact that \(T_{t}(u_{n})-T_{t}(g)\in W^{1,p} _{0}(\varOmega )\) again, we see that \(\{T_{t_{\varepsilon }}(u_{n})\}\) is a bounded sequence in \(W^{1,p}(\varOmega )\). Thus, up to a subsequence, \(\{T_{t_{\varepsilon }}(u_{n})\}\) converges strongly in \(L^{p}( \varOmega )\). Taking into account (22), there exists \(n_{0}=n_{0}(t _{\varepsilon },s)\geq 1\) such that

$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \bigl\vert T_{t_{\varepsilon }}(u_{n})-T_{t_{\varepsilon }}(u _{m}) \bigr\vert >s\bigr\} \bigr)< \frac{\varepsilon }{3}, \quad \forall n,m\geq n_{0}. \end{aligned}$$
(23)

Combining (18), (21), and (23), we obtain

$$\begin{aligned} \mathcal{L}^{N}\bigl(\bigl\{ \vert u_{n}-u_{m} \vert >s\bigr\} \bigr)< \varepsilon , \quad \forall n,m\geq n_{0}. \end{aligned}$$

Hence \(\{u_{n}\}\) is a Cauchy sequence in measure, and therefore there exists a measurable function u such that \(u_{n}\rightarrow u\) in measure. The remainder of the lemma is a consequence of the fact that \(\{T_{k}(u_{n})\}\) is a bounded sequence in \(W^{1,p}(\varOmega )\). □

Proposition 4

There exist a subsequence of \(\{u_{n}\}\) and a measurable function u such that, for each q given in (8), we have

$$\begin{aligned} u_{n}\rightarrow u \quad \textit{strongly in } W^{1,q}( \varOmega ). \end{aligned}$$

Furthermore, if \(0\leq \theta < \min \{ \frac{1}{N-p+1},\frac{N}{N-1}-\frac{1}{p-1},\frac{p-r}{p-1} \}\), then

$$\begin{aligned} \frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \rightarrow \frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}} \quad \textit{strongly in } \bigl(L^{1}(\varOmega )\bigr)^{N}. \end{aligned}$$

To prove Proposition 4, we need two preliminary lemmas.

Lemma 5

There exist a subsequence of \(\{u_{n}\}\) and a measurable function u such that, for each q given in (8), we have \(u_{n}\rightharpoonup u \) weakly in \(W^{1,q}(\varOmega )\), and \(u_{n}\rightarrow u \) strongly in \(L^{q}(\varOmega ) \).

Proof

Let \(k>0\) and \(n\geq 1\). Define \(D_{k}=\{ \vert u_{n} \vert \leq k\}\) and \(B_{k}=\{k\leq \vert u_{n} \vert < k+1\}\). Using Lemma 2 with \(v_{0}=g+(\psi -g)^{+}\), we get

$$\begin{aligned} \int _{D_{k}}\frac{ \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \,\mathrm{d}x\leq C \bigl(1+k^{r}\bigr), \end{aligned}$$
(24)

where C is a positive constant depending only on α, β, b, p, r, \(\Vert j \Vert _{p'}\), \(\Vert f \Vert _{1}\), \(\Vert \nabla v_{0} \Vert _{p}\), and \(\Vert v_{0} \Vert _{\infty }\).

Using the function \(T_{k}(u_{n})\) for \(k> \{ \Vert g \Vert _{\infty }, \Vert \psi \Vert _{\infty }\}\), as a test function for the problem associated with \((f_{n},\psi ,g)\), we obtain

$$\begin{aligned} & \int _{\varOmega }\frac{a(x,\nabla u_{n})\cdot \nabla (T_{1}(u_{n}-T_{k}(u _{n})))}{(1+ \vert u_{n} \vert )^{\theta (p-1)}}\,\mathrm{d}x+ \int _{\varOmega }b \vert u_{n} \vert ^{r-2}u _{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr)\,\mathrm{d}x \\ & \quad \leq \int _{\varOmega }f_{n}T_{1} \bigl(u_{n}-T_{k}(u_{n})\bigr)\, \mathrm{d}x, \end{aligned}$$

which and (2) give

$$\begin{aligned} \int _{B_{k}} \frac{\alpha \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{\theta (p-1)}}\,\mathrm{d}x+ \int _{\varOmega }b \vert u_{n} \vert ^{r-2} u_{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr)\,\mathrm{d}x \leq \Vert f_{n} \Vert _{1}\leq \Vert f \Vert _{1}+1. \end{aligned}$$

Note that on the set \(\{ \vert u_{n} \vert \geq k+1\} \), \(u_{n}\) and \(T_{1}(u_{n}-T _{k}(u_{n}))\) have the same sign. Then

$$\begin{aligned} \int _{\varOmega } \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr)\,\mathrm{d}x ={}& \int _{D_{k}} \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr)\,\mathrm{d}x \\ & {} + \int _{B_{k}} \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr)\,\mathrm{d}x \\ & {} + \int _{\{ \vert u_{n} \vert \geq k+1\}} \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u _{n})\bigr)\,\mathrm{d}x \\ \geq{}& \int _{B_{k}} \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u_{n}) \bigr) \,\mathrm{d}x. \end{aligned}$$

Thus we have

$$\begin{aligned} \int _{B_{k}} \frac{\alpha \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{\theta (p-1)}}\,\mathrm{d}x+ \leq{}& \Vert f \Vert _{1}+1- \int _{B_{k}}b \vert u_{n} \vert ^{r-2}u_{n}T_{1}\bigl(u_{n}-T_{k}(u _{n})\bigr)\,\mathrm{d}x \\ \leq{}& \Vert f \Vert _{1}+1+ \int _{B_{k}}b \vert u_{n} \vert ^{r-1}\,\mathrm{d}x \\ \leq{}& C \biggl(1+ \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{\frac{r-1}{q ^{*}}} \vert B_{k} \vert ^{1-\frac{r-1}{q^{*}}} \biggr) , \end{aligned}$$
(25)

where q is given in (8) and \(q^{*}=\frac{Nq}{N-q} \).

Let \(s=\frac{q\theta (p-1)}{p}\). Note that \(q< p\) and \(\frac{ps}{p-q}< q ^{*}\). For \(\forall k>0\), we estimate \(\int _{B_{k}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x\) as follows:

$$\begin{aligned} \int _{B_{k}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x={}& \int _{B_{k}}\frac{ \vert \nabla u _{n} \vert ^{q}}{(1+ \vert u_{n} \vert )^{s}}\cdot \bigl(1+ \vert u_{n} \vert \bigr)^{s}\,\mathrm{d}x \\ \leq{}& \biggl( \int _{B_{k}}\frac{ \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{ \theta (p-1)}}\,\mathrm{d}x \biggr)^{\frac{q}{p}} \biggl( \int _{B_{k}}\bigl(1+ \vert u _{n} \vert \bigr)^{\frac{ps}{p-q}}\,\mathrm{d}x \biggr)^{\frac{p-q}{p}} \\ \leq{}& C \biggl( \int _{B_{k}}\frac{ \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{ \theta (p-1)}}\,\mathrm{d}x \biggr)^{\frac{q}{p}} \biggl( \vert B_{k} \vert ^{ \frac{p-q}{p}}+ \biggl( \int _{B_{k}} \vert u_{n} \vert ^{\frac{ps}{p-q}}\,\mathrm{d}x \biggr)^{\frac{p-q}{p}} \biggr) \\ \leq{}& C \biggl( \int _{B_{k}}\frac{ \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{ \theta (p-1)}}\,\mathrm{d}x \biggr)^{\frac{q}{p}} \biggl( \vert B_{k} \vert ^{ \frac{p-q}{p}} + \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{\frac{s}{q ^{*}}} \vert B_{k} \vert ^{\frac{p-q}{p}-\frac{s}{q^{*}}} \biggr) \\ \leq{}& C \biggl( \vert B_{k} \vert ^{\frac{p-q}{p}}+ \vert B_{k} \vert ^{\frac{p-q}{p}-\frac{s}{q ^{*}}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{ \frac{s}{q^{*}}}+ \vert B_{k} \vert ^{1-p_{1}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \biggr)^{p_{1}} \\ & {} + \vert B_{k} \vert ^{1-p_{2}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{p _{2}} \biggr) \quad \text{by (25)} \\ ={}& C \biggl( \vert B_{k} \vert ^{\frac{p-q}{p}}+ \vert B_{k} \vert ^{\frac{p-q}{p}-\frac{s}{q ^{*}}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{\frac{s}{q^{*}}} \\ & {} + \vert B_{k} \vert ^{1-p_{1}-C_{1}} \vert B_{k} \vert ^{C_{1}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q ^{*}}\,\mathrm{d}x \biggr)^{p_{1}} \\ & {} + \vert B_{k} \vert ^{1-p_{2}-C_{2}} \vert B_{k} \vert ^{C_{2}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q ^{*}}\,\mathrm{d}x \biggr)^{p_{2}} \biggr), \end{aligned}$$

where \(p_{1}=\frac{q}{p}\frac{r-1}{q^{*}}\), \(p_{2}=\frac{s}{q^{*}}+ \frac{q}{p}\frac{r-1}{q^{*}}\), \(C_{1}\) and \(C_{2}\) are positive constants to be chosen later.

Note that \(\theta <\frac{p-r}{p-1}\), it follows

$$\begin{aligned} \frac{\theta (p-1)}{p}+\frac{r-1}{p}< \frac{p-1}{p}< 1- \frac{1}{N}=1- \frac{1}{q}+\frac{1}{q^{*}}. \end{aligned}$$

Thus

$$\begin{aligned} \frac{\theta q(p-1)}{p}+\frac{q(r-1)}{p}+1< q+\frac{q}{q^{*}} \quad &\Leftrightarrow\quad s+\frac{q(r-1)}{p}+1< q+\frac{q}{q^{*}} \\ & \Leftrightarrow\quad p_{2}+\frac{1-p _{2}}{q^{*}+1}< \frac{q}{q^{*}}. \end{aligned}$$

Note that \(p_{1}< p_{2}<1\). Then, for \(i=1,2\), we always have

$$\begin{aligned} p_{i}+\frac{1-p_{i}}{q^{*}+1}< \frac{q}{q^{*}}< 1. \end{aligned}$$

From this, we may find positive \(C_{i}\) (\(i=1,2\)) such that

$$\begin{aligned} p_{i}+\frac{1-p_{i}}{q^{*}+1}< p_{i}+C_{i}< \frac{q}{q^{*}}< 1, \quad i=1,2. \end{aligned}$$
(26)

It follows

$$\begin{aligned} \frac{1-p_{i}}{q^{*}+1}< C_{i}\quad \Leftrightarrow \quad 1-p_{i}-C_{i}< C_{i}q^{*}, \quad i=1,2, \end{aligned}$$

which implies

$$\begin{aligned} C_{i}\alpha _{i}q^{*}= \frac{C_{i}q^{*}}{1-p_{i}-C_{i}}>1, \quad i=1,2, \end{aligned}$$
(27)

with \(\alpha _{i}=\frac{1}{1-p_{i}-C_{i}}>1\), \(i=1,2\). Let \(\beta _{i}=\frac{1}{p _{i}+C_{i}}>1\), \(i=1,2\). Then we have \(\frac{1}{\alpha _{i}}+\frac{1}{ \beta _{i}}=1\) (\(i=1,2\)).

Since \(\vert B_{k} \vert \leq \frac{1}{k^{q^{*}}}\int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \), \(\vert B_{k} \vert ^{1-p_{1}-C_{1}} \leq \vert \varOmega \vert ^{1-p_{1}-C_{1}}\), and \(\vert B_{k} \vert ^{1-p_{2}-C_{2}} \leq \vert \varOmega \vert ^{1-p_{2}-C_{2}}\), we have, for \(k\geq k_{0}\geq 1\),

$$\begin{aligned} \int _{B_{k}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x\leq{}&\frac{C}{k^{q^{*} ( \frac{p-q}{p}-\frac{s}{q^{*}} )}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \biggr)^{\frac{p-q}{p}} \\ & {} +\frac{C}{k^{q^{*}C_{1}}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x \biggr)^{p_{1}+C_{1}}+\frac{C}{k^{q^{*}C_{2}}} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q ^{*}}\,\mathrm{d}x \biggr)^{p_{2}+C_{2}}. \end{aligned}$$

Summing up from \(k=k_{0}\) to \(k=K\) and using Hölder’s inequality, one has

$$\begin{aligned} \sum_{k=k_{0}}^{K} \int _{B_{k}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x\leq{}& C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}(\frac{p-q}{p}-\frac{s}{q ^{*}})\frac{p}{q}}} \Biggr)^{\frac{q}{p}}\cdot \Biggl(\sum _{k=k_{0}} ^{K} \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \Biggr)^{\frac{p-q}{p}} \\ & {} +C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C_{1}\alpha _{1}}} \Biggr)^{\frac{1}{ \alpha _{1}}} \cdot \Biggl(\sum _{k=k_{0}}^{K} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q ^{*}} \,\mathrm{d}x \biggr)^{\beta _{1}(p_{1}+C_{1})} \Biggr)^{\frac{1}{\beta _{1}}} \\ & {} +C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C_{2}\alpha _{2}}} \Biggr)^{\frac{1}{ \alpha _{2}}} \cdot \Biggl(\sum _{k=k_{0}}^{K} \biggl( \int _{B_{k}} \vert u_{n} \vert ^{q ^{*}} \,\mathrm{d}x \biggr)^{\beta _{2}(p_{2}+C_{2})} \Biggr)^{\frac{1}{\beta _{2}}} \\ ={}& C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}(\frac{p-q}{p}-\frac{s}{q ^{*}})\frac{p}{q}}} \Biggr)^{\frac{q}{p}}\cdot \Biggl(\sum _{k=k_{0}} ^{K} \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \Biggr)^{\frac{p-q}{p}} \\ & {} +C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C_{1}\alpha _{1}}} \Biggr)^{\frac{1}{ \alpha _{1}}} \cdot \Biggl(\sum _{k=k_{0}}^{K} \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \Biggr)^{p_{1}+C_{1}} \\ & {} +C \Biggl(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C_{2}\alpha _{2}}} \Biggr)^{\frac{1}{ \alpha _{2}}} \cdot \Biggl(\sum _{k=k_{0}}^{K} \int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \Biggr)^{p_{2}+C_{2}}. \end{aligned}$$
(28)

Note that

$$\begin{aligned} \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x= \int _{D_{k_{0}}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x+\sum_{k=k_{0}}^{K} \int _{B_{k}} \vert \nabla u _{n} \vert ^{q} \,\mathrm{d}x. \end{aligned}$$
(29)

To estimate the first integral in the right-hand side of (29), we compute by using Hölder’s inequality and (24), obtaining

$$\begin{aligned} \int _{D_{k_{0}}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x \leq{}& \biggl( \int _{D_{k _{0}}}\frac{ \vert \nabla u_{n} \vert ^{p}}{(1+ \vert u_{n} \vert )^{\theta (p-1)}}\,\mathrm{d}x \biggr)^{\frac{q}{p}} \biggl( \int _{D_{k_{0}}}\bigl(1+ \vert u_{n} \vert \bigr)^{ \frac{ps}{p-q}} \,\mathrm{d}x \biggr)^{\frac{p-q}{p}} \\ \leq{}& C, \end{aligned}$$
(30)

where C depends only on α, β, b, p, θ, \(\Vert j \Vert _{p'}\), \(\Vert f \Vert _{1}\), \(\Vert \nabla v_{0} \Vert _{p}\), \(\Vert v_{0} \Vert _{\infty }\), and \(k_{0}\).

Note that \(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}(\frac{p-q}{p}-\frac{s}{q ^{*}})\frac{p}{q}}}\) and \(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C _{i}\alpha _{i}}}\) converge as \(K\rightarrow \infty \) due to the fact that \(q^{*}(\frac{p-q}{p}-\frac{s}{q^{*}})\frac{p}{q}>1\) and \(q^{*}C_{i} \alpha _{i}>1\) by (27), respectively. Combining (28)–(30), we get for \(k_{0}\) large enough

$$\begin{aligned} \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x\leq{}& C+C \biggl( \int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \biggr)^{ \frac{p-q}{p}} \\ & {} + C \biggl( \int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \biggr)^{p _{1}+C_{1}} \\ & {} +C \biggl( \int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \biggr)^{p _{2}+C_{2}}. \end{aligned}$$
(31)

Since \(p>q\), \(T_{K}(u_{n})\in W^{1,q}(\varOmega )\), \(T_{K}(g)=g\in W^{1,q}( \varOmega )\) for \(K> \Vert g \Vert _{\infty }\). Hence \(T_{K}(u_{n})-g\in W^{1,q} _{0}(\varOmega )\). Using the Sobolev embedding \(W^{1,q}_{0}(\varOmega ) \subset L^{q^{*}}(\varOmega )\) and Poincaré’s inequality, we obtain

$$\begin{aligned} \bigl\Vert T_{K}(u_{n}) \bigr\Vert ^{q}_{q^{*}}\leq{}& 2^{q-1}\bigl( \bigl\Vert T_{K}(u_{n})-g \bigr\Vert ^{q}_{q ^{*}}+ \Vert g \Vert ^{q}_{q^{*}}\bigr) \\ \leq{}& C\bigl( \bigl\Vert \nabla \bigl(T_{K}(u_{n})-g \bigr) \bigr\Vert ^{q}_{q}+ \Vert g \Vert ^{q}_{q^{*}}\bigr) \\ \leq{}& C \bigl( \bigl\Vert \nabla T_{K}(u_{n}) \bigr\Vert ^{q}_{q}+ \Vert \nabla g \Vert ^{q}_{q}+ \Vert g \Vert ^{q}_{q^{*}} \bigr) \\ \leq{}& C \biggl(1+ \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x \biggr). \end{aligned}$$
(32)

Using the fact that

$$\begin{aligned} \int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x\leq \int _{\{ \vert u_{n} \vert \leq K\}} \bigl\vert T_{K}( u_{n}) \bigr\vert ^{q^{*}} \,\mathrm{d}x\leq \bigl\Vert T _{K}( u_{n}) \bigr\Vert ^{q^{*}}_{q^{*}}, \end{aligned}$$
(33)

we obtain from (31)–(32)

$$\begin{aligned} \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x\leq{}& C+C \biggl(1+ \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x \biggr)^{\frac{q ^{*}}{q}\frac{p-q}{p}} \\ & {} +C \biggl(1+ \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x \biggr)^{(p_{1}+C_{1})\frac{q^{*}}{q}} \\ & {} +C \biggl(1+ \int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x \biggr)^{(p_{2}+C_{2})\frac{q^{*}}{q}} . \end{aligned}$$
(34)

Note that \(p< N\Leftrightarrow \frac{q^{*}}{q}\frac{p-q}{p}<1\) and \((p_{i}+C_{i})\frac{q^{*}}{q}<1 \) by (26). It follows from (34) that, for \(k_{0}\) large enough, \(\int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x \) is bounded independently of n and K. Using (32) and (33), we deduce that \(\int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x\) is also bounded independently of n and K. Letting \(K\rightarrow \infty \), we deduce that \(\Vert \nabla u_{n} \Vert _{q}\) and \(\Vert u_{n} \Vert _{q^{*}}\) are uniformly bounded independently of n. Particularly, \(u_{n}\) is bounded in \(W^{1,q}(\varOmega )\). Therefore, there exist a subsequence of \(\{u_{n}\}\) and a function \(v\in W^{1,q}(\varOmega )\) such that \(u_{n}\rightharpoonup v\) weakly in \(W^{1,q}(\varOmega )\), \(u_{n}\rightarrow v\) strongly in \(L^{q}(\varOmega )\) and a.e. in Ω. By Lemma 3, \(u_{n}\rightarrow u\) in measure in Ω, we conclude that \(u=v\) and \(u\in W^{1,q}(\varOmega )\). □

Lemma 6

There exist a subsequence of \(\{u_{n}\}\) and a measurable function u such that \(\nabla u_{n}\) converges almost everywhere in Ω to u.

Proof

Define \(A(x,u,\xi )=\frac{a(x,\xi )}{(1+ \vert u \vert )^{\theta (p-1)}}\) (for the sake of simplicity, we omit the dependence of \(A(x,u,\xi )\) on x). Let \(h>0\), \(k> \max \{ \Vert g \Vert _{\infty }, \Vert \psi \Vert _{\infty }\}\), and \(n\geq h+k\). Take \(T_{k}(u)\) as a test function for (7), obtaining

$$\begin{aligned} I_{7}(n,k,h)\leq \int _{\varOmega }f_{n}T_{h} \bigl(u_{n}-T_{k}(u)\bigr)\,\mathrm{d}x+ \int _{\varOmega }b \vert u_{n} \vert ^{r-2}u_{n}T_{h}\bigl(u_{n}-T_{k}(u) \bigr)\,\mathrm{d}x, \end{aligned}$$

where

$$\begin{aligned} I_{7}(n,k,h)= \int _{\varOmega }A(u_{n},\nabla u_{n}) \cdot \nabla T_{h}\bigl(u_{n}-T_{k}(u) \bigr)\,\mathrm{d}x. \end{aligned}$$

Note that \(r-1< q^{*}\), and \(\int _{\varOmega } \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x\) is uniformly bounded (see the proof of Lemma 5), thus \(\vert u_{n} \vert \) converges strongly in \(L^{1}(\varOmega )\). Therefore we have

$$\begin{aligned} \lim_{n\rightarrow \infty } \int _{\varOmega } \vert u_{n} \vert ^{r-2}u_{n}T_{h}\bigl(u_{n}-T_{k}(u) \bigr)\,\mathrm{d}x= \int _{\varOmega } \vert u \vert ^{r-2}uT_{h} \bigl(u-T_{k}(u)\bigr)\,\mathrm{d}x. \end{aligned}$$

Then, using the strong convergence of \(f_{n}\) in \(L^{1}(\varOmega )\), one has

$$\begin{aligned} \lim_{n\rightarrow \infty }I_{7}(n,k,h) \leq \int _{\varOmega }-fT_{h}\bigl(u-T_{k}(u) \bigr)\,\mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}uT_{h} \bigl(u-T_{k}(u)\bigr)\,\mathrm{d}x. \end{aligned}$$

It follows

$$\begin{aligned} \lim_{k\rightarrow \infty }\lim_{n\rightarrow \infty }I_{7}(n,k,h) \leq 0. \end{aligned}$$

Thanks to Lemma 3 and Lemma 5, we can proceed exactly as [19, Lemma 6] to conclude that, up to subsequence, \(\nabla u_{n}\rightarrow \nabla u\) a.e. □

Proof of Proposition 4

We shall prove that \(\nabla u_{n}\) converges strongly to u in \(L^{q}(\varOmega )\) for each q being given by (8). To do that,we will apply Vitali’s theorem, using the fact that by Lemma 5, \(\nabla u_{n}\) is bounded in \(L^{q}(\varOmega )\) for each q given by (8). So let \(s\in (q,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)})\) and \(E\subset \varOmega \) be a measurable set. Then we have by Hölder’s inequality

$$\begin{aligned} \int _{E} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x\leq \biggl( \int _{E} \vert \nabla u_{n} \vert ^{r} \,\mathrm{d}x \biggr)^{\frac{q}{s}}\cdot \vert E \vert ^{\frac{s-q}{s}}\leq C \vert E \vert ^{ \frac{s-q}{s}}\rightarrow 0 \end{aligned}$$

uniformly in n, as \(\vert E \vert \rightarrow 0\). From this and from Lemma 6, we deduce that \(\nabla u_{n}\) converges strongly to u in \(L^{q}(\varOmega )\).

Now assume that \(0\leq \theta < \min \{\frac{1}{N-p+1},\frac{N}{N-1}- \frac{1}{p-1},\frac{p-r}{p-1}\}\). Note that since \(\nabla u_{n}\) converges to u a.e. in Ω, to prove the convergence

$$\begin{aligned} \frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \rightarrow \frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}} \quad \text{strongly in } \bigl(L^{1}(\varOmega )\bigr)^{N}, \end{aligned}$$

it suffices, thanks to Vitali’s theorem, to show that, for every measurable subset \(E\subset \varOmega \), \(\int _{E} \vert \frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \vert \,\mathrm{d}x\) converges to 0 uniformly in n, as \(\vert E \vert \rightarrow 0\). Note that \(p-1<\frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)})\) by assumptions. For any \(q\in (p-1, \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\), we deduce by Hölder’s inequality

$$\begin{aligned} \int _{E} \biggl\vert \frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \biggr\vert \,\mathrm{d}x\leq{}& \beta \int _{E}\bigl(j+ \vert \nabla u_{n} \vert ^{p-1}\bigr)\,\mathrm{d}x \\ \leq{}& \beta \Vert j \Vert _{p'} \vert E \vert ^{\frac{1}{p}}+\beta \biggl( \int _{E} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x \biggr)^{\frac{p-1}{q}} \vert E \vert ^{\frac{q-p+1}{q}} \\ \rightarrow &0 \quad \text{uniformly in } n \text{ as } \vert E \vert \rightarrow 0. \end{aligned}$$

 □

Lemma 7

There exists a subsequence of \(\{u_{n}\}\) such that, for all \(k>0\),

$$\begin{aligned} \frac{a(x, \nabla T_{k}(u_{n}))}{(1+ \vert T_{k}(u_{n}) \vert )^{\theta (p-1)}} \rightarrow \frac{a(x, \nabla T_{k}(u))}{(1+ \vert T_{k}(u) \vert )^{\theta (p-1)}} \quad \textit{strongly in } \bigl(L^{1}(\varOmega )\bigr)^{N}. \end{aligned}$$

Proof

See the proof of [19, Lemma 7]. □

Proof of the main result

Now we have gathered all the lemmas needed to prove the existence of an entropy solution to the obstacle problem associated with \((f,\psi ,g)\). In this part, let \(f_{n}\) be a sequence of smooth functions converging strongly to f in \(L^{1}(\varOmega )\), with \(\Vert f_{n} \Vert _{1}\leq \Vert f \Vert _{1}+1\). We consider the sequence of approximated obstacle problems associated with \((f_{n},\psi ,g)\). The proof can be proceeded in the same way as in [8] and [19]. We provide details for readers’ convenience.

Proof of Theorem 1

Let \(v\in K_{g, \psi }\cap L^{\infty }(\varOmega )\). Taking v as a test function in (7) associated with \((f_{n},\psi ,g)\), we get

$$\begin{aligned} & \int _{\varOmega }\frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \cdot \nabla \bigl(T_{t}(u_{n}-v)\bigr)\,\mathrm{d}x+ \int _{\varOmega }b \vert u_{n} \vert ^{r-2}u _{n}T_{t}(u_{n}-v)\, \mathrm{d}x \\ & \quad \leq \int _{\varOmega }f_{n}T_{t}(u_{n}-v) \,\mathrm{d}x. \end{aligned}$$

Since \(\{ \vert u_{n}-v \vert < t\}\subset \{ \vert u_{n} \vert < s\}\) with \(s=t+ \Vert v \Vert _{\infty }\), the previous inequality can be written as

$$\begin{aligned} \int _{\varOmega }\chi _{n}\nabla _{A} T_{s}(u_{n})\cdot \nabla v\,\mathrm{d}x \geq{}& \int _{\varOmega }-f_{n}T_{t}(u_{n}-v) \,\mathrm{d}x+ \int _{\varOmega }b \vert u _{n} \vert ^{r-2}u_{n}T_{t}(u_{n}-v)\, \mathrm{d}x \\ & {} + \int _{\varOmega }\chi _{n}\nabla _{A} T_{s}(u_{n})\cdot \nabla T_{s}(u _{n})\,\mathrm{d}x, \end{aligned}$$
(35)

where \(\chi _{n}=\chi _{\{ \vert u_{n}-v \vert < t\}}\) and \(\nabla _{A}u=\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\). It is clear that \(\chi _{n} \rightharpoonup \chi \) weakly* in \(L^{\infty }(\varOmega )\). Moreover, \(\chi _{n}\) converges a.e. to \(\chi _{\{ \vert u-v \vert < t\}}\) in \(\varOmega \setminus \{ \vert u-v \vert =t\}\). It follows that

$$\begin{aligned} \chi =\textstyle\begin{cases} 1, & \text{in } \{ \vert u-v \vert < t\}, \\ 0, & \text{in } \{ \vert u-v \vert >t\}. \end{cases}\displaystyle \end{aligned}$$

Note that we have \(\mathcal{L}^{N}(\{ \vert u-v \vert =t\})=0\) for a.e. \(t\in (0,\infty )\). So there exists a measurable set \(\mathcal{O} \subset (0,\infty )\) such that \(\mathcal{L}^{N}(\{ \vert u-v \vert =t\})=0\) for all \(t\in (0,\infty )\setminus \mathcal{O}\). Assume that \(t\in (0,\infty )\setminus \mathcal{O}\). Then \(\chi _{n}\) converges weakly* in \(L^{\infty }(\varOmega )\) and a.e. in Ω to \(\chi =\chi _{\{ \vert u-v \vert < t \}}\). Since \(\nabla T_{s}(u_{n})\) converges a.e. to \(\nabla T_{s}(u)\) in Ω (Proposition 4), we obtain by Fatou’s lemma

$$\begin{aligned} \liminf_{n\rightarrow \infty } \int _{\varOmega }\chi _{n}\nabla _{A} T_{s}(u _{n})\cdot \nabla T_{s}(u_{n}) \,\mathrm{d}x\geq \int _{\varOmega }\chi \nabla _{A} T_{s}(u) \cdot \nabla T_{s}(u)\,\mathrm{d}x. \end{aligned}$$
(36)

Using the strong convergence of \(\nabla _{A} T_{s}(u_{n})\) to \(\nabla _{A} T_{s}(u)\) in \(L^{1}(\varOmega )\) (Lemma 7) and the weak* convergence of \(\chi _{n}\) to χ in \(L^{\infty }(\varOmega )\), we obtain

$$\begin{aligned} \lim_{n\rightarrow \infty } \int _{\varOmega }\chi _{n}\nabla _{A} T_{s}(u _{n})\cdot \nabla v\,\mathrm{d}x= \int _{\varOmega }\chi \nabla _{A} T_{s}(u) \cdot \nabla v\,\mathrm{d}x. \end{aligned}$$
(37)

Moreover, due to the strong convergence of \(f_{n}\) to f and \(\vert u_{n} \vert ^{r-2}u_{n} \) to \(\vert u \vert ^{r-2}u\) (by \(r-1< q^{*}\) and the boundedness of \(\Vert u_{n} \Vert _{q^{*}}\)) in \(L^{1}(\varOmega )\), and the weak* convergence of \(T_{t}(u_{n}-v)\) to \(T_{t}(u-v)\) in \(L^{\infty }( \varOmega )\), by passing to the limit in (35) and taking into account (36)–(37), we obtain

$$\begin{aligned} \int _{\varOmega }\chi \nabla _{A} T_{s}(u) \cdot \nabla v\,\mathrm{d}x- \int _{\varOmega }\chi \nabla _{A} T_{s}(u) \cdot \nabla T_{s}(u)\,\mathrm{d}x \geq{}& \int _{\varOmega }-fT_{t}(u-v)\,\mathrm{d}x \\ & {} + \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v) \,\mathrm{d}x, \end{aligned}$$

which can be written as

$$\begin{aligned} \int _{\{ \vert v-u \vert \leq t\}}\chi \nabla _{A} T_{s}(u)\cdot (\nabla v-\nabla u)\,\mathrm{d}x\geq{}& \int _{\varOmega }-fT_{t}(u-v)\,\mathrm{d}x \\ & {} + \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v) \,\mathrm{d}x, \end{aligned}$$

or since \(\chi =\chi _{\{ \vert u-v \vert < t\}}\) and \(\nabla (T_{t}(u-v))= \chi _{\{ \vert u-v \vert < t\}}\nabla (u-v)\)

$$\begin{aligned} & \int _{\varOmega }\nabla _{A} u\cdot \nabla T_{t}( u-v)\,\mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v) \,\mathrm{d}x \\ & \quad \leq \int _{\varOmega }fT_{t}(u-v)\,\mathrm{d}x,\forall t\in (0,\infty ) \setminus \mathcal{O}. \end{aligned}$$

For \(t\in \mathcal{O}\), we know that there exists a sequence \(\{t_{k}\}\) of numbers in \((0,\infty )\setminus \mathcal{O}\) such that \(t_{k}\rightarrow t\) due to \(\vert \mathcal{O} \vert =0\). Therefore, we have

$$\begin{aligned} \int _{\varOmega }\nabla _{A} u\cdot \nabla T_{t_{k}}( u-v)\,\mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t_{k}}(u-v) \,\mathrm{d}x\leq \int _{\varOmega }fT _{t_{k}}(u-v)\,\mathrm{d}x. \end{aligned}$$
(38)

Since \(\nabla (u-v)=0\) a.e. in \(\{ \vert u-v \vert =t\} \), the left-hand side of (38) can be written as

$$\begin{aligned} \int _{\varOmega }\nabla _{A} u\cdot \nabla T_{t_{k}}( u-v)\,\mathrm{d}x= \int _{\varOmega \setminus \{ \vert u-v \vert =t\}}\chi _{\{ \vert u-v \vert < t_{k}\}}\nabla _{A} u \cdot \nabla ( u-v)\,\mathrm{d}x. \end{aligned}$$

The sequence \(\chi _{\{ \vert u-v \vert < t_{k}\}}\) converges to \(\chi _{\{ \vert u-v \vert < t \}}\) a.e. in \(\varOmega \setminus \{ \vert u-v \vert =t\} \) and therefore converges weakly* in \(L^{\infty }(\varOmega \setminus \{ \vert u-v \vert =t\})\). We obtain

$$\begin{aligned} \lim_{k\rightarrow \infty } \int _{\varOmega }\nabla _{A} u\cdot \nabla T _{t_{k}}( u-v)\,\mathrm{d}x={}& \int _{\varOmega \setminus \{ \vert u-v \vert =t\}} \chi _{\{ \vert u-v \vert < t\}}\nabla _{A} u\cdot \nabla ( u-v)\,\mathrm{d}x \\ ={}& \int _{\varOmega }\chi _{\{ \vert u-v \vert < t\}}\nabla _{A} u \cdot \nabla ( u-v) \,\mathrm{d}x \\ ={}& \int _{\varOmega }\nabla _{A} u\cdot \nabla T_{t}(u-v)\,\mathrm{d}x. \end{aligned}$$
(39)

For the right-hand side of (38), we have

$$\begin{aligned} \biggl\vert \int _{\varOmega }fT_{t_{k}}(u-v)\,\mathrm{d}x- \int _{\varOmega }fT_{t}(u-v) \,\mathrm{d}x \biggr\vert \leq \vert t_{k}-t \vert \cdot \Vert f \Vert _{1}\rightarrow 0 \quad \text{as }k\rightarrow \infty . \end{aligned}$$
(40)

Similarly, we have

$$\begin{aligned} \biggl\vert \int _{\varOmega } \vert u \vert ^{r-2}uT_{t_{k}}(u-v) \,\mathrm{d}x- \int _{\varOmega } \vert u \vert ^{r-2}uT _{t}(u-v)\,\mathrm{d}x \biggr\vert &\leq \vert t_{k}-t \vert \cdot \bigl\Vert \vert u \vert ^{r-1} \bigr\Vert _{1} \\ &\rightarrow 0 \quad \text{as } k\rightarrow \infty . \end{aligned}$$
(41)

It follows from (38)–(41) that we have the inequality

$$\begin{aligned} & \int _{\varOmega }\nabla _{A} u\cdot \nabla T_{t}( u-v)\,\mathrm{d}x+ \int _{\varOmega }b \vert u \vert ^{r-2}uT_{t}(u-v) \,\mathrm{d}x \\ & \quad \leq \int _{\varOmega }fT_{t}(u-v)\,\mathrm{d}x,\quad \forall t\in (0,\infty ). \end{aligned}$$

Hence, u is an entropy solution of the obstacle problem associated with \((f,\psi ,g)\). The dependence of the entropy solution on the data \(f\in L^{1}(\varOmega )\) is guaranteed by Proposition 4. □

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Acknowledgements

The author would like to thank the reviewers for their valuable suggestions and comments to improve the quality of this paper.

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This research was partially supported by the Fundamental Research Funds for the Central Universities: No. 2682019LK01.

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This paper was completed by JZ independently. All authors read and approved the final manuscript.

Correspondence to Jun Zheng.

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MSC

  • 35J87
  • 35J70
  • 35B45

Keywords

  • Obstacle problem
  • Non-coercive equation
  • Entropy solution
  • \(L^{1}\)-data
  • Lower order term