1 Introduction

We consider the degenerate nonlinear elliptic problem

$$\begin{aligned} ({\mathcal {P}})\left\{ \begin{array}{ll} -\text{ div }\>b(x,v,\nabla v)=g&{} \quad \text{ in }\ \Omega , \\ v=0 &{}\quad \text{ on }\ \partial \Omega , \end{array}\right. \end{aligned}$$

where \(b: \Omega \times {\mathbb{R}}\times {{\mathbb{R}}^N}\rightarrow {\mathbb{R}}^N\) is a Carathéodory function such that for \(\xi \ne \xi ',\alpha >0\) and \(0\le \theta (\cdot )\ll p(\cdot )-1, (\theta \in {\mathcal {C}}({\overline{\Omega }})),\) the function b satisfies

$$\begin{aligned} b(x,s,\xi )\xi \ge \alpha \frac{|\xi |^{p(x)}}{(1+|s|)^{\theta (x)}},\quad \text{ and }\quad (b(x,s,\xi )-b(x,s,\xi '))(\xi -\xi ')>0, \end{aligned}$$
(1)

and the natural growth

$$\begin{aligned} |b(x,s,\xi )|\le \beta |\xi |^{p(x)-1}, \end{aligned}$$
(2)

where \(\beta >0.\)

Our goal in this paper is first to prove the existence of entropy solutions when the second member g belongs to \(L^1(\Omega ),\) moreover we prove that such solutions are also weak solutions under the hypothesis \(p(\cdot )\gg \theta ^++2-\frac{1}{N}.\) In a second step, we will deal with the regularity of entropy solutions when g belongs to \(L^{q(\cdot )}(\Omega ),\) with \(q(x)\ge 1, \forall x\in \Omega ,\) and \(q^-<\frac{p_*^-(p^--\theta ^-)}{p_*^-(p^--\theta ^-)-p^-}.\) Noting that the last condition is equivalent to \(q<(p_*)'\) if \(\theta =0\) and \(p(\cdot )=p\) in the problem \(({\mathcal {P}}).\)

We recall that in the classical case, i.e., when \(p(\cdot )=p\) and \(\theta (\cdot )=\theta \) the problem \(({\mathcal {P}}),\) was studied for example in [23], where the authors established the regularity of entropy solutions when the second member g belongs to the Marcinkiewicz spaces \(M^m(\Omega )\) using the generalized Stampacchia Lemma. We also refer to [12], where the authors proved the existence of a distributive or entropic solution for a degenerate problem according to the growth assumptions on a lower order term. For more results in this topic, see for example [12, 13, 24].

For \(\theta =0\) in (1), i.e., when the principal part of problem \(({\mathcal {P}})\) is coercive, we have \(\gamma =(p^-)', m_0(\cdot )=\frac{p_*(\cdot )}{(p^-)'}\) and \(m_1(\cdot )=\frac{m_0(\cdot )}{m_0(\cdot )+1}p(\cdot )\) which are the same quantities obtained in [27, 29], where the authors established the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth and a second member \(L^1.\) For more details in the framework of Sobolev spaces with variable exponent, see [2,3,4,5, 10, 19, 20, 31].

The difficulty presented for studying this problem is that the coercivity can degenerate when u is too big, so we cannot apply the standard Leray–Lions surjectivity theorem for the establishment of existence of solutions. To overcome this difficulty, we consider the approximate problem \(({\mathcal {P}}_n)\) of which the differential operator on \(W_0^{1,p(\cdot )}(\Omega )\) is coercive and we can establish a priori estimates on approximating solutions. Then the existence of entropy solutions to problem \(({\mathcal {P}})\) can be obtained by passing to the limit in the approximate problem. Moreover the entropy solutions are also weak solutions under additional assumptions on exponent \(p(\cdot ).\) The method using the approximate problem is widely studied in the literature, see for example [1, 8, 9, 14,15,16, 22, 29].

The study of the Partial Differential Equations with non-standard (variable exponent) growth received wide attention in recent years due to their applications in image processing, elasticity theory and fluid mechanics. In fact, when \(b(x,v,\nabla v)=b(x,\nabla v),\) the equation is studied (in a broader framework) in [28], where Lewy–Stampacchia inequalities are used to derive regularity of solutions under coercivity and truncated monotonicity (T-monotonicity) conditions. Moreover, in [18] also are established estimates on the second order derivatives of solutions. A more particular case, that is, when \(b(x,v,\nabla v) =b(\nabla v)\) is studied in [17]. The study of problems governed by these type of operators goes as far as understanding the limit case with rapidly oscillating coefficients (homogenization), as in [26, 33].

The paper is organized as follows. In Sect. 2, we recall some rearrangement properties and the definitions of Sobolev and Marcinkiewicz spaces with variable exponent. In Sect. 3, we obtain a priori estimates, the existence of entropy solutions and then the weak solutions are proved. In the last section we prove some regularity results of solutions.

2 Preliminaries

2.1 Sobolev spaces with variable exponent

Let \(p:{\overline{\Omega }} \rightarrow {\mathbb{R}}\) a real-valued continuous function and \(c>0.\) If

$$\begin{aligned} -|p(x) - p(y)|\log |x-y|<c, \quad \forall x,y\in {\overline{\Omega }} \text{ such that } |x-y|<\frac{1}{2}, \end{aligned}$$

we say that \(p(\cdot )\) verifies the log-Lipschitz condition.

We denote

$$\begin{aligned} {\mathcal {C}}_{+}({\overline{\Omega }})= \big \{\text{log-Lipschitz} ~ p:{\overline{\Omega }} \rightarrow {\mathbb{R}} \text{ with } 1< p^{-} \le p^{+} < N\big \}, \end{aligned}$$

where \( \rho ^{-} = \inf _{x\in \Omega }\rho (x)\ \text{ and }\ \rho ^{+} = \sup _{x\in \Omega }\rho (x)~~ \forall \rho \in {\mathcal {C}}({\overline{\Omega }}). \)

For \( \rho _1(\cdot ) \) and \( \rho _2(\cdot ) \) in \({\mathcal {C}}({\overline{\Omega }})\)

$$\begin{aligned} \text{ we means by }\ \rho _1(\cdot )\ll \rho _2(\cdot )\ \text{ that }\ \inf _{x\in \Omega }(\rho _2(x)-\rho _1(x))>0. \end{aligned}$$
(3)

Let \(\rho \) be the function defined by

$$\begin{aligned} \rho (v) = \int _{\Omega } |v(x)|^{p(x)} \>dx \quad \quad \forall v\in L^{p(\cdot )}(\Omega ). \end{aligned}$$

For \( p\in {\mathcal {C}}_{+}({\overline{\Omega }}),\) we define the Lebesgue space with variable exponent by

$$\begin{aligned} L^{p(\cdot )}(\Omega ) =\left\{ v : \Omega \rightarrow {\mathbb{R}} \text{ measurable}: \rho (v) < \infty \right\} , \end{aligned}$$

the space \((L^{p(\cdot )}(\Omega ),\Vert v\Vert _{p(\cdot )})\) is reflexive with

$$\begin{aligned} \Vert v\Vert _{p(\cdot )} = \inf \left\{ \lambda > 0 : \rho \left( \frac{v}{\lambda }\right) \le 1 \right\} . \end{aligned}$$

We denote by \( L^{p'(\cdot )}(\Omega ) \) the conjugate space of \(L^{p(\cdot )}(\Omega ),\) where \(\frac{1}{p(x)} + \frac{1}{p'(x)} = 1,\) for all \(x\in \Omega .\)

Proposition 2.1

(Hölder inequality [21, 32])

  1. (i)

    For all \( (v,v')\in L^{p(\cdot )}(\Omega )\times L^{p'(\cdot )}(\Omega ),\) we have

    $$\begin{aligned} \left| \int _{\Omega } v(x) v'(x) \>dx \right| \le \Big (\frac{1}{p^{-}} + \frac{1}{p'^{-}}\Big ) \Vert v\Vert _{p(\cdot )} \Vert v'\Vert _{p'(\cdot )} . \end{aligned}$$
  2. (ii)

    For all \(p_1, p_{2}\in {\mathcal {C}}_{+}({\overline{\Omega }}) \) such that \( p_1(x) \le p_{2}(x) \) for all x in \(\Omega ,\) we have

    $$\begin{aligned} L^{p_2(\cdot )}(\Omega ) \hookrightarrow L^{p_1(\cdot )}(\Omega ), \end{aligned}$$

    moreover the embedding is continuous.

Proposition 2.2

[21, 32] The following assertions hold

  1. (1)

    \(\Vert v\Vert _{p(\cdot )} < 1\) (resp. \(= 1,\) \(> 1\)) \(\Longleftrightarrow \) \(\rho (v) < 1\) (resp. \(= 1,> 1);\)

  2. (2)

    We have the following implication

    $$\begin{aligned}&\Vert v\Vert _{p(\cdot )} > 1 \Longrightarrow \Vert v\Vert _{p(\cdot )}^{p^{-}} \le \rho (v) \le \Vert v\Vert _{p(\cdot )}^{p^{+}}, \\ &\Vert v\Vert _{p(\cdot )} < 1 \Longrightarrow \Vert v\Vert _{p(\cdot )}^{p^{+}} \le \rho (v) \le \Vert v\Vert _{p(\cdot )}^{p^{-}}. \end{aligned}$$
  3. (3)

    The following equivalents hold true

    $$\begin{aligned}&\Vert v\Vert _{p(\cdot )} \rightarrow 0 \Longleftrightarrow \rho (v) \rightarrow 0,\\ &\Vert v\Vert _{p(\cdot )} \rightarrow \infty \Longleftrightarrow \rho (v) \rightarrow \infty . \end{aligned}$$

We define Sobolev space with variable exponent by

$$\begin{aligned} W^{1,p(\cdot )}(\Omega ) = \big \{ v\in L^{p(\cdot )}(\Omega ) \text{ and } |\nabla v|\in L^{p(\cdot )}(\Omega ) \big \}, \end{aligned}$$

with the norm

$$\begin{aligned} \Vert v\Vert _{1,p(\cdot )} = \Vert v\Vert _{p(\cdot )} + \Vert \nabla v\Vert _{p(\cdot )} \quad \forall v \in W^{1,p(\cdot )}(\Omega ). \end{aligned}$$

We denote by \( W_0^{1,p(\cdot )}(\Omega )\) the closure of \( {\mathcal {D}}(\Omega )\) in \(W^{1,p(\cdot )}(\Omega ),\) and we define the Sobolev exponent by \( p^{*}(x)=\frac{Np(x)}{N-p(x)}\) with \(p(x) < N.\)

Proposition 2.3

[21]

  1. (i)

    The spaces \( W^{1, p(\cdot )}(\Omega ) \) and \( W_0^{1, p(\cdot )}(\Omega ) \) are Banach spaces, separable and reflexive.

  2. (ii)

    The embedding \( W^{1, p(\cdot )}_0(\Omega ) \hookrightarrow \hookrightarrow L^{m(\cdot )}(\Omega )\) is continuous and compact, if \(m(x) < p^{*}(x), \forall x \in \Omega .\)

  3. (iii)

    (Poincaré inequality). For all \(v\in W_0^{1,p(\cdot )}(\Omega )\) there exists a constant \(c>0,\) such that \( \Vert v\Vert _{p(\cdot )} \le c \Vert \nabla v\Vert _{p(\cdot )}. \)

  4. (iv)

    (Sobolev–Poincaré inequality). For all \(v\in W_0^{1,p(\cdot )}(\Omega )\) there exists a constant \(c>0,\) such that \( \Vert v\Vert _{p_*(\cdot )} \le c \Vert \nabla v\Vert _{p(\cdot )}. \)

Remark 1

We conclude that the norms \(\Vert \nabla v\Vert _{p(\cdot )}\) and \(\Vert v\Vert _{1,p(\cdot )}\) are equivalents in \(W^{1,p(\cdot )}_0(\Omega )\) using (iii) of Proposition 2.3.

The truncation function \(T_k: {\mathbb{R}}\rightarrow {\mathbb{R}}\) is defined by

$$\begin{aligned} T_k(r)=\left\{ \begin{array}{ll} r &{} \quad \text{ if }\ |r|\le k \\ k\cdot \text{ sign }(r) &{} \quad \text{ if }\ |r|>k. \end{array}\right. \end{aligned}$$

2.2 Marcinkiewicz spaces

In this section we review some properties of rearrangements and Marcinkiewicz spaces with variable exponents, for more details, see [7, 25, 29, 30].

First we recall the definition of decreasing rearrangement of functions. Let \(v:\Omega \rightarrow {\mathbb{R}}\) a measurable function.

Definition 2.4

We define the distribution function of v as follows

$$\begin{aligned} \mu _v(t)=\text{ meas }\{x\in \Omega : |v(x)|>t\},\quad t\ge 0. \end{aligned}$$

\(\mu _v\) is right continuous and decreasing function.

Definition 2.5

We define the decreasing rearrangement of v as follows

$$\begin{aligned} v_*(s):=\sup \{t\ge 0 :\mu _v(t)>s\},\quad s\ge 0. \end{aligned}$$

Definition 2.6

A measurable function \(v:\Omega \rightarrow {\mathbb{R}}\) belongs to the Marcinkiewicz space \(M^p(\Omega )\)(or weak-\(L^p\)) if

$$\begin{aligned} \mu _v(t)\le \frac{c}{t^r},\ \forall t>0, \quad \text{ or }\quad v_{*}(s)\le \frac{c}{s^{1/r}},\ \forall s>0, \end{aligned}$$

for some constant c.

Let \(m(\cdot )\) be a measurable function such that \(m^->0.\) We say that a measurable function v belongs to the Marcinkiewicz space \(M^{m(\cdot )}\) if there exists a positive constant C such that

$$\begin{aligned} \int _{\{|v|>t\}}t^{m(x)}\>dx\le C,\quad \text{ for } \text{ all }\ t>0. \end{aligned}$$

When \(m(\cdot )\) is constant i.e. \(m(\cdot )\equiv m\) this definition is coincides with the classical definition of the Marcinkiewicz space \(M^m(\Omega ).\) Moreover we have

$$\begin{aligned} \int _{\{|v|>t\}}t^{m(x)}\>dx\le \int _{\Omega }|v|^{m(x)}\>dx,\quad \text{ for } \text{ all }\ t>0. \end{aligned}$$

Thus if \(|v|^{m(\cdot )}\in L^1(\Omega ),\) we have \(v\in M^{m(\cdot )}(\Omega )\) and \(L^{m(\cdot )}(\Omega )\subset M^{m(\cdot )}(\Omega ),\) for all \(m(\cdot )\ge 1.\)

In the Marcinkiewicz space with constant exponent, if \(v\in M^r(\Omega ),\) then \(|v|^m\in L^1(\Omega ),\) for all \(0<m<r.\)

This claim is extended to the nonconstant setting by the following lemma, whose proof is given in [29].

Lemma 2.7

Let \(r(\cdot )\) and \(m(\cdot )\) be bounded functions such that \(0\ll m(\cdot )\ll r(\cdot )\) and let \(\epsilon :=(r-m)^->0.\) If \(v\in M^{r(\cdot )}(\Omega ),\) then

$$\begin{aligned} \int _{\Omega }|v|^{m(x)}\>dx\le 2|\Omega |+c\frac{r^+-\epsilon }{\epsilon }, \end{aligned}$$

where c is a positive constant. In particular, \(M^{r(\cdot )}(\Omega )\subset L^{m(\cdot )}(\Omega )\) for all \(1\le m(\cdot )\ll r(\cdot ).\)

3 Main results

3.1 A priori estimate

Definition 3.1

A measurable function v is an entropy solution of problem \(({\mathcal {P}})\) if for every \(t>0,\) \(T_t(v)\in W_0^{1,p(\cdot )}(\Omega )\) and

$$\begin{aligned} \int _{\Omega }b(x,v,\nabla v)\nabla T_t(v-\varphi )\>dx\le \int _{\Omega }g(x)T_t(v-\varphi )\>dx, \end{aligned}$$
(4)

for all \(\varphi \in W_0^{1,p(\cdot )}(\Omega )\cap L^{\infty }(\Omega ).\)

It is well known in [11], that for a measurable function v such that \(T_t(v)\in W_0^{1,p(\cdot )}(\Omega )\) there exists a unique measurable function \( w: \Omega \rightarrow {\mathbb{R}}^{N} \) such that \( w\chi _{\{|v|\le t\}} = \nabla T_t(v)\) for a.e. x in \(\Omega \) and for all \(t> 0.\) We will define the gradient of v as the function w,  and we will denote it by \(w=\nabla v.\)

Theorem 3.2

Under assumptions (1) and (2). If v is an entropy solution of problem \(({\mathcal {P}}),\) then there exists a positive constant C,  depending only on \(p^\pm , N,\) and \(\Omega ,\) such that for all \(t>0\)

$$\begin{aligned} \int _{\{|v|> t\}}t^{\frac{p_*(x)}{\gamma }}\>dx\le C, \quad \text{ with }\ \gamma =\left( \frac{p(\cdot )}{p(\cdot )-(\theta (\cdot )+1)}\right) ^+. \end{aligned}$$

Proof

We denote by C a constant that varies from line to line.

Since v is an entropy solution to the problem \(({\mathcal {P}}),\) \(\text{ for } \text{ all }~\varphi \in W_0^{1,p(\cdot )}(\Omega )\cap L^{\infty }(\Omega )\) we have

$$\begin{aligned} \int _{\Omega }b(x,v,\nabla v)\nabla T_t(v-\varphi )\>dx\le \int _{\Omega }g(x)T_t(v-\varphi )\>dx, \end{aligned}$$

for \(\varphi =0,\) we obtain

$$\begin{aligned} \int _{\Omega }b(x,v,\nabla v)\nabla T_t(v)\>dx\le t\int _{\Omega }|g(x)|\>dx, \end{aligned}$$
(5)

by using (1) it follows that

$$\begin{aligned} \int _{\Omega }\frac{|\nabla T_t(v)|^{p(x)}}{(1+|v|)^{\theta (x)}}\>dx\le t\frac{\Vert g\Vert _1}{\alpha }. \end{aligned}$$
(6)

Using the Sobolev inequality in Propositions 2.3 and 2.2, we have for \(t>1\)

$$\begin{aligned} \int _{\{|v|>t\}}t^{\frac{p_*(x)}{\gamma }}\>dx&= \int _{\{|v|>t\}}t^{\frac{p_*(x)}{\gamma }}\bigg |\frac{T_t(v)}{t}\bigg |^{p_*(x)}\>dx \\& \le \int _{\Omega }\left( t^{\frac{1}{\gamma }-1}|T_t(v)|\right) ^{p_*(x)}\>dx \\& \le \big \Vert t^{\frac{1}{\gamma }-1}T_t(v)\big \Vert _{p_*(\cdot )}^{\alpha _1} \\& \le C\big \Vert \nabla (t^{\frac{1}{\gamma }-1}T_t(v))\big \Vert _{p(\cdot )}^{\alpha _1} \\& \le C\left( \int _{\Omega }|\nabla (t^{\frac{1}{\gamma }-1}T_t(v))|^{p(x)}\>dx\right) ^{\frac{\alpha _1}{\alpha _2}}, \end{aligned}$$

using (6) we get

$$\begin{aligned} \int _{\{|v|>t\}}t^{\frac{p_*(x)}{\gamma }}\>dx&\le C\left( \int _{\Omega } t^{(\frac{1}{\gamma }-1){p(x)}}\frac{|\nabla T_t(v)|^{p(x)}}{(1+|v|)^{\theta (x)}}(1+|v|)^{\theta (x)}\>dx\right) ^{\frac{\alpha _1}{\alpha _2}}\\ &\le C\left( \int _{\Omega } t^{(\frac{1}{\gamma }-1){p(x)}+1}\frac{|\nabla T_t(v)|^{p(x)}}{t(1+|v|)^{\theta (x)}}(1+t)^{\theta (x)}\>dx\right) ^{\frac{\alpha _1}{\alpha _2}}\\ &\le C\left( \int _{\Omega } t^{(\frac{1}{\gamma }-1){p(x)}+1}\frac{|\nabla T_t(v)|^{p(x)}}{t(1+|v|)^{\theta (x)}}(2t)^{\theta (x)}\>dx\right) ^{\frac{\alpha _1}{\alpha _2}}\\ &\le 2^{\theta ^+}C\left( \int _{\Omega } t^{(\frac{1}{\gamma }-1){p(x)}+\theta (x)+1}\frac{|\nabla T_t(v)|^{p(x)}}{t(1+|v|)^{\theta (x)}}\>dx\right) ^{\frac{\alpha _1}{\alpha _2}}\\ &\le C\left( \frac{\Vert g\Vert _1}{\alpha }\right) ^{\frac{\alpha _1}{\alpha _2}}, \end{aligned}$$

where

$$\begin{aligned} \alpha _1=\left\{ \begin{array}{ll} (p_*)^+ &{}\quad \text{ if }\ \big \Vert t^{\frac{1}{\gamma }-1}T_t(v)\big \Vert _{p_*(\cdot )}\ge 1\\ (p_*)^- &{}\quad \text{ if }\ \big \Vert t^{\frac{1}{\gamma }-1}T_t(v)\big \Vert _{p_*(\cdot )}\le 1 \end{array}\right. \quad \text{ and }\quad \alpha _2=\left\{ \begin{array}{ll} p^+ &{}\quad \text{ if }\ \big \Vert t^{\frac{1}{\gamma }-1}\nabla T_t(v)\big \Vert _{p(\cdot )}\le 1\\ p^- &{}\quad \text{ if }\ \big \Vert t^{\frac{1}{\gamma }-1}\nabla T_t(v)\big \Vert _{p(\cdot )}\ge 1. \end{array}\right. \end{aligned}$$

For \(t\le 1\) we have

$$\begin{aligned} \int _{\{|v|>t\}}t^{\frac{p_*(x)}{\gamma }}\>dx\le |\Omega |. \end{aligned}$$

By combining the estimates in both cases, the result follows. \(\square \)

Remark 2

Let \(m_0(\cdot )=\frac{p_*(\cdot )}{\gamma },\) from Theorem 3.2 we have \(u\in M^{m_0(\cdot )}(\Omega ).\) For the constant exponent case (\(p(\cdot )\equiv p\)) we have

$$\begin{aligned} u\in M^{m_0}(\Omega )\quad \text{ with }\ m_0=\frac{N(p-1-\theta )}{N-p}, \end{aligned}$$

which the same regularity obtained by the authors in [23].

Theorem 3.3

Assume that the assumptions (1) and (2) hold true, if v is an entropy solution of a problem \(({\mathcal {P}}),\) then \(|v|^{m(\cdot )}\in L^1(\Omega )\) for all \(m(\cdot )\) such that \(0\ll m(\cdot )\ll m_0(\cdot ).\)

Proof

Let \(m(\cdot )\) and \(m_0(\cdot )\) such that \(0\ll m(\cdot )\ll m_0(\cdot )\) and \(\epsilon =(m_0(\cdot )-m(\cdot ))^->0.\) By Theorem 3.2, we get

$$\begin{aligned} \int _{\{|v|>t\}}t^{m_0(x)}\>dx\le c,\quad \text{ for } \text{ all }\ t>0. \end{aligned}$$

From Lemma 2.7, we conclude that

$$\begin{aligned} \int _{\Omega }|v|^{m(x)}\>dx\le 2|\Omega |+c\left( \frac{m_0(\cdot )-\epsilon }{\epsilon }\right) ^+, \end{aligned}$$

which implies that \(|v|^{m(\cdot )}\in L^1(\Omega ).\) \(\square \)

Theorem 3.4

Let \(\alpha (x)=\frac{p(x)}{\theta (x)+m_0(x)+1}\) for all x in \(\Omega ,\) under assumptions (1) and (2), if v an entropy solution of a problem \(({\mathcal {P}}),\) then \(|\nabla v|^{\alpha (\cdot )}\in M^{m_0(\cdot )}(\Omega ).\) Moreover

$$\begin{aligned} \int _{\{|\nabla v|^{\alpha (\cdot )}>t\}}t^{m_0(x)}\>dx\le c,\quad \text{ for } \text{ all }\ t>0. \end{aligned}$$

Proof

Using Theorem 3.2, for \(t>1\) we have

$$\begin{aligned} \int _{\{|\nabla v|^{\alpha (\cdot )}>t\}}t^{m_0(x)}\>dx&\le \int _{\{|\nabla v|^{\alpha (\cdot )}>t\}\cap \{|v|\le t\}}t^{m_0(x)}\>dx+\int _{\{|v|>t\}}t^{m_0(x)}\>dx\\ &\le \int _{\{|v|\le t\}}t^{m_0(x)}\bigg (\frac{|\nabla v|^{\alpha (x)}}{t}\bigg )^{p(x)/\alpha (x)}\>dx+c\\ &=\int _{\{|v|\le t\}}t^{m_0(x)+1-\frac{p(x)}{\alpha (x)}}\frac{|\nabla T_t(v)|^{p(x)}}{t(1+|v|)^{\theta (x)}}(1+|v|)^{\theta (x)}\>dx+c\\ &\le \int _{\{|v|\le t\}}t^{m_0(x)+1-\frac{p(x)}{\alpha (x)}}\frac{|\nabla T_t(v)|^{p(x)}}{t(1+|v|)^{\theta (x)}}(1+t)^{\theta (x)}\>dx+c\\ &\le \int _{\Omega }\left( \frac{t+1}{t}\right) ^{\theta (x)}\frac{|\nabla T_t(v)|^{p(x)}}{t(1+|v|)^{\theta (x)}}\>dx+c \\ &\le 2^{\theta ^+}\int _{\Omega }\frac{|\nabla T_t(v)|^{p(x)}}{t(1+|v|)^{\theta (x)}}\>dx+c \le C', \end{aligned}$$

where c and \(C'\) are positive constants, for \(t\le 1\) we have

$$\begin{aligned} \int _{\{|\nabla v|^{\alpha (\cdot )}>t\}}t^{m_0(x)}\>dx\le |\Omega |, \end{aligned}$$

which gives the required result according to the cases \(t>1\) and \(t\le 1.\) \(\square \)

Theorem 3.5

Let \(m_1(x)=\frac{m_0(x)p(x)}{m_0(x)+\theta (x)+1}\) for all x in \(\Omega ,\) under assumptions stated in Theorem 3.4, we have

$$\begin{aligned} |\nabla v|^{m(\cdot )}\in L^{1}(\Omega )\quad \text{ for } \text{ all }\ m(\cdot )\ \text{ such } \text{ that }\ 0\ll m(\cdot )\ll m_1(\cdot ). \end{aligned}$$

Proof

By Theorem 3.4, we have

$$\begin{aligned} |\nabla v|^{\alpha (\cdot )}\in M^{m_0(\cdot )}(\Omega ),\quad with\ \alpha (\cdot )=\frac{ p(\cdot )}{m_0(\cdot )+\theta (\cdot )+1}. \end{aligned}$$

Let \(0\ll m(\cdot )\ll m_1(\cdot )\) and \(r(\cdot )=m(\cdot )/\alpha (\cdot )\) then \(r(\cdot )\ll m_0(\cdot ).\)

By using Lemma 2.7 we obtain

$$\begin{aligned} \int _{\Omega }|\nabla v|^{m(x)}\>dx=\int _{\Omega }|\nabla v|^{\alpha (x)r(x)}\>dx\le C. \end{aligned}$$

\(\square \)

3.2 Existence of entropy solutions

In this section we prove the existence and regularity of entropy solutions, extending some results known in the constant exponent case.

Theorem 3.6

Under assumptions (1), (2) and \(g\in L^1(\Omega ).\) There exists an entropy solutions v of problem \(({\mathcal {P}}).\) Moreover

  1. (1)

    \(|v|^{m(\cdot )}\in L^1(\Omega )\ \text{ for }\ 0\ll m(\cdot )\ll m_0(\cdot ),\ \text{ with }\ m_0(\cdot )=\frac{p_*(\cdot )}{\gamma }.\)

  2. (2)

    \(|\nabla v|^{m(\cdot )}\in L^1(\Omega )\ \text{ for }\ 0\ll m(\cdot )\ll m_1(\cdot ),\ \text{ with }\ m_1(\cdot )=\frac{m_0(\cdot )p(\cdot )}{m_0(\cdot )+\theta (\cdot )+1},\)

where

$$\begin{aligned} \gamma =\left( \frac{p(\cdot )}{p(\cdot )-(\theta (\cdot )+1)}\right) ^+. \end{aligned}$$

Remark 3

In the case \(p(\cdot )=p\) and \(\theta (\cdot )=\theta ,\) the exponents \(m_0(\cdot )\) and \(m_1(\cdot )\) are respectively of the form \(m_0=\frac{N(p-(\theta +1))}{N-p}\) and \(m_1=\frac{N(p-(\theta +1))}{N-(\theta +1)}\) which are the same quantities obtained by the authors in [23].

Let \((g_n)_n\subset L^{\infty }(\Omega )\) a sequence that converge strongly to g in \(L^{1}(\Omega ),\) and \(\Vert g_n\Vert _{1}\le \Vert g\Vert _{1}, \) for all n. Let \(({\mathcal {P}}_n)\) the approximate problem defined by

$$\begin{aligned} ({\mathcal {P}}_n)\left\{ \begin{array}{ll} -\text{ div }\>b_n(x,v,\nabla v)=g_n,&{} \quad \text{ in }\ \Omega , \\ v=0,&{} \quad \text{ on }\ \partial \Omega , \end{array}\right. \end{aligned}$$

where \(b_n(x,s,\xi )=b(x,T_n(s),\xi ).\)

We remark that the operator \(b_n\) is coercive. Indeed we have

$$\begin{aligned} b_n(x,s,\xi )\cdot \xi&=b(x,T_n(s),\xi )\cdot \xi \\ &\ge \alpha \frac{|\xi |^{p(x)}}{(1+|T_n(s)|)^{\theta (x)}}\\ &\ge \frac{\alpha }{(1+n)^{\theta ^+}}|\xi |^{p(x)}. \end{aligned}$$

The problem \(({\mathcal {P}}_n)\) has a weak energy solutions \(v_n \in W_0^{1,p(\cdot )}(\Omega )\) as a result of a standard modification of the arguments in [21]. Our goal is to prove that \(v_n\) tend to a measurable function v as n tend to infinity, and we prove that v is an entropy solution of problem \(({\mathcal {P}}).\) We will divide the proof in two steps and we employ the a priori estimates for \(v_n\) and its gradient derived in the preceding section as our main tool. We follow the standard method used in the several paper as [6, 11, 27].

We prove in first step the almost everywhere convergence of the gradient.

First we prove that the sequence \((v_n)_n\) of solutions to problem \(({\mathcal {P}}_n)\) converges in measure to a measurable function v.

Define the \({\mathcal {I}}_1, {\mathcal {I}}_2,\) and \({\mathcal {I}}_3\) sets as follows

$$\begin{aligned} {\mathcal {I}}_1=\{|v_n|>t\},\quad {\mathcal {I}}_2=\{|v_m|>t\},\quad \text{ and }\quad {\mathcal {I}}_3=\{|T_t(v_n)-T_t(v_m)|>s\}, \end{aligned}$$

for \(s>0\) and \(t>0.\) Since

$$\begin{aligned} \{|v_n-v_m|>s\}\subset {\mathcal {I}}_1\cup {\mathcal {I}}_2\cup {\mathcal {I}}_3, \end{aligned}$$

it follows that

$$\begin{aligned} \text{ meas }\{|v_n-v_m|>s\}\le \text{ meas }({\mathcal {I}}_1)+\text{ meas }({\mathcal {I}}_2)+\text{ meas }({\mathcal {I}}_3). \end{aligned}$$

Let \(\epsilon >0,\) by Theorem 3.2, \(v_n\) is uniformly bounded sequence, thus there exists \(t_{\epsilon },\) such that for \(t\ge t_{\epsilon }\) we have

$$\begin{aligned} \text{ meas }({\mathcal {I}}_1)\le \epsilon /3\quad \text{ and }\quad \text{ meas }({\mathcal {I}}_2)\le \epsilon /3. \end{aligned}$$

In the approximate problem \(({\mathcal {P}}_n),\) we take \(T_t(v_n)\) as test function and following the outlines of Theorem 3.2, we get

$$\begin{aligned} \int _{\Omega }\frac{|\nabla T_t(v_n)|^{p(x)}}{(1+|v_n|)^{\theta (x)}}\>dx \le \frac{\Vert g\Vert _1}{\alpha }t,\quad \text{ for } \text{ all }\ n\ge 0 \ \text{ and }\ t>0, \end{aligned}$$

which implies that for all \(n\ge 0\) and \(t>0,\)

$$\begin{aligned} \int _{\Omega }|\nabla T_t(v_n)|^{p(x)}\>dx&\le \frac{\Vert g\Vert _1}{\alpha }t(1+t)^{\theta ^+}\\ &\le \frac{\Vert g\Vert _1}{\alpha }(1+t)^{\theta ^++1}. \end{aligned}$$

Sobolev embedding imply that there exists a subsequence still denoted by \((T_t(v_n))_n\) such that

$$\begin{aligned} \begin{array}{l} T_t(v_n)\rightharpoonup T_t(v)\ \text{ weakly } \text{ in }\ W_0^{1,p(\cdot )}(\Omega ), \\ T_t(v_n)\rightarrow T_t(v)\ \text{ strongly } \text{ in }\ L^{m(\cdot )}(\Omega ),\ \text{ for }\ 1\le m(\cdot )< p_*(\cdot ), \\ T_t(v_n)\rightarrow T_t(v)\ \text{ a.e. } \text{ in }\ \Omega , \end{array} \end{aligned}$$
(7)

for all \(t>0.\) Thus there exists \(n_0(s,\epsilon )\in {\mathbb{N}}\) such that for all \(n,m\ge n_0(s,\epsilon )\) we have

$$\begin{aligned} \text{ meas }({\mathcal {I}}_3)&=\int _{\{|T_t(v_n)-T_t(v_m)|>s\}}\>dx\\ &\le \int _{\Omega }\bigg (\frac{|T_t(v_n)-T_t(v_m)|}{s}\bigg )^{m(x)}\>dx\\ &\le \frac{1}{s^{m^\pm }}\int _{\Omega }|T_t(v_n)-T_t(v_m)|^{m(x)}\>dx \le \epsilon /3. \end{aligned}$$

Finely for all \(n,m\ge n_0(s,\epsilon )\) we have

$$\begin{aligned} \text{ meas }\{|v_n-v_m|>s\}\le \text{ meas }({\mathcal {I}}_1)+\text{ meas }({\mathcal {I}}_2)+\text{ meas }({\mathcal {I}}_3)\le \epsilon , \end{aligned}$$

which implies that \((v_n)_n\) is a Cauchy sequence in measure.

Following the standard argument as in [16], proving that \((\nabla v_n)_n\) is a Cauchy sequence in measure is an easy task.

In the second step we passing to the limit.

Let \(v_n\) be a solution of approximate problem \(({\mathcal {P}}_n),\) for \(w\in W_0^{1,p(\cdot )}(\Omega )\) we have

$$\begin{aligned} \int _{\Omega }b(x,T_n(v_n),\nabla v_n)\nabla w\>dx=\int _{\Omega }g_n(x)w\>dx. \end{aligned}$$

Taking \(w=T_t(v_n-\varphi )\) with \(\varphi \in W_0^{1,p(\cdot )}(\Omega )\cap L^{\infty }(\Omega )\) we get

$$\begin{aligned} \int _{\Omega }b(x,T_n(v_n),\nabla v_n)\nabla T_t(v_n-\varphi )\>dx=\int _{\Omega }g_n(x)T_t(v_n-\varphi )\>dx. \end{aligned}$$

For the term in the right hand side, since \(g_n\) converge strongly to g in \(L^1(\Omega )\) and \(T_t(v_n-\varphi )\) converge weakly-* to \(T_t(v-\varphi )\) in \(L^{\infty }(\Omega ),\) and a.e. in \(\Omega \) we have

$$\begin{aligned} \int _{\Omega }g_n(x)T_t(v_n-\varphi )\>dx\longrightarrow \int _{\Omega }g(x)T_t(v-\varphi )\>dx. \end{aligned}$$

For the left hand side we have

$$\begin{aligned} \int _{\Omega }b(x,T_n(v_n),\nabla v_n)\cdot \nabla T_t(v_n-\varphi )\>dx=&\int _{\{|v_n-\varphi |\le t\}}b(x,T_n(v_n),\nabla v_n)\cdot \nabla v_n\>dx\\ &-\int _{\{|v_n-\varphi |\le t\}}b(x,T_n(v_n),\nabla v_n)\cdot \nabla \varphi \>dx\\ =&\int _{\{|v_n-\varphi |\le t\}}b(x,v_n,\nabla v_n)\cdot \nabla v_n\>dx\\ &-\int _{\{|v_n-\varphi |\le t\}}b(x,v_n,\nabla T_r(v_n))\cdot \nabla \varphi \>dx \end{aligned}$$

with \(r=t+\Vert \varphi \Vert _{\infty }.\)

By (2) and (7), we can prove that \(b(x,v_n,\nabla T_r(v_n))\) is uniformly bounded in \((L^{p'(\cdot )}(\Omega ))^N,\) and converges weakly to \(b(x,v,\nabla T_r(v))\) in \((L^{p'(\cdot )}(\Omega ))^N.\) Therefore we have

$$\begin{aligned} \int _{\{|v_n-\varphi |\le t\}}b(x,v_n,\nabla T_r(v_n))\cdot \nabla \varphi \>dx\longrightarrow \int _{\{|v-\varphi |\le t\}}b(x,v,\nabla v)\cdot \nabla \varphi \>dx. \end{aligned}$$
(8)

Since \(b(x,v_n,\nabla v_n)\cdot \nabla v_n\) converge almost everywhere to \(b(x,v,\nabla v)\cdot \nabla v\) in \(\Omega ,\) by Fatou’s lemma we have

$$\begin{aligned} \lim \inf _n\int _{\{|v_n-\varphi |\le t\}}b(x,T_n(v_n),\nabla v_n)\cdot \nabla v_n\>dx\ge \int _{\{|v-\varphi |\le t\}}b(x,v,\nabla v)\cdot \nabla v\>dx. \end{aligned}$$
(9)

By (8) and (9), for all \(\varphi \in W_0^{1,p(\cdot )}(\Omega )\cap L^{\infty }(\Omega )\) we have

$$\begin{aligned} \int _{\Omega }b(x,v,\nabla v)\nabla T_t(v-\varphi )\>dx\le \int _{\Omega }g(x)T_t(v-\varphi )\>dx. \end{aligned}$$

The results of regularity follow from Theorems 3.3 and 3.5.

3.3 Existence of weak solutions

In this section, first we find a sufficient conditions to have \(m_0(\cdot )\gg 1\) and \(m_1(\cdot )\gg 1,\) and then we prove that the entropy solutions to the problem \(({\mathcal {P}})\) are also weak solutions under assumption \(m_1(\cdot )\gg p(\cdot )-1.\)

Lemma 3.7

Let the exponents \(m_0(\cdot )\) and \(m_1(\cdot )\) as defined in Theorem 3.6, if \( p(\cdot )\gg \theta ^++2-\frac{1}{N}.\) Then

$$\begin{aligned} m_0(\cdot )\gg 1\quad \text{ and }\quad m_1(\cdot )\gg 1. \end{aligned}$$

Proof

First, we prove that \(m_0(\cdot )\gg 1.\)

For all x in \(\Omega ,\) we have

$$\begin{aligned} \gamma =\left( \frac{p(x)}{p(x)-(\theta (x)+1)}\right) ^+\le \left( \frac{p(x)}{p(x)-(\theta ^++1)}\right) ^+=\frac{p^-}{p^--(\theta ^++1)}. \end{aligned}$$
(10)

On the other hand, by a simple computations, we get

$$\begin{aligned} (p_*(\cdot ))^-=\frac{Np^-}{N-p^-}>\frac{p^-}{p^--(\theta ^++1)}, \end{aligned}$$

which gives according to (10) that \((p_*(\cdot ))^->\gamma \) and then \(m_0(\cdot )\gg 1.\)

Now we prove that \(m_1(\cdot )\gg 1.\)

By definition of \(m_1(\cdot )\) we have for all \(x\in \Omega \)

$$\begin{aligned} m_1(x)&=\frac{Np^2(x)}{Np(x)+\gamma (\theta (x)+1)(N-p(x))}\\ &\ge \frac{Np^2(x)}{Np(x)+\gamma (\theta ^++1)(N-p(x))}. \end{aligned}$$

Using that \(\frac{Np^2(\cdot )}{Np(\cdot )+\gamma (\theta ^++1)(N-p(\cdot ))}\) is increasing in \(p(\cdot ),\) \( p(\cdot )\gg \theta ^++2-\frac{1}{N}\) and (17) we have

$$\begin{aligned} \left( m_1(\cdot )\right) ^-&\ge \left( \frac{Np^2(\cdot )}{Np(\cdot )+ \gamma (\theta ^++1)(N-p(\cdot ))}\right) ^-\\ &=\frac{Np^-{^2}}{Np^-+\gamma (\theta ^++1)(N-p^-)}\\ &\ge \frac{Np^-(p^--(\theta ^++1))}{Np^--N(\theta ^++1)+(\theta ^++1)(N-p^-)}\\ &=\frac{N(p^--(\theta ^++1))}{N-(\theta ^++1)}>1. \end{aligned}$$

\(\square \)

Theorem 3.8

Assume that (1)–(2) hold true and \(p(\cdot )\gg \theta ^++2-\frac{1}{N},\) if \(m_1(\cdot )\gg p(\cdot )-1\) then the entropy solutions of problem \(({\mathcal {P}})\) are also weak solutions.

Proof

Let \(v_n\) be a solution to the approximate problem \(({\mathcal {P}}_n),\) we have

$$\begin{aligned} \int _{\Omega }b(x,T_n(v_n),\nabla v_n)\nabla \varphi \>dx=\int _{\Omega }g_n(x)\varphi \>dx, \quad \forall ~ \varphi \in {\mathcal {D}}(\Omega ). \end{aligned}$$
(11)

Let E be a subset of \(\Omega ,\) by (2) we have

$$\begin{aligned} \int _E|b(x,T_n(v_n),\nabla v_n)|\>dx\le \beta \int _E|\nabla v_n|^{p(x)-1}\>dx. \end{aligned}$$
(12)

By Theorem 3.6, Lemma 3.7 and \(m_1(\cdot )\gg p(\cdot )-1\) the terms \(|\nabla v_n|^{p(x)-1}\) is uniformly bounded in \(L^{m(\cdot )}(\Omega ),\) for some \( m(\cdot )\gg 1,\) so the terms in the right hand side of (12) goes to zero when a measure of E is small enough.

Since \(b(x,T_n(v_n),\nabla v_n)\) converge almost everywhere to \(b(x,v,\nabla v),\) by Vitali theorem we conclude that \(b(x,T_n(v_n),\nabla v_n)\) converge strongly to \(b(x,v,\nabla v)\) in \(L^1(\Omega ).\) Now passing to the limit in (11), by using the previous results and \(g_n\) converge strongly to g in \(L^1(\Omega )\) we obtain

$$\begin{aligned} \int _{\Omega }b(x,v,\nabla v)\nabla \varphi \>dx=\int _{\Omega }g(x)\varphi \>dx,\quad \forall ~ \varphi \in {\mathcal {D}}(\Omega ). \end{aligned}$$

\(\square \)

Remark 4

We can deal with the following degenerate elliptic problem and obtain the same results as above

$$\begin{aligned} \left\{ \begin{array}{ll} -\text{ div }\>b(x,v,\nabla v)+h(x,v)=g&{} \quad \text{ in }\ \Omega , \\ v=0 &{}\quad \text{ on }\ \partial \Omega , \end{array}\right. \end{aligned}$$

with the lower order term \(h(\cdot ,v(\cdot ))\) is a Carathéodory function that verifies a sign condition and with a natural growth.

4 Regularity results

In this section we assume that \(g\in L^{q(\cdot )}(\Omega )\) with \(q(x)\ge 1,~ \forall x\in \Omega \) and \(q^-<\frac{p_*^-(p^--\theta ^-)}{p_*^-(p^--\theta ^-)-p^-}.\) We prove some regularity of entropy solutions for a problem \(({\mathcal {P}}).\)

Remark 5

For \(\gamma >0, v\in M^{\frac{p_*(\cdot )}{\gamma }}(\Omega )\) implies that \(v\in M^{\frac{p_*^{-}}{\gamma }}(\Omega ),\) which gives \(\mu _v(t)\le c_1t^{-\frac{p_*^{-}}{\gamma }},\) with \(c_1\) is a positive constant.

Remark 6

Since \(g\in L^{m(\cdot )}(\Omega )\subset M^{m(\cdot )}(\Omega )\) we have \(g\in M^{m^-}(\Omega ),\) which gives \(g_{*}(t)\le c_2 t^{-\frac{1}{m^-}},\) with \(c_2\) is a positive constant.

Theorem 4.1

Under assumptions (1) and (2). If v is a solution in the sense of Definition 3.1that belongs to \(M^{\frac{p_*(\cdot )}{\gamma }}(\Omega ),\) then there exists a positive constant C,  depending only on \(p(\cdot ), N,\) and \(\Omega ,\) such that

$$\begin{aligned} \int _{\{|v|\le t\}}\frac{|\nabla v|^{p(x)}}{(1+|v|)^{\theta (x)}}\>dx\le C t^{1-\frac{(p_*)^{-}}{\gamma (q^{-})'}}, \end{aligned}$$

for all \(t>0\) and \(\gamma >\frac{(p_*)^{-}}{(q^{-})'}.\)

Proof

Since v is an entropy solution to the problem \(({\mathcal {P}}),\) we have

$$\begin{aligned} \int _{\Omega }a(x,v,\nabla v)\nabla T_t(v-\varphi )\>dx\le \int _{\Omega }g(x)T_t(v-\varphi )\>dx \end{aligned}$$

for all \(\varphi \in W_0^{1,p(\cdot )}(\Omega )\cap L^{\infty }(\Omega ),\) by taking \(\varphi =T_s(v)\) with \(s\ge 1,\) we have

$$\begin{aligned} \int _{\{s<|v|\le t+s\}}a(x,v,\nabla v)\nabla v\>dx\le t\int _{\{|v|>s\}}|g(x)|\>dx. \end{aligned}$$
(13)

By Young inequality and dividing in the both sides by t,  (1) gives

$$\begin{aligned} \frac{\alpha }{t}\int _{\{s<|v|\le t+s\}}\frac{|\nabla v|^{p(x)}}{(1+|v|)^{\theta (x)}}\>dx\le \int _{\{|v|>s\}}|g(x)|\>dx. \end{aligned}$$
(14)

Passing to the limit in (14), for t goes to zero we have

$$\begin{aligned} \alpha \frac{d}{ds}\int _{\{|v|\le s\}}\frac{|\nabla v|^{p(x)}}{(1+|v|)^{\theta (x)}}\>dx\le \int _0^{\mu _v(s)}g_*(\tau )\>d\tau . \end{aligned}$$

Integrating between 0 and t,  we get

$$\begin{aligned} \alpha \int _{\{|v|\le t\}}\frac{|\nabla v|^{p(x)}}{(1+|v|)^{\theta (x)}}\>dx\le \int _0^t\int _0^{\mu _v(s)}g_*(\tau )\>d\tau \>ds. \end{aligned}$$
(15)

Now by using Remarks 5 and 6 the term in the right hand side of the previous inequality becomes

$$\begin{aligned} \int _0^t\int _0^{\mu _v(s)}g_*(\tau )\>d\tau \>ds&\le c_2\int _0^t\int _0^{\mu _v(s)}\tau ^{-\frac{1}{q^-}}\>d\tau \>ds\\ &\le c_2\int _0^t\mu _v(s)^{1-\frac{1}{q^-}} \>ds\\ &\le c_2c_1^{1-\frac{1}{q^-}}\int _0^t s^{-\frac{(p_*)^-}{(q^-)'\gamma }} \>ds\\ &\le 2^{\theta ^+}c_2c_1^{1-\frac{1}{q^-}}\int _0^t s^{-\frac{(p_*)^-}{(q^-)'\gamma }} \>ds\\ &=\frac{2^{\theta ^+}c_2c_1^{\frac{1}{(q^-)'}}}{1-\frac{(p_*)^-}{(q^-)'\gamma }}\times t^{1-\frac{(p_*)^-}{(q^-)'\gamma }}, \end{aligned}$$

which implies that there exists a constant \(c>0\) such that

$$\begin{aligned} \int _{\Omega }\frac{|\nabla T_t(v)|^{p(x)}}{(1+|v|)^{\theta (x)}}\>dx\le c t^{1-\frac{(p_*)^-}{(q^-)'\gamma }}. \end{aligned}$$

\(\square \)

Theorem 4.2

Assume that the assumptions (1) and (2) hold true. If v is an entropy solution of \(({\mathcal {P}}),\) then there exists a positive constant c,  depending only on \(p^\pm ,\) N,  and \(\Omega ,\) such that

  1. (1)

    \(\int _{\{|v|>t\}}t^{\frac{p_*(x)}{\gamma _q}}\>dx\le c,\) for all \(t>0,\) with \(\gamma _q=\left( \frac{q^-p(\cdot )-(p_*)^-(q^--1)}{q^-(p(\cdot )-(\theta (\cdot )+1))}\right) ^+.\)

  2. (2)

    Let \(m_{0,q}(x)=\frac{p_*(x)}{\gamma _q},\) for all \(m(\cdot )\) such that \(0\ll m(\cdot )\ll m_{0,q}(\cdot ),\) we have \(|v|^{m(\cdot )}\in L^1(\Omega ).\) Moreover there exists a constant positive \(c_0\) such that \(\int _{\Omega }|v|^{m(x)}\>dx\le c_0.\)

Remark 7

We remark that \(\gamma _q>\frac{p_*^-}{(q^-)'}.\) Ended by using \(\theta ^-\le \theta (x), ~\forall x\in \Omega \) we have

$$\begin{aligned} \gamma _q=\left( \frac{q^-p(x)-(p_*)^-(q^--1)}{q^-(p(x)-(\theta (x)+1))}\right) ^+\ge \left( \frac{q^-p(x)-(p_*)^-(q^--1)}{q^-(p(x)-(\theta ^-+1))}\right) ^+, \end{aligned}$$

and since \(q^-<\frac{p_*^-(p^--\theta ^-)}{p_*^-(p^--\theta ^-)-p^-}\) we have

$$\begin{aligned} \frac{q^-p(x)-(p_*)^-(q^--1)}{q^-(p(x)-(\theta ^-+1))}\ \text{ is } \text{ non-increasing } \text{ in }\ p(\cdot ). \end{aligned}$$

Moreover we have

$$\begin{aligned} \gamma _q=\left( \frac{q^-p(x)-(p_*)^-(q^--1)}{q^-(p(x)-(\theta ^-+1))}\right) ^+=\frac{q^-p^--(p_*)^-(q^--1)}{q^-(p^--(\theta ^-+1))}>\frac{p_*^-}{(q^-)'}. \end{aligned}$$

Proof

  1. (1)

    We find \(\gamma _q\) such that \(v\in M^{\frac{p_*(x)}{\gamma _q}}.\)

    Case 1: \(t\ge 1,\) by using Proposition 2.2 and Sobolev embedding we have

    $$\begin{aligned} \int _{\{|v|>t\}}t^{\frac{p_*(x)}{\gamma _q}}\>dx&=\int _{\{|v|>t\}}t^{\frac{p_*(x)}{\gamma _q}}\bigg |\frac{T_t(v)}{t}\bigg |^{p_*(x)}\>dx\\ &=\int _{\{|v|>t\}}\big |t^{(\frac{1}{\gamma _q}-1)}T_t(v)\big |^{p_*(x)}\>dx\\ &\le \bigg \Vert t^{(\frac{1}{\gamma _q}-1)}T_t(v)\bigg \Vert _{p_*(x)}^{\alpha _1}\le c\bigg \Vert \nabla (t^{(\frac{1}{\gamma _q}-1)}T_t(v))\bigg \Vert _{p(\cdot )}^{\alpha _1}\\ &\le c\bigg (\int _{\Omega }t^{(\frac{1}{\gamma _q}-1)}|\nabla T_t(v)|^{p(x)}\>dx\bigg )^{\alpha _1/\alpha _2} \end{aligned}$$
    (16)

    by using Theorem 4.1 we obtain

    $$\begin{aligned}&\int _{\{|v|>t\}}t^{\frac{p_*(x)}{\gamma _q}}\>dx\\ &\quad \le c\bigg (\int _{\Omega }t^{(\frac{1}{\gamma _q}-1)p(x)}\frac{|\nabla T_t(v)|^{p(x)}}{(1+|v|)^{\theta (x)}t^{1-\frac{(p_*)^-}{(q^-)'\gamma _q}}}t^{1-\frac{(p_*)^-}{(q^-)'\gamma _q}}(1+|v|)^{\theta (x)}\>dx\bigg )^{\alpha _1/\alpha _2}\\ &\quad \le 2^{\theta ^+}c\bigg (\int _{\Omega }t^{(\frac{1}{\gamma _q}-1)p(x)}\frac{|\nabla T_t(v)|^{p(x)}}{(1+|v|)^{\theta (x)}t^{1-\frac{(p_*)^-}{(q^-)'\gamma _q}}}t^{1+\theta (x)-\frac{(p_*)^-}{(q^-)'\gamma _q}}\>dx\bigg )^{\alpha _1/\alpha _2}\\ &\quad \le 2^{\theta ^+}c\bigg (\int _{\Omega }|t^{(\frac{1}{\gamma _q}-1)p(x)+1+\theta (x)-\frac{(p_*)^-}{(q^-)'\gamma _q}}\>dx\bigg )^{\alpha _1/\alpha _2}, \end{aligned}$$
    (17)

    if we choose \(\gamma _q\) such that \((\frac{1}{\gamma _q}-1)p(x)+1+\theta (x)-\frac{(p_*)^-}{(q^-)'\gamma _q} \le 0\)

    i.e.

    $$\begin{aligned} \gamma _q\ge \frac{q^-p(x)-(p_*)^-(q^--1)}{q^-(p(x)-(\theta (x)+1))},\quad \forall x\in \Omega , \end{aligned}$$

    and by taking \(\gamma _q=\bigg (\frac{q^-p(x)-(p_*)^-(q^--1)}{q^-(p(x)-(\theta (x)+1))}\bigg )^+,\) (17) gives

    $$\begin{aligned} \int _{\{|v|>t\}}t^{\frac{p_*(x)}{\gamma _q}}\>dx\le c. \end{aligned}$$

    Case 2: \(0<t<1\)

    $$\begin{aligned} \int _{\{|v|>t\}}t^{\frac{p_*(x)}{\gamma _q}}\le |\Omega |. \end{aligned}$$

    Combining the estimates in both cases, the result follows.

  2. (2)

    Let \(0\ll m(\cdot )\ll m_{0,q}(\cdot )\) and \(\epsilon =(m_{0,q}(\cdot )-m(\cdot ))^->0.\) By Theorem 4.1, we have

    $$\begin{aligned} \int _{\{|v|>t\}}t^{m_{0,q}(x)}\>dx\le c,\quad \text{ for } \text{ all }\ t>0. \end{aligned}$$

    From Lemma 2.7, we get

    $$\begin{aligned} \int _{\Omega }|v|^{m(x)}\>dx\le 2|\Omega |+c\big (\frac{m_{0,q}(\cdot )-\epsilon }{\epsilon }\big )^+,\ \text{ which } \text{ gives } \text{ the } \text{ results }. \end{aligned}$$

\(\square \)

Theorem 4.3

Assume that the assumptions (1) and (2) hold true. Let v be an entropy solution of \(({\mathcal {P}}).\) If there exists a positive constant c such that \(\int _{\{|v|>t\}}t^{m_{0,q}(x)}\>dx\le c,\) for all \(t>0,\) then \(|\nabla v|^{\alpha (\cdot )}\in M^{m_{0,q}(\cdot )}(\Omega ),\) where \(\alpha (\cdot )=\frac{\gamma _q(q^-)' p(\cdot )}{\gamma _q(q^-)'[m_{0,q}(\cdot )+\theta (\cdot )+1]-p_*(\cdot )}.\) Moreover there exists a positive constant \(C'\) such that

$$\begin{aligned} \int _{\{|\nabla v|^{\alpha (\cdot )}>t\}}t^{m_{0,q}(x)}\>dx\le C',\quad \text{ for } \text{ all }\ t>0. \end{aligned}$$

Proof

Using Theorem 4.1, and the definition of \(\alpha (\cdot ),\) for \(t>1\) we have

$$\begin{aligned} \int _{\{|\nabla v|^{\alpha (\cdot )}>t\}}t^{m_{0,q}(x)}\>dx&\le \int _{\{|\nabla v|^{\alpha (\cdot )}>t\}\cap \{|v|\le t\}}t^{m_{0,q}(x)}\>dx+\int _{\{|\nabla v|^{\alpha (\cdot )}>t\}\cap \{|v|>t\}}t^{m_{0,q}(x)}\>dx\\ &\le \int _{\{|v|\le t\}}t^{m_{0,q}(x)}\bigg (\frac{|\nabla v|^{\alpha (x)}}{t}\bigg )^{p(x)/\alpha (x)}\>dx+c'\\ &=\int _{\{|v|\le t\}}t^{m_{0,q}(x)+1-\frac{(p_*)^-}{\gamma _q (q^-)'}-\frac{p(x)}{\alpha (x)}}\frac{|\nabla T_t(v)|^{p(x)}(1+|v|)^{\theta (x)}}{t^{1-\frac{(p_*)^-}{\gamma _q (q^-)'}}(1+|v|)^{\theta (x)}}\>dx+c'\\ &\le 2^{\theta ^+}\int _{\{|v|\le t\}}t^{m_{0,q}(x)+1+\theta (x)-\frac{(p_*)^-}{\gamma _q (q^-)'}-\frac{p(x)}{\alpha (x)}}\frac{|\nabla T_t(v)|^{p(x)}}{t^{1-\frac{(p_*)^-}{\gamma _q (q^-)'}}(1+|v|)^{\theta (x)}}\>dx+c'\\ &\le 2^{\theta ^+}\int _{\{|v|\le t\}}\frac{|\nabla T_t(v)|^{p(x)}}{t^{1-\frac{(p_*)^-}{\gamma _q (q^-)'}}}\>dx+c'\\ &\le C'. \end{aligned}$$

where \(c'\) and \(C'\) are positive constants.

For \(t\le 1\) we have

$$\begin{aligned} \int _{\{|\nabla v|^{\alpha (\cdot )}>t\}}t^{m_{0,q}(x)}\>dx\le |\Omega |. \end{aligned}$$

\(\square \)

Theorem 4.4

Assume that the assumptions (1) and (2) hold true. Let \(m_{0,q}(\cdot )\) be defined in Theorem 4.2and \(m_{1,q}(\cdot )=m_{0,q}(\cdot )\alpha (\cdot ).\)

If v is an entropy solution of problem \(({\mathcal {P}}),\) then \(|\nabla v|^{m(\cdot )}\in L^1(\Omega ),\) for all \(m(\cdot )\) such that \(0\ll m(\cdot )\ll m_{1,q}(\cdot ).\) Moreover there exists a constant C such that

$$\begin{aligned} \int _{\Omega }|\nabla v|^{m(x)}\>dx\le C. \end{aligned}$$

Proof

By Theorem 4.3, we have

$$\begin{aligned} |\nabla v|^{\alpha (\cdot )}\in M^{m_{0,q}(\cdot )}(\Omega ),\quad with\ \alpha (\cdot )=\frac{p(\cdot )}{m_{0,q}(\cdot )+\theta (\cdot )+1-\frac{(p_*)^-}{\gamma _q(q^-)'}}. \end{aligned}$$

Let \(0\ll m(\cdot )\ll m_{1,q}(\cdot )\) and \(r(\cdot )=m(\cdot )/\alpha (\cdot )\ll m_{0,q}(\cdot ).\)

Using the Theorem 4.2 we obtain

$$\begin{aligned} \int _{\Omega }|\nabla v|^{m(x)}\>dx=\int _{\Omega }|\nabla v|^{\alpha (x)r(x)}\>dx\le C. \end{aligned}$$

\(\square \)

Remark 8

If \(q(\cdot )\equiv 1,\) i.e., \(g\in L^1(\Omega ),\) we remark that \(m_0(\cdot )\) coincide with \(m_{0,q}(\cdot )\) and \(m_1(\cdot )\) coincide with \(m_{1,q}(\cdot ),\) which implies that the regularity results obtained in the current section are a generalization of those obtained in Theorem 3.6 of Sect. 3.