Lipschitz continuity of minimizers in a problem with nonstandard growth

To our dear friend Sandro Salsa on the occasion of his 70th birthday.

1.   Introduction

In this paper we study the regularity properties of nonnegative, local minimizers of the functional

under nonstandard growth conditions of the energy function $F(x, s, \eta)$ and $0 < \lambda_{\min}\le \lambda(x)\le \lambda_{\max} < \infty$.

There has been a great deal of interest in these type of problems. Their study started with the seminal paper of Alt and Caffarelli ^[2] where the case $F(x, s, \eta) = \frac12|\eta|^2$ was considered. Later on, ^[3] considered the case $F(x, s, \eta) = G(|\eta|^2)$ under uniform ellipticity assumptions. The general power case $F(x, s, \eta) = \frac1p|\eta|^p$ with $1 < p < \infty$ was studied in ^[8], and $F(x, s, \eta) = G(|\eta|)$ with $G$ convex under the assumption that $G'$ satisfies Lieberman's condition namely, $G''(t)\sim G'(t)/t$, was analyzed in ^[19]. The linear inhomogeneous case $F(x, s, \eta) = \frac12|\eta|^2+f(x)s$ was addressed in ^[12] and ^[15].

The minimization problem for the functional (1.1) with $F(x, s, \eta) = \frac1{p(x)}|\eta|^{p(x)}$ was first considered in ^[6] for $p(x)\ge2$ and then, in ^[16] and ^[17] in the inhomogeneous case $F(x, s, \eta) = \frac1{p(x)}|\eta|^{p(x)}+f(x)s$, for $1 < p(x) < \infty$ and $f\in L^{\infty}(\Omega)$. In ^[17], among other results, we proved that nonnegative local minimizers $u$ are locally Lipschitz continuous and satisfy

The operator $\Delta_{p(x)}$, called the $p(x)$-Laplacian, extends the Laplacian, where $p(x)\equiv 2$ and the $p$-Laplacian, where $p(x)\equiv p$. This is a prototype operator with nonstandard growth. The functional setting for the study of this type of operators are the variable exponent Lebesgue and Sobolev spaces $L^{p(\cdot)}$ and $W^{1, p(\cdot)}$.

Functionals and PDEs with nonstandard growth have a wide range of applications, such as the modelling of non-Newtonian fluids, as for instance, electrorheological ^[21] or thermorheological fluids ^[4]. Other areas of application include non-linear elasticity ^[24], image reconstruction ^[1,7], the modelling of electric conductors ^[25], as well as processes of filtration of gases in non-homogeneous porous media ^[5].

As far as we know, no result on the minimization of (1.1) with $F(x, s, \eta)$ a general function with nonstandard growth has been obtained.

The main purpose of our work is to prove the local Lipschitz continuity of nonnegative local minimizers of such an energy. We stress that this is the optimal regularity since it is known from the particular cases refered to above that the gradient of a minimizer $u$ jumps across $\Omega\cap \partial\{u > 0\}$.

We prove that nonnegative minimizers of (1.1) are solutions to the associated equation in their positivity sets. That is, a local minimizer $u\ge0$ satisfies

in $\{u > 0\}$, where

Under our assumptions, the governing equation (1.2) is given by $A(x, s, \eta)$ satisfying

and has a right hand side given by $B(x, s, \eta)\not\equiv 0$ of $p(x)$-type growth in $\eta$. This equation is singular in the regions where $1 < p(x) < 2$ and degenerate in the ones where $p(x) > 2$.

Our study thus presents new features, needed in order to overcome the deep technical difficulties arising due to the nonlinear degenerate/singular nature and the $x$ and $s$ dependence of this general operator associated to our energy functional (1.1).

The first part of the paper is devoted to the study of equation (1.2) in a domain $\Omega$, under nonstandard growth conditions of $p(x)$-type. We prove existence results, a comparison principle, a uniqueness result, a maximum principle and other local $L^\infty$ bounds of solutions of this equation. These delicate results are of independent interest.

Some of these results are obtained under the growth assumption (3.14). We remark that this hypothesis on the functions $A$ and $B$ allows to consider very general equations. This condition not only enables us to get the inequality in Proposition 3.3 that is a main tool for all the proofs in the paper, but also it is invariant under rescalings. All these results are included in Section 3.

In the second part of the paper we deal with the minimization problem for the functional (1.1). In fact, in Section 4 we first get an existence result for minimizers. We also prove nonnegativity and boundedness, under suitable assumptions. Then, we prove the local Hölder and Lipschitz continuity of nonnegative local minimizers (Theorems 4.3 and 4.5).

The proofs in Section 4 involve delicate rescalings. One of the main difficulties this problem presents is that it is not invariant under the rescaling $u(x) \longmapsto \frac{u(tx)}{k}$, if $t\neq k$ —rescaling that is a crucial tool in dealing with this type of problems. The rescaled functionals lose the uniform properties and nontrivial modifications are needed to get through the proofs. Even after these modifications, there is in general no limit equation for the rescaled problems due to the growth we are allowing to the function $B(x, s, \eta)$. Novel arguments are used to complete the proof of Theorem 4.4. In fact, we are able to show that, although there is in general no limit equation for the rescaled problems, there is a limit function and it satisfies Harnack's inequality (see (4.58)).

A thorough follow up of the dependence of the bounds found in Section 3 with respect to the structural conditions on $F, A$ and $B$ is of most importance as well.

Let us point out that the results in the paper are new even in the case $p(x)\equiv p$ constant.

Finally, in Section 5 we present some examples of functionals (1.1) where our results can be used.

Our examples include functionals (1.1) involving energy functions of the form

A possible example of admissible functions $a(x, s), f(x, s)$ is given by

for $s$ in the range where the nonnegative local minimizer takes values, $q_0(x)$ a function depending on $p(x)$ and

with $\tau(x)$ satisfying (2.7).

Our results also apply to functionals (1.1) involving energy functions of the form

Some admissible $G(x, \eta), f(x, s)$ are

Also,

is an admissible function if both $F_1(x, s, \eta)$ and $F_2(x, s, \eta)$ are admissible.

We begin our paper with a section where we state the hypotheses on $F, A$, $B$, $\lambda$ and $p(x)$ that will be used throught the article. And we end it with an Appendix where we state some properties of the function spaces $L^{p(\cdot)}$ and $W^{1, p(\cdot)}$ where the problem is well posed.

1.1.   Preliminaries on Lebesgue and Sobolev spaces with variable exponent

Let $p :\Omega \to [1, \infty)$ be a measurable bounded function, called a variable exponent on $\Omega$ and denote $p_{\max} = {\rm ess sup} \, p(x)$ and $p_{\min} = {\rm ess inf} \, p(x)$. We define the variable exponent Lebesgue space $L^{p(\cdot)}(\Omega)$ to consist of all measurable functions $u :\Omega \to {\mathbb R}$ for which the modular $\varrho_{p(\cdot)}(u) = \int_{\Omega} |u(x)|^{p(x)}\, dx$ is finite. We define the Luxemburg norm on this space by

This norm makes $L^{p(\cdot)}(\Omega)$ a Banach space.

There holds the following relation between $\varrho_{p(\cdot)}(u)$ and $\|u\|_{L^{p(\cdot)}}$:

Moreover, the dual of $L^{p(\cdot)}(\Omega)$ is $L^{p'(\cdot)}(\Omega)$ with $\frac{1}{p(x)}+\frac{1}{p'(x)} = 1$.

Let $W^{1, p(\cdot)}(\Omega)$ denote the space of measurable functions $u$ such that $u$ and the distributional derivative $\nabla u$ are in $L^{p(\cdot)}(\Omega)$. The norm

makes $W^{1, p(\cdot)}(\Omega)$ a Banach space.

The space $W_0^{1, p(\cdot)}(\Omega)$ is defined as the closure of the $C_0^{\infty}(\Omega)$ in $W^{1, p(\cdot)}(\Omega)$.

For the sake of completeness we include in an Appendix at the end of the paper some additional results on these spaces that are used throughout the paper.

1.2.   Notation

$\bullet \; N$ spatial dimension

$\bullet$ $|S|$ $N$-dimensional Lebesgue measure of the set $S$

$\bullet$ $B_r(x_0)$ open ball of radius $r$ and center $x_0$

$\bullet$ $B_r$ open ball of radius $r$ and center $0$

$\bullet \; \chi_{{}_S}$ characteristic function of the set $S$

$\bullet \; u^{+} = \text{ max}(u, 0)$, $u^{-} = \text{ max}(-u, 0)$

$\bullet \; \langle\, \xi\, , \, \eta\, \rangle$ and $\xi \cdot \eta$ both denote scalar product in ${\mathbb R} ^{N}$

2.   Assumptions

In this section we collect all the assumptions that will be made along the paper.

Throughout the paper $\Omega$ will denote a $C^1$ bounded domain in ${\mathbb R}^N$. In addition, the following assumptions will be made:

2.1.   Assumptions on $p(x)$

We assume that the function $p(x)$ is measurable in $\Omega$ and verifies

We assume further that $p(x)$ is Lipschitz continuous in $\Omega$ and we denote by $L$ the Lipschitz constant of $p(x)$, namely, $\|\nabla p\|_{L^{\infty}(\Omega)}\leq L$.

When we are restricted to a ball $B_r$ we use ${p^-}_r$ and ${p^+}_r$ to denote the infimum and the supremum of $p(x)$ over $B_r$.

2.2.   Assumptions on $\lambda(x)$

We assume that the function $\lambda(x)$ is measurable in $\Omega$ and verifies

2.3.   Assumptions on $F$

We assume that $F$ is measurable in $\overline\Omega\times {\mathbb R}\times {\mathbb R}^N$, and for every $x\in \overline\Omega$, $F(x, \cdot, \cdot)\in C^1({\mathbb R}\times{\mathbb R}^N)\cap C^2({\mathbb R}\times{\mathbb R}^N\setminus\{0\})$.

We denote $A(x, s, \eta) = \nabla_\eta F(x, s, \eta)$ and $B(x, s, \eta) = F_s(x, s, \eta)$.

2.4.   Assumptions on $A$

We assume that $A\in C(\overline\Omega\times {\mathbb R}\times {\mathbb R}^N, {\mathbb R}^N)$ and for every $x\in \overline\Omega$, $A(x, \cdot, \cdot)\in C^1({\mathbb R}\times{\mathbb R}^N\setminus\{0\}, {\mathbb R}^N)$. Moreover, there exist positive constants $\lambda_0$ and $\Lambda_0$, and $\beta\in (0, 1)$ such that for every $x, x_1, x_2\in\overline\Omega$, $s, s_1, s_2\in{\mathbb R}$, $\eta\in {\mathbb R}^N\setminus\{0\}$ and $\xi\in{\mathbb R}^N$, the following conditions are satisfied:

2.5.   Assumptions on $B$

We assume that $B$ is measurable in $\overline\Omega\times {\mathbb R}\times {\mathbb R}^N$ and for every $x\in \overline\Omega$, $B(x, \cdot, \cdot)\in C^1({\mathbb R}\times{\mathbb R}^N)$, and for every $(x, s, \eta)\in\overline\Omega\times {\mathbb R}\times {\mathbb R}^N$,

where $\Lambda_0$ is as in the assumptions on $A$ and

Remark 2.1. From (2.1) and (2.3) we get

so that

From (2.1) and (2.2) we have

so that

3.   Existence, uniqueness and bounds of solutions to equation (1.2)

In this section we consider $A$ and $B$ as in Section 2 and we prove results for solutions of the equation

Namely, existence, comparison principle, uniqueness, maximum principle and bounds of solutions.

Our first result is Proposition 3.1, were we prove existence of a solution to (3.1) with given boundary data. In order to prove the existence of a solution to (3.1) we show that, given $u\in W^{1, p(\cdot)}(\Omega)$, there exists a minimizer of the functional

in $u+W_0^{1, p(\cdot)}(\Omega)$, where $F$ is as in Section 2, $A(x, s, \eta) = \nabla_\eta F(x, s, \eta)$ and $B(x, s, \eta) = F_s(x, s, \eta)$.

Then, in Proposition 3.2 we get an existence result under a growth assumption on the function $F$ stronger than (3.3) in Proposition 3.1, but without the small oscillation hypothesis there.

In Proposition 3.4 and Corollary 3.2 we prove comparison and uniqueness for this problem, assuming that condition (3.14) below holds. In Proposition 3.5 we prove that solutions to (3.1) with bounded boundary data are bounded and in Proposition 3.6 we prove a maximum principle for this problem, under suitable assumptions. In Proposition 3.7 we give another existence result of a bounded solution.

We start with the definition of solution to (3.1).

Definition 3.1. Let $p$, $A$ and $B$ be as in Section 2. We say that $u$ is a solution to (3.1) if $u\in W^{1, p(\cdot)}(\Omega)$ and, for every $\varphi \in C_0^{\infty}(\Omega)$, there holds that

We are using that, under the conditions in (2.7), the embedding theorem (see Theorem A.5) applies.

Our first existence result is

Proposition 3.1. Let $p, F, A, B$ as in Section 2 and let ${\Omega'}\subset\Omega$ be a $C^1$ domain. Let $u\in W^{1, p(\cdot)}({\Omega'})$ and let us call $p^+ = \sup_{{\Omega'}}p(x)$, $p^- = \inf_{{\Omega'}}p(x)$. Assume that there exist $\mu, c_1\in{\mathbb R}_+$, $p_{\min} > \delta > 0$ and $g\in L^1(\Omega)$ such that

Assume, moreover that $\delta > {p^+}-{p^-}$ and that

with $\tau$ satisfying (2.7).

Then, there exists a solution $v\in u+W_0^{1, p(\cdot)}({\Omega'})$ to (3.1) in ${\Omega'}$.

Moreover, $\|v\|_{W^{1, p(\cdot)}({\Omega'})}\le C$, for a constant $C$ depending only $\|u\|_{W^{1, p(\cdot)}({\Omega'})}$, $\|g\|_{L^1({\Omega'})}$, $|{\Omega'}|$, $\rm{diam}({\Omega'})$, $N$, ${p^-}$, ${p^+}$, $\delta$, $L$, $\mu$, $c_1$, $||\tau||_{L^{\infty}({\Omega'})}$ and the $C^1$ norm of $\partial{\Omega'}$.

Proof. We will show that there is a minimizer of $\mathcal J_{{\Omega'}}$ in $u+W_0^{1, p(\cdot)}({\Omega'})$ where

This minimizer is a solution to the associated Euler-Lagrange equation (3.1) in ${\Omega'}$.

We will use the embedding theorem (see Theorem A.5) that states that, under the conditions in (2.7), $W^{1, p(\cdot)}(\Omega')\hookrightarrow L^{\tau(\cdot)}(\Omega')$ continuously.

So, let $v_n$ be a minimizing sequence. That is, $v_n\in u+W_0^{1, p(\cdot)}({\Omega'})$ and

Let us show that there is a constant $\kappa > 0$ such that $\|v_n\|_{L^{p(\cdot)}({\Omega'})}\le\kappa$. In fact, by (3.3), for $n$ large,

By Poincare's inequality (Theorem A.4)

Hence, recalling Proposition A.1,

with $\bar C$ depending on $\|u\|_{W^{1, p(\cdot)}({\Omega'})}, \|g\|_{L^1({\Omega'})}, N, {p^-}$, ${p^+}$, $\delta$, $|{\Omega'}|$, $\rm{diam}({\Omega'})$, $L$, $||\tau||_{L^{\infty}({\Omega'})}$, the $C^1$ norm of $\partial{\Omega'}$, and the constants in (3.3).

Observe that in case $u\equiv M$, there holds that $\int_{{\Omega'}}F(x, u, \nabla u)\, dx$ is bounded by a constant that depends only on $M$, $\|\tau\|_{L^\infty({\Omega'})}$ and $|{\Omega'}|$. Hence, in that case $\bar C$ is independent of the regularity of ${\Omega'}$.

Since we want to find a uniform bound of $\|v_n\|_{L^{p(\cdot)}({\Omega'})}$, we may assume that this norm is larger than 1. Let $q$ be the middle point of the interval $[{p^+}-\delta, {p^-}]$. By Young's inequality with $r(x) = \frac {q}{p(x)-\delta}$,

for $0 < {\varepsilon} < 1$ with $C_{\varepsilon}$ depending only on $|{\Omega'}|, {\varepsilon}, {p^-}, {p^+}$ and $\delta$. On the other hand, since $\|v_n\|_{L^q({\Omega'})}\le C\|v_n\|_{L^{p(\cdot)}({\Omega'})}$ with $C$ depending only on $|{\Omega'}|, {p^-}, {p^+}$ and $\delta$,

So that

By choosing ${\varepsilon}$ small enough, we find that

with $C$ depending on $|{\Omega'}|$, $\rm{diam}({\Omega'})$, $\|u\|_{W^{1, p(\cdot)}({\Omega'})}, {p^-}, {p^+}$, $N$, $\delta$, $\|g\|_{L^1({\Omega'})}$, $L$, $||\tau||_{L^{\infty}({\Omega'})}$, the $C^1$ norm of $\partial{\Omega'}$, $\mu$ and $c_1$.

From the computations above we find that $\int_{{\Omega'}} |v_n|^{p(x)-\delta}\, dx\le C_1$. So that we have that $I > -\infty$ and

with $C_2$ depending on $|{\Omega'}|$, $\rm{diam}({\Omega'})$, $\|u\|_{W^{1, p(\cdot)}({\Omega'})}, {p^-}, {p^+}$, $N$, $\delta$, $\|g\|_{L^1({\Omega'})}$, $L$, $||\tau||_{L^{\infty}({\Omega'})}$, the $C^1$ norm of $\partial{\Omega'}$, $\mu$ and $c_1$.

From our comment above, we have that in case $u\equiv M$ in ${\Omega'}$, the constant $C_2$ is independent of the regularity of $\partial{\Omega'}$.

Let us proceed with the proof of the existence of a minimizer. By the estimates above, for a subsequence that we still call $v_n$, there holds that there exists $v\in u+W_0^{1, p(\cdot)}({\Omega'})$, such that

and such that the bounds (3.5) and (3.6) also hold for $v$.

By Egorov's Theorem, for every ${\varepsilon} > 0$ there exists $\Omega_{\varepsilon}$ such that $|{\Omega'}\setminus\Omega_{\varepsilon}| < {\varepsilon}$ and $v_n\to v$ uniformly in $\Omega_{\varepsilon}$.

On the other hand, if we set $\Omega_K = \{x\in{\Omega'}\, /\, |v|+|\nabla v|\le K\}$, there holds that $|{\Omega'}\setminus\Omega_K|\to0$ as $K\to\infty$.

Let $\Omega_{{\varepsilon}, K} = \Omega_{\varepsilon}\cap\Omega_K$. Then, $|{\Omega'}\setminus\Omega_{{\varepsilon}, K}|\to0$ as ${\varepsilon}\to0$ and $K\to\infty$.

There holds

Let us prove that

In fact,

On the one hand, ${\mathcal B}\to0$ since $F(x, v_n, \nabla v)-F(x, v, \nabla v)\to0$ uniformly in $\Omega_{{\varepsilon}, K}$ and it is uniformly bounded. On the other hand, by the convexity assumption on $F(x, s, \eta)$ with respect to $\eta$,

since $A(x, v_n, \nabla v)\to A(x, v, \nabla v)$ uniformly in $\Omega_{{\varepsilon}, K}$, they are uniformly bounded and $\nabla v_n\rightharpoonup \nabla v$ weakly in $L^{p(\cdot)}(\Omega_{{\varepsilon}, K})$.

Hence, for every ${\varepsilon}, K$,

Now, by letting ${\varepsilon}\to0$ and $K\to\infty$, we get

and therefore, $v$ is a minimizer of $\mathcal J_{{\Omega'}}$ in $u+W_0^{1, p(\cdot)}({\Omega'})$ and a solution to (3.1).

As a corollary of Proposition 3.1 we have the following existence result that will be used in the next section.

Corollary 3.1. Let $p, F, A, B$ as in Section 2 and let ${\Omega'}\subset\Omega$ be a $C^1$ domain. Let $u\in W^{1, p(\cdot)}({\Omega'})$ and let us call $p^+ = \sup_{{\Omega'}}p(x)$, $p^- = \inf_{{\Omega'}}p(x)$. Assume that there exist $\mu, c_1\in{\mathbb R}_+$ and $p_{\min} > \delta > 0$ such that

Assume, moreover that $\delta > {p^+}-{p^-}$ and that

with $\tau(x)$ satisfying (2.7).

Then, there exists a solution $v\in u+W_0^{1, p(\cdot)}(\Omega')$ to (3.1) in ${\Omega'}$ and $\|v\|_{W^{1, p(\cdot)}({\Omega'})}\le C$, for a constant $C$ depending only $\|u\|_{W^{1, p(\cdot)}({\Omega'})}$, $|{\Omega'}|$, $\rm{diam}({\Omega'})$, $N$, ${p^-}$, ${p^+}$, $\delta$, $L$, $\mu$, $c_1$, $||\tau||_{L^{\infty}({\Omega'})}$ and the $C^1$ norm of $\partial{\Omega'}$.

With a stronger growth assumption on the $s$ variable for the function $F(x, s, \eta)$ we get an existence result without the small oscillation assumption of the function $p$.

Proposition 3.2. Let $p, F, A, B$ as in Section 2 and let ${\Omega'}\subset\Omega$ be a $C^1$ domain. Let $u\in W^{1, p(\cdot)}({\Omega'})$. Assume that there exist $\mu, c_1\in{\mathbb R}_+$, $g\in L^1(\Omega)$ and $1\le q < p_{\min}$ such that

Assume, moreover that

with $\tau$ satisfying (2.7).

Then, there exists a solution $v\in u+W_0^{1, p(\cdot)}(\Omega')$ to (3.1) in ${\Omega'}$ and $\|v\|_{W^{1, p(\cdot)}({\Omega'})}\le C$, for a constant $C$ depending only $\|u\|_{W^{1, p(\cdot)}({\Omega'})}$, $\|g\|_{L^1({\Omega'})}$, $|{\Omega'}|$, $\rm{diam}({\Omega'})$, $N$, $p_{\min}$, $p_{\max}$, $q$, $L$, $\mu$, $c_1$, $||\tau||_{L^{\infty}({\Omega'})}$ and the $C^1$ norm of $\partial{\Omega'}$.

Proof. We proceed as in the proof of Proposition 3.1 and we prove that a minimizing sequence $\{v_n\}$ satisfies

We want to prove that there is a constant such that $\int_{{\Omega'}}|\nabla v_n|^{p(x)}\, dx\le C$. So, we can assume that $\int_{{\Omega'}}|\nabla v_n|^{p(x)}\, dx > 1$.

Thus,

where $C$ depends on $q, p_{\min}, p_{\max}$, $N$, $L$ and $|\Omega'|$, $\rm{diam}({\Omega'})$. Hence, as $q < p_{\min}$,

with $C$ depending only on $q, p_{\min}, p_{\max}, N, |{\Omega'}|, {\rm{diam}({\Omega'})}, L, \|u\|_{W^{1, p(\cdot)}({\Omega'})}$, and $\widetilde C$ depending on the same constants and also on ${\varepsilon}$.

Thus, by (3.12) and (3.13),

with $\widehat C$ depending only on $q, p_{\min}, p_{\max}, N, \mu, |{\Omega'}|, {\rm{diam}({\Omega'})}, L, \int_{{\Omega'}}g(x)\, dx$, $c_1$, $||\tau||_{L^{\infty}({\Omega'})}$, the $C^1$ norm of $\partial{\Omega'}$ and $\|u\|_{W^{1, p(\cdot)}({\Omega'})}$.

Now, as in the proof of Proposition 3.1, we get that there exists a subsequence that we still call $\{v_n\}$ and a function $v\in u+W_0^{1, p(\cdot)}({\Omega'})$ such that

Now, the proof follows as that of Proposition 3.1.

We next prove a result valid for solutions of equation (3.1) that will be of use in the proofs of Hölder and Lipschitz continuity of minimizers of the energy functional (1.1)

Proposition 3.3. Let $p, F, A$ and $B$ be as in Section 2. Assume moreover that

for every $(x, s, \eta)\in \overline\Omega\times{\mathbb R}\times{\mathbb R}^N\setminus\{0\}$, $\xi\in{\mathbb R}^N$ and $w\in {\mathbb R}$.

Let $u\in W^{1, p(\cdot)}(\Omega)\cap L^{\infty}(\Omega)$ and let $v\in W^{1, p(\cdot)}(\Omega)\cap L^{\infty}(\Omega)$ be such that

Then,

where $\alpha = \alpha(p_{\min}, p_{\max})$ and $\lambda_0$ is as in (2.2).

Proof. For $0\le \sigma\le 1$, let $u^\sigma = v+\sigma(u-v)$. Then, denoting $\nabla_\eta F = A$ and $F_s = B$, we obtain

where we have used (3.15). Moreover,

Now, using (2.2), and the inequality

we get

where $\alpha = \alpha(p_{\min}, p_{\max})$ and $\lambda_0$ is as in (2.2). On the other hand,

Finally, using that $A_s(x, s, \eta) = \nabla_{\eta}B(x, s, \eta)$, the assumption (3.14) and estimates (3.17), (3.18), (3.20) and (3.21), we get (3.16).

We now prove a comparison principle for equation (3.1), which holds under assumption (3.14).

Proposition 3.4. Let $p, A$ and $B$ be as in Section 2. Assume moreover that condition (3.14) holds. Let $u, v \in W^{1, p(\cdot)}(\Omega)$ be such that

Then,

Proof. We will use arguments similar to those in Proposition 3.3. In fact, for $R > 0$ we consider the nonnegative function $w_R\in W_0^{1, p(\cdot)}(\Omega)\cap L^\infty(\Omega)$ given by

and by (3.22) we have

Then, denoting $\Omega_R = \Omega\cap\{0 < u-v < R\}$ and, for $0\le \tau\le 1$, $u^\tau = v+\tau(u-v)$, we get

Now, proceeding as in Proposition 3.3, we obtain

where $\tilde\alpha = \tilde\alpha(p_{\min}, p_{\max})$ and $\lambda_0$ is as in (2.2).

On the other hand, we observe that the evaluation of (3.14) in $\xi = 0$ implies that $B_s(x, s, \eta)\ge 0$. Then, we get

Now, using that $A_s(x, s, \eta) = \nabla_{\eta}B(x, s, \eta)$, (2.3), (3.19), assumption (3.14) and estimates (3.25), (3.26), (3.27) and (3.28), we get

where $\hat\alpha = \hat\alpha(p_{\min}, p_{\max})$ and $\Lambda_0$ is as in (2.3). Since $R > 0$ is arbirtrary, we can use that $u, v \in W^{1, p(\cdot)}(\Omega)$ and let $R\to\infty$ and we obtain

which implies that $\nabla (u-v)^+ = 0$ in $\Omega$. Since $(u-v)^+\in W_0^{1, p(\cdot)}(\Omega)$, Poincare's inequality (Theorem A.4) gives $(u-v)^+ = 0$ in $\Omega$. That is, (3.23) holds.

As a corollary of Propostion 3.4 we obtain the following uniqueness result

Corollary 3.2. Let $p, A$ and $B$ be as in Section 2. Assume moreover that condition (3.14) holds. Let $\varphi \in W^{1, p(\cdot)}(\Omega)$ and let $u_1, u_2 \in W^{1, p(\cdot)}(\Omega)$ be such that

for $i = 1, 2$. Then, $u_1 = u_2$ in $\Omega$.

We next prove that solutions to (3.1) with bounded boundary data are bounded, under the assumptions of Proposition 3.1.

Proposition 3.5. Let $p, A$ and $B$ be as in Section 2 and let ${\Omega'}\subset\Omega$ be a $C^1$ domain. Assume moreover, that conditions (3.3), (3.4) and (3.14) hold in ${\Omega'}$ for some ${p^+}-{p^-} < \delta < p_{\min}$ where ${p^+} = \sup_{{\Omega'}} p$ and ${p^-} = \inf_{{\Omega'}} p$ and with $\tau$ satisfying (2.7). Let us also assume that there exists a positive constant $\Lambda_0$ such that the following condition holds:

for every $(x, s, \eta)\in\overline{\Omega'}\times{\mathbb R}\times{\mathbb R}^N$. Let $u \in W^{1, p(\cdot)}({\Omega'})$ be such that

for some positive constant $M$. Then, there exists C such that $|u|\le C$ in ${\Omega'}$, where $C$ depends only on $M$, $|{\Omega'}|$, $\rm{diam}({\Omega'})$, $N, \lambda_0, \Lambda_0, L, {p^-}, {p^+}$, $\delta$, $\|g\|_{L^1({\Omega'})}$, $||\tau||_{L^{\infty}({\Omega'})}$, $\mu$ and $c_1$.

Proof. Let $v^+$ be the solution to (3.1) with boundary data $M$. Then, from the proof of Proposition 3.1 it follows that $||v^+||_{W^{1, p(\cdot)}({\Omega'})}$ depends only on the constants in the structural conditions, on $|{\Omega'}|$, $\rm{diam}({\Omega'})$ and $M$. Since (recall Remark 2.1) we are under the assumptions of Theorem 4.1 in ^[11], then $v^+\in L^{\infty}({\Omega'})$ with bounds depending only on the constants in the structural conditions, on $|{\Omega'}|$, $\rm{diam}({\Omega'})$ and $M$. Now, the comparison principle (Proposition 3.4) implies that $u\le v^+$ in ${\Omega'}$ and the upper bound follows. Proceeding in an analogous way with $v^-$ the solution to (3.1) with boundary data $-M$, we obtain the lower bound, thus concluding the proof.

As a corollary of Propositions 3.1 and 3.5 we get

Corollary 3.3. Let $p, F, A$ and $B$ as in Section 2 and let ${\Omega'}\subset\Omega$ be a $C^1$ domain. Assume, moreover that $F$ satisfies (3.8) and (3.9) with $\tau$ satisfying (2.7) and $A$ and $B$ satisfy (3.14) and (3.32) in ${\Omega'}$ for some ${p^+}-{p^-} < \delta < p_{\min}$ where ${p^+} = \sup_{{\Omega'}} p$ and ${p^-} = \inf_{{\Omega'}} p$.

Let $u\in W^{1, p(\cdot)}({\Omega'})\cap L^\infty({\Omega'})$. Then, there exists $v\in u+W_0^{1, p(\cdot)}({\Omega'})$ a solution to

Moreover, $v\in L^\infty({\Omega'})$ and $\|v\|_{L^\infty({\Omega'})}$ is bounded by a constant $C$ that depends only on $\|u\|_{L^\infty({\Omega'})}$, $|{\Omega'}|$, $\rm{diam}({\Omega'})$, $N, \lambda_0, \Lambda_0, L, {p^-}, {p^+}$, $\delta$, $||\tau||_{L^{\infty}({\Omega'})}$, $\mu$ and $c_1$.

We also prove the following maximum principle

Proposition 3.6. Let $p, A$ and $B$ be as in Section 2. Assume moreover that condition (3.14) holds. We also assume that $B(x, 0, 0)\equiv 0$ for every $x\in\overline\Omega$. Let $u \in W^{1, p(\cdot)}(\Omega)$ be such that

for some nonnegative constants $M_1, M_2$. Then, $-M_1\le u\le M_2$ in $\Omega$.

Proof. Since condition (3.14) implies that $B_s(x, s, \eta)\ge 0$ in $\overline\Omega\times{\mathbb R}\times{\mathbb R}^N\setminus\{0\}$, we have $B(x, M_2, 0)\ge 0$ and also $B(x, -M_1, 0)\le 0$, for every $x\in\Omega$. Recalling (2.1), we take $v^+\equiv M_2$ and $v^-\equiv -M_1$ and observe that $\mbox{div} A(x, v^+, \nabla v^+) \le B(x, v^+, \nabla v^+)$ and $\mbox{div} A(x, v^-, \nabla v^-) \ge B(x, v^-, \nabla v^-)$ in $\Omega$. Then, we can apply the comparison principle (Proposition 3.4) and obtain $-M_1\equiv v^-\le u\le v^+\equiv M_2$ in $\Omega$ and the conclusion follows.

As a corollary of Propositions 3.1 and 3.6 we get

Corollary 3.4. Let $p, F, A$ and $B$ as in Section 2 and let ${\Omega'}\subset\Omega$ be a $C^1$ domain. Assume, moreover that $F$ satisfies (3.8) and (3.9) with $\tau$ satisfying (2.7) and $A$ and $B$ satisfy (3.14) in ${\Omega'}$ for some ${p^+}-{p^-} < \delta < p_{\min}$ where ${p^+} = \sup_{{\Omega'}} p$ and ${p^-} = \inf_{{\Omega'}} p$. We also assume that $B(x, 0, 0)\equiv 0$ for every $x\in\overline{\Omega'}$.

Let $u\in W^{1, p(\cdot)}({\Omega'})\cap L^\infty({\Omega'})$. Then, there exists $v\in u+W_0^{1, p(\cdot)}({\Omega'})$ a solution to

Moreover, $v\in L^\infty({\Omega'})$ and $\|v\|_{L^\infty({\Omega'})}\le \|u\|_{L^\infty({\Omega'})}$.

We also have the following existence result of a bounded solution

Proposition 3.7. Let $p$ as in Section 2. Assume that $F(x, \cdot, \cdot)$ is locally Lipschitz in ${\mathbb R}\times{\mathbb R}^N$ for almost every $x\in \Omega$ and that $F(x, s, \cdot)\in C^1({\mathbb R}^N)\cap C^2({\mathbb R}^N\setminus\{0\})$ for $s\in {\mathbb R}$ and almost every $x\in \Omega$. Let $A = \nabla_\eta F$, $B = F_s$. Assume that $A$ satisfies (2.2) and (2.5),

and $F$ satisfies (3.3) and (3.4), where $\tau$ satisfies (2.7). Assume moreover that

and,

Then, for every ${\Omega'}\subset\Omega$ of class $C^1$ there holds that, if ${p^+}-{p^-} < \delta < p_{\min}$ where ${p^+} = \sup_{{\Omega'}} p$ and ${p^-} = \inf_{{\Omega'}} p$ for $\delta$ in (3.3), given $u\in W^{1, p(\cdot)}({\Omega'})$ such that $0\le u\le M$ in ${\Omega'}$ there exists $v$ that minimizes the functional ${\mathcal J}_{{\Omega'}}(v)$ in $u+W_0^{1, p(\cdot)}({\Omega'})$. Moreover, $0\le v\le M$ in ${\Omega'}$.

In addition, if there exists ${\varepsilon}_0 > 0$ such that for almost every $x\in\Omega$, $F(x, \cdot, \cdot)\in C^1((-{\varepsilon}_0, M+{\varepsilon}_0)\times{\mathbb R}^N)$, then there holds that $v$ is a solution to

Proof. To begin with, the existence of a minimizer $v$ follows proceeding as in Proposition 3.1. Let us prove that a minimizer satisfies $0\le v\le M$. In fact, both $w_1 = v-(v-M)^+$ and $w_2 = v+v^-$ are admissible functions. So that on the one hand,

Hence, $G(x, v, \nabla v) = 0$ in $\{v > M\}$. So that, $\nabla (v-M)^+ = 0$ in ${\Omega'}$. As $(v- M)^+ = 0$ on $\partial{\Omega'}$, we deduce that $v\le M$ in ${\Omega'}$.

On the other hand, proceeding in a similar way with $w_2$,

and we deduce as before that $v^- = 0$. This is, $v\ge0$ in ${\Omega'}$.

Now, in order to proceed with the proof we assume further regularity of $F$ for $-{\varepsilon}_0\le s\le M+{\varepsilon}_0$. Let $0\le\varphi\in C_0^\infty({\Omega'})$ and $0 < {\varepsilon} < {\varepsilon}_0/\|\varphi\|_{L^\infty}$. Then, $w = v+{\varepsilon}\varphi$ is an admissible function, $-{\varepsilon}_0 < w < M+{\varepsilon}_0$ and we deduce that

Replacing $\varphi$ by $-\varphi$ we reverse the inequality. So that, $v$ is a solution to (3.38).

4.   Energy minimizers of energy functional (1.1)

In this section we prove properties of nonnegative local minimizers of the energy functional (1.1). We prove that nonnegative local minimizers are locally Hölder continuous (Theorem 4.3) and are solutions to

where $A(x, s, \eta) = \nabla_\eta F(x, s, \eta)$ and $B(x, s, \eta) = F_s(x, s, \eta)$. In particular we prove our main result which is the local Lipschitz continuity on nonnegative local minimizers (Theorem 4.5).

We start with a definition, some related remarks and an existence result of a minimizer. We also prove nonnegativity and boundedness, under suitable assumptions.

Definition 4.1. Let $p, F$ and $\lambda$ be as in Section 2. Assume that $F$ satisfies (3.3) and (3.4) with $\tau$ satisfying (2.7). We say that $u\in W^{1, p(\cdot)}(\Omega)$ is a local minimizer in $\Omega$ of

if for every $\Omega'\subset\subset\Omega$ and for every $v\in W^{1, p(\cdot)}(\Omega)$ such that $v = u$ in $\Omega\setminus{\Omega'}$ there holds that $J(v)\ge J(u)$.

We point out that the energy $J$ is well defined in $W^{1, p(\cdot)}(\Omega)$ since, under the conditions in (2.7), the embedding theorem (see Theorem A.5) applies.

Remark 4.1. Let $u$ be as in Definition 4.1. Let $\Omega'\subset\subset\Omega$ and $w-u\in W_0^{1, p(\cdot)}(\Omega')$. If we define

then $\bar w\in W^{1, p(\cdot)}(\Omega)$ and therefore $J(\bar w)\ge J(u)$. If we now let

it follows that $J_{\Omega'}(w)\ge J_{\Omega'}(u)$.

Remark 4.2. Let $J$ be as in Definition 4.1. If $u\in W^{1, p(\cdot)}(\Omega)$ is a minimizer of $J$ among the functions $v\in u+W_0^{1, p(\cdot)}(\Omega)$, then $u$ is a local minimizer of $J$ in $\Omega$.

We start with an existence result of a minimizer to (1.1).

Theorem 4.1. Let $p, F, A, B$ and $\lambda$ be as in Section 2. Let $\phi\in W^{1, p(\cdot)}(\Omega)$ and assume moreover that $F$ satisfies (3.10) and (3.11) with $\tau$ satisfying (2.7).

Then, there exists a minimizer $u\in \phi +W_0^{1, p(\cdot)}(\Omega)$ to (1.1) and there holds that $\|u\|_{W^{1, p(\cdot)}(\Omega)}\le C$, for a constant $C$ depending only on $\|\phi\|_{W^{1, p(\cdot)}(\Omega)} \; \|g\|_{L^1(\Omega)}$, $\lambda_{\max}$, $|\Omega|$, $\rm{diam}(\Omega)$, $N$, $p_{\min}$, $p_{\max}$, $q$, $L$, $\mu$, $c_1$, $||\tau||_{L^{\infty}(\Omega)}$ and the $C^1$ norm of $\partial\Omega$.

Proof. The proof is immediate from the computations in the proof of Proposition 3.2.

We also have,

Theorem 4.2. Let $p$ and $\lambda$ be as in Section 2. Let $F, A$ and $B$ be as in Propostion 3.7, except for the fact that we require that $F$ satisfies (3.10) and (3.11) with $\tau$ satisfying (2.7), instead of (3.3) and (3.4), and with no oscillation assumption on $p$. Let $\phi\in W^{1, p(\cdot)}(\Omega)$ such that $0\le \phi\le M$, for some $M > 0$.

Then, there exists a minimizer $u\in \phi +W_0^{1, p(\cdot)}(\Omega)$ to (1.1) and $0\le u\le M$ in $\Omega$.

Proof. Proceeding as in the proof of Proposition 3.2 we obtain that there exists a minimizer $u\in \phi +W_0^{1, p(\cdot)}(\Omega)$ to (1.1). The proof that $0\le u\le M$ is similar to that of Proposition 3.7. We only have to observe that

For local minimizers of (1.1) we first have

Lemma 4.1. Let $p, F, A, B$ and $\lambda$ be as in Section 2. Assume that $F$ satisfies (3.3) and (3.4) with $\tau$ satisfying (2.7). Let $u\in W^{1, p(\cdot)}(\Omega)$ be a local minimizer of

Then

where $A(x, s, \eta) = \nabla_\eta F(x, s, \eta)$ and $B(x, s, \eta) = F_s(x, s, \eta)$.

Proof. In fact, let $t > 0$ and $0\le\xi\in C^{\infty}_0(\Omega)$. Using the minimality of $u$ we have

and if we take $t\rightarrow 0$, we obtain

which gives (4.1).

From now on we will deal with nonnegative, bounded, local minimizers of (1.1). Next we will prove that they are locally Lipschitz continuous.

We first prove that nonnegative, bounded, local minimizers are locally Hölder continuous.

Theorem 4.3. Let $p, F, A, B$ and $\lambda$ be as in Section 2. Assume that $F$ satisfies (3.3) and (3.4) with $\tau$ satisfying (2.7). Let $x_0\in\Omega$, $\hat r_0 > 0$ such that $B_{\hat r_0}(x_0)\subset\subset\Omega$. Assume that $A, B$ satisfy condition (3.14) in $B_{\hat r_0}(x_0)$ and either $B(x, 0, 0)\equiv0$ for $x\in B_{\hat r_0}(x_0)$ or $B$ satisfies (3.32) for $x\in B_{\hat r_0}(x_0)$. Let $u\in W^{1, p(\cdot)}(\Omega)\cap L^\infty(\Omega)$ be a nonnegative local minimizer of (1.1). Then, there exist $0 < \gamma < 1$, $\gamma = \gamma(N, p_{\min})$ and $0 < \hat\rho_0 < \hat r_0$, such that $u\in C^{\gamma}(\overline{B_{\hat\rho_0}(x_0)})$. Moreover, $\|u\|_{C^{\gamma}(\overline{B_{\hat\rho_0}(x_0)})}\leq C$ with $\hat \rho_0$ and $C$ depending only on $\beta, p_{\max}, p_{\min}, N, L, \hat r_0, \lambda_0, \Lambda_0$, $\|g\|_{L^1(B_{\hat r_0}(x_0))}$, $\mu$, $c_1$, $\lambda_{\max}$, $\|u\|_{L^\infty(B_{\hat r_0}(x_0))}$, $||\tau||_{L^{\infty}(B_{\hat r_0}(x_0))}$ and $\delta$.

Proof. We will prove that there exist $0 < \gamma < 1$ and $0 < \rho_0 < r_0 < \hat r_0$ such that, if $B_{r_0}(y)\subset B_{\hat r_0}(x_0)$ and $\rho\leq \rho_0$, then

where $p_- = \inf\{p(x), x\in B_{r_0}(y)\}$. Without loss of generality we will assume that $y = 0$.

In fact, let $0 < r_0\le \min\{\frac{\hat r_0}{2}, 1\}$, $0 < r\leq r_0$ and $v$ the solution of

Observe that, under our assumptions we can apply either Proposition 3.1 and Proposition 3.5 or Proposition 3.6 and deduce that such a solution exists and it is bounded in $\overline{B}_r$ if $r_0$ is small enough depending on $\delta$ and $L = \|\nabla p\|_{L^\infty(\Omega)}$. Hence, by Proposition 3.3, we have

where $\alpha = \alpha(p_{\min}, p_{\max})$ and $\lambda_0$ is as in (2.2).

By the minimality of $u$, we have (if $A_1 = B_r\cap\{p(x) < 2\}$ and $A_2 = B_r\cap\{p(x)\geq 2\}$)

where $C = C(p_{\min}, p_{\max}, N, {\lambda_{\max}}, \lambda_0)$.

Let ${\varepsilon} > 0$. Take $\rho = r^{1+{\varepsilon}}$ and suppose that $r^{{\varepsilon}}\leq 1/2$. Take $0 < \eta < 1$ to be chosen later. Then, by Young's inequality, the definition of $A_1$ and (4.7), we obtain

Therefore, by (4.6) and (4.8), we get

where $C = C(p_{\min}, p_{\max}, N, {\lambda_{\max}}, \lambda_0)$.

Since, $|\nabla u|^q\leq C(|\nabla u-\nabla v|^q+|\nabla v|)^q)$, for any $q > 1$, with $C = C(q)$, we have, by (4.9), choosing $\eta$ small, that

where $C = C(p_{\min}, p_{\max}, N, {\lambda_{\max}}, \lambda_0)$.

Now let $M\ge 1$ such that $||v||_{L^{\infty}(B_{r})}\le M$ and define

Observe that $M$ depends only on $\|u\|_{L^\infty(B_{\hat r_0}(x_0))}$ if $B(x, 0, 0)\equiv 0$ or it depends also on the structural conditions on $F$, $A$ and $B$, on $\hat r_0$ and on the bound $L$ of $\|\nabla p\|_{L^\infty}$ if not.

There holds that,

where

Now, let

Observe that $w\in W^{1, \bar p(\cdot)}(B_1)\cap L^\infty(B_1)$ satisfies

where $\bar p(x) = p(rx)$.

Let us see that (4.11) is under the conditions of Theorem 1.1 in ^[10].

First, we clearly have $\widetilde A(x, s, 0) = 0$. Moreover, as $1\le r^{{p^-}_r-{p^+}_r}\le C_L < \infty$ if $r\le 1$ and we have assumed that $M\ge1$,

On the other hand,

Then, assuming without loss of generality that $p(rx_1)\ge p(r x_2)$,

if $r\le r_{M, \beta}$.

Similarly,

with $\Lambda_4 = \Lambda_0C_L M^{p_{\max}-p_{\min}+1}$.

On the other hand, denoting $\bar\tau(x) = \tau (rx)$,

with $\Lambda_5$ depending on $\Lambda_0$, $L$, $p_{\max}$, $p_{\min}$, $M$ and $||\tau||_{L^{\infty}(B_{\hat r_0}(x_0))}$.

Since $|w|\le 1$, we may assume that

with $\Lambda_6$ depending on $\Lambda_0$, $L$, $p_{\min}$, $p_{\max}$, $M$ and $||\tau||_{L^{\infty}(B_{\hat r_0}(x_0))}$.

From Theorem 1.1 in ^[10], it follows that $w\in C^{1, \alpha}_{\rm loc}(B_1)$ for some $0 < \alpha < 1$ and that

with $C$ depending only on $\beta, p_{\max}, p_{\min}, N, L, \lambda_0, \Lambda_0$, $M$ and $||\tau||_{L^{\infty}(B_{\hat r_0}(x_0))}$, which implies

Therefore, from (4.10) and (4.15), we deduce that if $r$ is small depending on $M$ and $\beta$,

with $p_+ = \sup\{p(x), x\in B_{r_0}\}$ and $C$ depending on $\beta, p_{\max}, p_{\min}, N, L, \lambda_0, \Lambda_0$, $\lambda_{\max}$, $M$ and $||\tau||_{L^{\infty}(B_{\hat r_0}(x_0))}$.

Then, if we take ${\varepsilon}\le\frac{p_{\min}}{N}$, we have by (4.16) and by our election of $\rho$, that

Now let $r_0\le r_0({\varepsilon}, p_{\min}, L)$ so that

and small enough so that, in addition, $r_0^{{\varepsilon}}\leq 1/2$. Then, if $\rho\leq \rho_0 = r_0^{1+{\varepsilon}}$ and moreover, $r_0$ is small depending on $M$ and $\beta$,

where $\gamma = \frac{\frac{{\varepsilon}}{2}}{(1+{\varepsilon})} = \gamma(N, p_{\min})$. That is, if $\rho\leq \rho_0 = r_0^{1+{\varepsilon}}$

Thus (4.3) holds, with $C$ depending only on $\beta, p_{\max}, p_{\min}, N, L, \hat r_0, \lambda_0, \Lambda_0$, $\|g\|_{L^1(B_{\hat r_0}(x_0))}$, $\mu$, $c_1$, $\lambda_{\max}$, $\|u\|_{L^\infty(B_{\hat r_0}(x_0))}$, $||\tau||_{L^{\infty}(B_{\hat r_0}(x_0))}$ and $\delta$.

Applying Morrey's Theorem, see e.g., ^[18], Theorem 1.53, we conclude that $u\in C^{\gamma}(B_{\rho_0}(x_0))$ and $\|u\|_{C^{\gamma}(\overline{B_{{\rho_0}/2}(x_0)})}\leq {C}$ for $C$ depending only on $\beta, p_{\max}, p_{\min}, N, L, \hat r_0, \lambda_0, \Lambda_0$, $\|g\|_{L^1(B_{\hat r_0}(x_0))}$, $\mu$, $c_1$, $\lambda_{\max}$, $\|u\|_{L^\infty(B_{\hat r_0}(x_0))}$, $||\tau||_{L^{\infty}(B_{\hat r_0}(x_0))}$ and $\delta$.

As a corollary we obtain

Corollary 4.1. Let $p, F, A, B$ and $\lambda$ be as in Section 2. Assume that $F$ satisfies (3.8) and (3.9) with $\tau$ satisfying (2.7). Assume that $A, B$ satisfy condition (3.14) and either $B(x, 0, 0)\equiv0$ for $x\in \Omega$ or $B$ satisfies (3.32) for $x\in \Omega$. Let $u\in W^{1, p(\cdot)}(\Omega)\cap L^\infty(\Omega)$ be a nonnegative local minimizer of (1.1). Then, there exists $0 < \gamma < 1$, $\gamma = \gamma(N, p_{\min})$ such that $u\in C^{\gamma}(\Omega)$. Moreover, if $\Omega'\subset\subset\Omega$, then $\|u\|_{C^{\gamma}(\overline{\Omega'})}\leq C$ with $C$ depending only on ${\rm dist}(\Omega', \partial\Omega)$, $\beta$, $N$, $p_{\min}$, $p_{\max}$, $L$, ${\lambda_{\max}}$, $\lambda_0$, $\Lambda_0$, $\mu$, $c_1$, $\|u\|_{L^{\infty}(\Omega)}$, $||\tau||_{L^{\infty}(\Omega)}$ and $\delta$.

Then, under the assumptions of the previous corollary we have that $u$ is continuous in $\Omega$ and therefore, $\{u > 0\}$ is open. We can now prove the following property for nonnegative local minimizers of (1.1).

Lemma 4.2. Let $p, F, A, B$ and $\lambda$ be as in Corollary 4.1. If $u\in W^{1, p(\cdot)}(\Omega)\cap L^{\infty}(\Omega)$ is a nonnegative local minimizer of

there holds that,

where $A(x, s, \eta) = \nabla_\eta F(x, s, \eta)$ and $B(x, s, \eta) = F_s(x, s, \eta)$.

Proof. From Lemma 4.1 we already know that (4.1) holds. In order to obtain the opposite inequality in $\{u > 0\}$, we let $0\le\xi\in C^{\infty}_0(\{u > 0\})$ and consider $u-t\xi$, for $t < 0$, with $|t|$ small.

Using the minimality of $u$ we have

and if we take $t\rightarrow 0$, we obtain

which gives the desired inequality, so (4.17) follows.

We will next prove the Lipschitz continuity of nonnegative local minimizers of (1.1).

Before getting the Lipschitz continuity we prove the following result

Theorem 4.4. Let $p, F, A, B$, $\lambda$ and $u$ be as in Corollary 4.1. Let $\Omega'\subset\subset\Omega$. There exist constants $C > 0$, $r_0 > 0$ such that if $x_0\in \Omega'\cap\partial\{u > 0\}$ and $r\le r_0$ then

The constants depend only on ${\rm dist}(\Omega', \partial\Omega)$, $\beta$, $N$, $p_{\min}$, $p_{\max}$, $L$, ${\lambda_{\max}}$, $\lambda_0$, $\Lambda_0$, $\mu$, $c_1$, $\|u\|_{L^{\infty}(\Omega)}$, $||\tau||_{L^{\infty}(\Omega)}$ and $\delta$.

Proof. Let us suppose by contradiction that there exist a sequence of nonnegative local minimizers $u_k$ corresponding to functionals $J_k$ given by

with $u_k\in W^{1, p_k(\cdot)}(\Omega)\cap L^{\infty}(\Omega)$, $p_{\min}\leq p_k(x)\leq p_{\max}$, $\|\nabla p_k\|_{L^{\infty}}\leq L$, $0\le\lambda_k(x)\le{\lambda_{\max}}$, $||u_k||_{L^{\infty}(\Omega)}\le M$, for some $M\ge 1$, and points $\bar x_k\in\Omega'\cap\partial\{u_k > 0\}$, such that

We denote $A_k(x, s, \eta) = \nabla_\eta F_k(x, s, \eta)$ and $B_k(x, s, \eta) = (F_k)_s(x, s, \eta)$ and we also suppose that $p_k, F_k, A_k, B_k$ and $\tau_k$ satisfy the assumptions in Section 2 with constants $\lambda_0$, $\Lambda_0$ and $\beta$, we assume that $A_k, B_k$ satisfy condition (3.14) and $F_k$ satisfy (3.8) and (3.9) with ${\tau}_k$ satisfying (2.7) and either and $B_k(x, 0, 0)\equiv0$ for $x\in \Omega$ or $B_k$ satisfy (3.32) for $x\in \Omega$. All these conditions with exponent $p_k$ and constants independent of $k$ and with $||{\tau}_k||_{L^{\infty}(\Omega)}\le \tau_0$, for some $\tau_0 > 0$.

Without loss of generality we will assume that $\bar x_k = 0$.

Let us define in $B_1$, for $k$ large, $\bar u_k(x) = \frac1{r_k}u_k(r_k x)$, ${\bar p}_k(x) = p_k(r_k x)$ and ${\bar \lambda}_k(x) = {\lambda}_k(r_k x)$. Then $p_{\min}\leq {\bar p}_k(x)\leq p_{\max}$, $\|\nabla {\bar p}_k\|_{L^{\infty}(B_1)}\leq L r_k$ and $0\le{\bar\lambda}_k(x)\le{\lambda_{\max}}$. Moreover, $\bar u_k$ is a nonnegative minimizer in $\bar u_k + W_0^{1, \bar p_k(\cdot)}(B_1)$ of the functional

where

with

Let $d_k(x) = { }\mbox{dist}(x, \{\bar u_k = 0\})$ and $\mathcal{O}_k = { }\Big\{x\in B_1: d_k(x)\leq \frac{1-|x|}{3}\Big\}$. Since $\bar u_k(0) = 0$ then $\overline{B}_{1/4}\subset \mathcal{O}_k$, therefore

For each fixed $k$, $\bar u_k$ is bounded, then $(1-|x|) \bar u_k(x)\rightarrow 0 \mbox{ when } |x|\rightarrow 1$ which means that there exists $x_k\in {\mathcal{O}_k}$ such that $(1-|x_k|) \bar u_k(x_k) = \sup_{\mathcal{O}_k}(1-|x|) \bar u_k(x)$, and then

as $x_k\in \mathcal{O}_k$. Observe that $\delta_k: = d_k(x_k)\leq \frac{1-|x_k|}{3}$. Let $y_k\in \partial\{\bar u_k > 0\}\cap B_1$ such that $|y_k-x_k| = \delta_k$. Then,

By (2) we have

where in the last inequality we are using (3). Then,

As $B_{\delta_k}(x_k)\subset \{\bar u_k > 0\}$, then $B_{r_k\delta_k}(r_k x_k)\subset \{u_k > 0\}$. Hence, ${\rm div} {A}_k(x, u_k, \nabla u_k) = {B}_k(x, u_k, \nabla u_k)$ in $B_{r_k\delta_k}(r_k x_k)$. Recalling that $||u_k||_{L^{\infty}(B_{r_k\delta_k}(r_k x_k))}\le {M}$, we can replace $|s|^{{\tau}_k(x)}$ in (2.6) for $B_k$ by $M^{{\tau}_0}$. Then we can apply Harnack's inequality (Theorem 3.2 in ^[23]) and we thus have

with $C$ a positive constant depending only on $N, p_{\min}, p_{\max}, L$, $M$, $\lambda_0$, $\Lambda_0$ and ${\tau}_0$.

It follows that

Recalling (4.19), we get from (4.22), for $k$ large,

with $c$ a positive constant depending only on $N, p_{\min}, p_{\max}, L \; M$, $\lambda_0$, $\Lambda_0$ and ${\tau}_0$. As $\overline{B_{\frac{3}{4}\delta_k}}(x_k)\cap \overline{B_{\frac{\delta_k}{4}}}(y_k)\neq \emptyset$ we have by (4.23)

Let $w_k(x) = { }\frac{\bar u_k(y_k+\frac{\delta_k}{2} x)}{\bar u_k(x_k)}$. Then, $w_k(0) = 0$ and, by (4.20) and (4.24), we have

Now, recalling that $\bar u_k$ is a nonnegative minimizer in $\bar u_k + W_0^{1, \bar p_k(\cdot)}(B_1)$ of the functional ${\bar J}_k$ in (4.18) and that $B_{\frac{\delta_k}{2}}(y_k)\subset B_1$, we see that $w_k$ is a nonnegative minimizer of $\hat J_k$ in $w_k + W^{1, {\bar p}_k(y_k+\frac{\delta_k}{2} x)}_0(B_1)$, where

We let $c_k = \frac{2 \bar u_k(x_k)}{\delta_k}$ and we notice that $c_k\rightarrow\infty$. So we define ${{\tilde p}_k(x)} = {\bar p}_k(y_k+\frac{\delta_k}{2} x)$ and divide the functional $\hat J_k$ by $c_k^{{\tilde p}_k^{-}}$, with ${{\tilde p}_k^{-}} = \inf_{B_1}{\tilde p}_k$. Then, it follows that $w_k$ is a nonnegative minimizer of $\tilde J_k$ in $w_k + W^{1, \tilde p_k(\cdot)}_0(B_1)$, where

We claim that

up to a subsequence, for some constants $M_1$ and $p_0$, where $M_1 = M_1(M, L)$.

On the one hand, $0 < {{\tilde\lambda}_k}(x)\le {\lambda_{\max}} c_k^{-1}\to 0$ gives (4.26).

In addition, in $B_1$ there holds, for $k$ large, that $1\le {c_k^{{{\tilde p}_k(x)}-{{\tilde p}_k^{-}}}}\le e^{2 \|\nabla \tilde p_k\|_{L^{\infty}}\log c_k}$. But we have $\|\nabla \tilde p_k\|_{L^{\infty}}\log c_k\le L r_k\frac{\delta_k}{2}\log \big(\frac{2M}{r_k \delta_k}\big)\to 0$, which implies (4.27).

To see (4.28) we observe that $p_{\min}\leq p_k(x)\leq p_{\max}$ and $\|\nabla p_k\|_{L^{\infty}(\Omega)}\leq L$ and then, for a subsequence, ${p}_k\rightarrow p$ uniformly on compacts of $\Omega$, so ${\tilde p}_k(x) = {p}_k(r_k(y_k+\frac{\delta_k}{2} x))\rightarrow p_0 = p(0)$ uniformly in $B_1$.

We define $\tilde{A}_k = \nabla_\eta \tilde{F}_k$ and $\tilde{B}_k = (\tilde{F}_k)_s$ and we observe that

There holds that ${\tilde p}_k$, ${\tilde F}_k$, ${\tilde A}_k$, ${\tilde B}_k$ and ${\tilde\tau}_k$ are under the assumptions of Section 2, with constants independent of $k$. In fact, recalling (4.27), we get for $k$ large

Assuming, without loss of generality, that ${\tilde p}_k(x_1)\ge {\tilde p}_k(x_2)$ and using that $(r_k\frac{\delta_k}{2})^{{\beta}_1}\log c_k\le (r_k\frac{\delta_k}{2})^{{\beta}_1}\log \big(\frac{2M}{r_k \delta_k}\big)\to 0$, we get

Finally, recalling that $r_k\bar u_k(x_k)\le M$, we obtain

On the other hand, ${\tilde A}_k$ and ${\tilde B}_k$ satisfy condition (3.14). In fact, since ${ A}_k$ and ${ B}_k$ satisfy condition (3.14),

Also, since $F_k$ satisfy (3.8) and (3.9) with $\tau_k$ satisfying (2.7), with exponent $p_k$ and constants independent of $k$, then $\tilde F_k$ satisfy (3.8) and (3.9) with ${\tilde\tau}_k$ satisfying (2.7), with exponent $\tilde p_k$ and constants independent of $k$. In fact,

Analogously,

If $B_k(x, 0, 0)\equiv0$ for $x\in \Omega$, then ${\tilde B_k(x, 0, 0)}\equiv0$ for $x\in B_1$.

On the other hand, if $B_k$ satisfy (3.32) for $x\in \Omega$ with exponent $p_k$ and constant independent of $k$, then $\tilde B_k$ satisfy (3.32) for $x\in B_1$ with exponent $\tilde p_k$ and constant independent of $k$. In fact,

We now take $v_k$ the solution of

In fact, from Corollaries 3.3, 3.4 and 3.2 and the upper bound in (4.25), it follows that if $k$ is large enough

where $\bar C$ depends only on $N$, $p_{\min}$, $p_{\max}$, $L$, $\lambda_0$, $\Lambda_0$, $\mu$, $c_1$, $\delta$, $M$ and $\tau_0$. Here we have used that $\sup_{B_{3/4}}{\tilde p}_k-\inf_{B_{3/4}} {\tilde p}_k \le \|\nabla \tilde p_k\|_{L^{\infty}}\frac32\le 3L r_k\frac{\delta_k}{4} < \delta$ in (3.8), for $k$ large.

Then, by (4.38), we can replace $|s|^{{\bar\tau}_k(x)}$ in (4.33) by $1+{\bar C}^{\tau_0}$ and applying Theorem 1.1 in ^[10] we obtain that, for $k$ large,

where $\hat C$ depends only on $\beta$, $N$, $p_{\min}$, $p_{\max}$, $L$, $\lambda_0$, $\Lambda_0$, $\mu$, $c_1$, $\delta$, $M$ and $\tau_0$. Therefore, there is a function $v_0\in C^{1, \alpha}(\overline{B_{1/2}})$ such that, for a subsequence,

Let us now show that

From the minimality of $w_k$ we have

which together with Proposition 3.3 gives

where $A_1^k = B_{3/4}\cap\{\tilde p_k(x) < 2\}$, $A_2^k = B_{3/4}\cap\{\tilde p_k(x)\geq 2\}$ and $C = C(p_{\min}, p_{\max}, N, \lambda_0)$.

Applying Hölder's inequality (Theorem A.3) with exponents $\frac{2}{\tilde p_k(x)}$ and $\frac{2}{2-\tilde p_k(x)}$, we get

where

Since

then, from (4.44), (4.26) and Proposition A.1, we get, for $k$ large,

$C = C(p_{\min}, p_{\max}, N, \lambda_0)$. On the other hand, (4.37) and the bounds (4.34), (4.35) and (4.38) give

This implies

for some $\tilde C\ge 1$, depending only on $p_{\min}$, $p_{\max}$ and the uniform constants and functions in (4.34), (4.35) and (4.38). Now (4.47) and Proposition A.1 give

Let us show that the right hand side in (4.48) can be bounded independently of $k$.

In fact, let $\tilde v_k$ be the solution of

Then, similar arguments to those leading to (4.38) and (4.39), give, for $k$ large enough,

and

where $\bar C$ and $\hat C$ depend only $\beta$, $N$, $p_{\min}$, $p_{\max}$, $L$, $\lambda_0$, $\Lambda_0$, $\mu$, $c_1$, $\delta$, $M$ and $\tau_0$.

Since $w_k$ is a nonnegative minimizer of $\tilde J_k$ in $B_1$, then we can argue as in the proof of Theorem 4.3 and get estimate (4.10) for $u = w_k$, $v = \tilde v_k$, $p(x) = \tilde p_k(x)$, $\lambda(x) = \tilde\lambda_k(x)$, $r = 7/8$ and $\rho = 3/4$. That is,

where $C = C(p_{\min}, p_{\max}, N, {\lambda_{\max}}, \lambda_0)$. Therefore (4.52) and (4.51) give, for $k$ large, a uniform bound for the right hand side in (4.48). That is,

with $\bar C$ a constant depending only on $\beta$, $N$, $p_{\min}$, $p_{\max}$, $L$, $\lambda_0$, $\Lambda_0$, $\mu$, $c_1$, $\delta$, $M$ and $\tau_0$.

Now, putting together (4.43), (4.45), (4.46), (4.53) and (4.26), we obtain

Thus, using Poincare's inequality (Theorem A.4) and Theorem A.2, we get (4.41).

In order to conclude the proof, we now observe that, since ${\tilde p}_k$, ${\tilde F}_k$, ${\tilde A}_k$, ${\tilde B}_k$, ${\tilde\tau}_k$, ${\tilde \lambda}_k$ and $w_k$ fall (uniformly) under the assumption of Corollary 4.1 in $B_1$, there exists $0 < \gamma < 1$, $\gamma = \gamma(N, p_{\min})$, such that

with $C$ depending only on $\beta$, $N$, $p_{\min}$, $p_{\max}$, $L$, ${\lambda_{\max}}$, $\lambda_0$, $\Lambda_0$, $\mu$, $c_1$, $\tau_0$ and $\delta$ (recall that $\|w_k\|_{L^\infty(B_1)}\le 2$).

Therefore, there is a function $w_0\in C^{\gamma}(\overline{B_{1/2}})$ such that, for a subsequence,

In addition, recalling (4.40) and (4.41), we get $v_0 = w_0$ in $\overline{B_{1/2}}$.

We then observe that, since there holds that $w_k\ge 0$, $w_k(0) = 0$ and (4.25), then (4.55) implies

That is,

Let us show that (4.56) gives a contradiction. We will divide the proof in two cases.

Case I. Assume that ${\tilde B_k(x, 0, 0)}\equiv0$ for $x\in B_1$.

We first observe that, since $w_k\ge 0$, from Proposition 3.6 we deduce that $v_k\ge 0$.

Recalling (4.39), we choose $M_0 > 0$ such that, for every $k$,

and define

where

Then,

From (4.29) and (4.30) (recall Remark 2.1) we deduce

for some constant $\tilde\Lambda_0 > 0$ independent of $k$.

Let us now fix ${\varepsilon} > 0$. Then, if $k\ge k_0({\varepsilon})$, (4.57), (4.33) and (4.28) give, for large $k$,

for some positive constants ${\tilde{\tilde{\Lambda}}}_0$ and $c$ (independent of ${\varepsilon}$ and $k$).

Applying Harnack's inequality (see ^[22], Theorems 5 and 6 and Section 5), we get for any $0 < r < 1$

with $C_r$ a positive constant.

Now, letting $k\to\infty$ first, and then ${\varepsilon}\to 0$, we get

with

Since $0 < r < 1$ is arbitrary, we get $v_0\equiv 0$ in $B_{1/2}$. This is in contradiction with (4.56) and concludes the proof of Case I.

Case II. Assume that $\tilde B_k$ satisfy (3.32) for $x\in B_1$ with exponent $\tilde p_k$ and constant independent of $k$. Then, (4.30), (4.31), (4.32) and (4.36) imply that, for a subsequence,

and from (4.29) and (4.30) (recall Remark 2.1) we deduce

for some constant $\tilde\Lambda_0 > 0$. Then, (4.37) and (4.40) imply that

Applying Harnack's inequality (see ^[22], Theorems 5 and 6 and Section 5), we get again, that (4.58) and (4.59) holds for any $0 < r < 1$. This contradicts once more (4.56) and concludes the proof.

We can now prove the Lipschitz continuity of nonnegative local minimizers of (1.1)

Theorem 4.5. Let $p, F, A, B, \lambda$ and $u$ be as in Corollary 4.1. Then $u$ is locally Lipschitz continuous in $\Omega$. Moreover, for any $\Omega'\subset\subset \Omega$ the Lipschitz constant of $u$ in $\Omega'$ can be estimated by a constant $C$ depending only on ${\rm dist}(\Omega', \partial\Omega)$, $\beta$, $N$, $p_{\min}$, $p_{\max}$, $L$, ${\lambda_{\max}}$, $\lambda_0$, $\Lambda_0$, $\mu$, $c_1$, $\|u\|_{L^{\infty}(\Omega)}$, $||\tau||_{L^{\infty}(\Omega)}$ and $\delta$.

Proof. The result is a consequence of Corollary 4.1, Lemma 4.2 and Theorem 4.4 above, and Proposition 2.1 in ^[16]. We point out that, although the proof of Proposition 2.1 in ^[16] is written for the particular case in which $A(x, s, \eta) = |\eta|^{p(x)-2}\eta$ and $B(x, s, \eta) = f(x)$, this same proof is valid for general $A$ and $B$ under the present assumptions, without changes.

5.   Examples

In this section we present some examples of application of our results.

Theorem 5.1. Let $f(x, s)$ be a measurable function such that $f(x, \cdot)\in C^2({\mathbb R})$ for every $x\in\Omega$. Let $a(x, s)$ be a Hölder continuous function with exponent $\alpha$, $a(x, \cdot)\in C^2({\mathbb R})$ for every $x\in\Omega$. Let $p$, $\tau$ and $\lambda$ as in Section 2 and $0 < \delta < p_{\min}$. Assume that there exist positive constants $a_0, a_1, a_2, c_1$ and $\Lambda_0$ such that

f1 $-c_1(1+|s|^{p(x)-\delta})\le f(x, s)\le c_1(1+|s|^{\tau(x)})$ in $\Omega\times{\mathbb R}$.

f2 $f_s(x, 0)\equiv 0$ in $\Omega$.

f3 $f_{ss}(x, s)\ge0$ in $\Omega\times{\mathbb R}$.

f4 $|f_s(x, s)|\le \Lambda_0(1+|s|^{\tau(x)})$ in $\Omega\times{\mathbb R}$.

And

a1 $0 < a_0\le a(x, s)\le a_1 < \infty$ in $\Omega\times{\mathbb R}$.

a2 $|a_s(x, s)|\le a_2$ in $\Omega\times{\mathbb R}$.

a3 $\big(a(x, s)^{1-\gamma(x)}\big)_{ss}\le0$ in $\Omega\times{\mathbb R}$ with $\gamma(x) = \frac{2p(x)}{\min\{1, p(x)-1\}} > 1$.

Let

and let $u\in W^{1, p(\cdot)}(\Omega)\cap L^\infty(\Omega)$ a nonnegative, local minimizer of (1.1). Then, $u$ is locally Lipschitz continuous in $\Omega$.

Proof. We only have to see that $F, A, B$ satisfy the hypotheses of Theorem 4.5.

There holds that

And

Moreover,

$(1). \; A(x, s, 0) = 0$.

$(2). \; \sum_{i, j}\frac{\partial A_i}{\partial\eta_j}(x, s, \eta)\xi_i\xi_j\ge\lambda_0|\eta|^{p(x)-2}|\xi|^2$. In fact,

with $\lambda_0 = a_0\min\{1, p_{\min}-1\}$.

$(3). \; \sum_{i, j}\Big|\frac{\partial A_i}{\partial\eta_j}(x, s, \eta)\Big|\le \Lambda_0|\eta|^{p(x)-2}$ if $\Lambda_0\ge a_1N(p_{\max}+3)$.

$(4). \; |A(x_1, s, \eta)-A(x_2, s, \eta)|\le \Lambda_0|x_1-x_2|^\alpha\big(|\eta|^{p(x_1)-1}+|\eta|^{p(x_2)-1}\big)\big|\big(1+\big|\log|\eta|\big|\big)$ for a big enough constant $\Lambda_0$. In fact, without loss of generality we may assume that $p(x_1)\ge p(x_2)$. There holds,

Now, if $|\eta|\ge1$,

A similar inequality holds if $|\eta|\le1$. So that,

where $C_a$ is the Holder constant of the function $a$. And the result follows if $\Lambda_0\ge a_1{L} d(\Omega)^{1-\alpha}+C_a$ with $d(\Omega)$ the diameter of $\Omega$.

$(5). \; |A(x, s_1, \eta)-A(x, s_2, \eta)|\le a_2|\eta|^{p(x)-1}|s_1-s_2|$.

We clearly have,

$(1).$ $|B(x, s, \eta)|\le \Lambda_0(1+|\eta|^{p(x)}+|s|^{\tau(x)})$ (as we may assume, without loss of generality that $\Lambda_0\ge \frac{a_2}{p_{\min}}$).

$(2). \; B(x, 0, 0) = 0$.

Finally, let us see that

In fact, let

Then,

By (5.1), we only have to check that

Since $f_{ss}(x, s)\ge0$ it is enough to check that

And, (5.2) holds by hypothesis a3.

If $a(x, s)$ is smooth in $-M_1 < s < M_2$ with $M_1, M_2 > 0$, condition a3 only holds in $0\le s\le M < M_2$ and the local minimizer $u$ satisfies that $0\le u\le M$, we can still apply the results in this paper and get that $u$ is locally Lipschitz continuous.

Theorem 5.2. Let $f(x, s)$ be a measurable function such that $f(x, \cdot)\in C^2({\mathbb R})$ for every $x\in\Omega$. Let $a(x, s)$ be a Hölder continuous function with exponent $\alpha$, $a(x, \cdot)\in C^2(-M_1, M_2)\cap Lip({\mathbb R})$ for almost every $x\in\Omega$ with $M_1, M_2 > 0$. Let $p$, $\tau$ and $\lambda$ as in Section 2 and $0 < \delta < p_{\min}$. Assume that there exist positive constants $a_0, a_1, a_2, c_1, \Lambda_0$ and $0 < M < M_2$ such that

f1 $-c_1(1+|s|^{p(x)-\delta})\le f(x, s)\le c_1(1+|s|^{\tau(x)})$ in $\Omega\times{\mathbb R}$.

f2 $f_s(x, 0)\equiv 0$ in $\Omega$.

f3 $f_{ss}(x, s)\ge0$ in $\Omega\times{\mathbb R}$.

f4 $|f_s(x, s)|\le \Lambda_0(1+|s|^{\tau(x)})$ in $\Omega\times{\mathbb R}$.

And

a1 $0 < a_0\le a(x, s)\le a_1 < \infty$ in $\Omega\times{\mathbb R}$.

a2 $|a_s(x, s)|\le a_2$ \ \ a.e. in $\Omega\times{\mathbb R}$.

a3' $\big(a(x, s)^{1-\gamma(x)}\big)_{ss}\le0$ in $\Omega\times[0, M]$ with $\gamma(x) = \frac{2p(x)}{\min\{1, p(x)-1\}} > 1$.

Let

and let $u\in W^{1, p(\cdot)}(\Omega)\cap L^\infty(\Omega)$ be a local minimizer of (1.1) such that $0\le u\le M$. Then, $u$ is locally Lipschitz continuous in $\Omega$.

Proof. By Proposition 3.7 for such a function $f$ and with $a$ satisfying a1 and a2, for every ball $B_r(x_0)\subset\Omega$ with $r$ small enough there exists a solution $v\in u+W^{1, p(\cdot)}_0(B_r(x_0))$ of (1.2) such that $0\le v\le \|u\|_{L^\infty(B_r(x_0))}$. And this result also holds for all the rescaled equations and functions that appear in the proofs of Section 4. Hence, condition (3.14) is only needed for $s\in(0, M)$ and this is a consequence of a3'.

Example 5.1. A possible example of functions $a$ and $f$ satisfying the assumptions of Theorem 5.2 is

with $M_2 > 0$ and $q\in L^\infty(\Omega)$ a Hölder continuous function such that $0 < q(x) < \frac1{\gamma(x)-1}$ and

with $0\le b\in L^\infty(\Omega)$ and $\tau(x)\ge 2$ in $\Omega$ satisfying (2.7).

Another possible choice of $f$ is

with $0\le b\in L^\infty(\Omega)$ and

where $\tau(x)$ satisfies (2.7) and the functions $\widetilde a, \widetilde b, \widetilde c \in L^\infty(\Omega)$ are chosen in such a way that $\widetilde f(x, \cdot)\in C^2({\mathbb R})$ for every $x\in\Omega$.

With this choice of $a$ and $f$, for every $0 < M < M_2$ there holds that any local minimizer $u$ such that $0\le u\le M$ is locally Lipschitz continuous in $\Omega$.

Observe that, by Theorem 4.2, if $\phi\in W^{1, p(\cdot)}(\Omega)$ is such that $0\le\phi\le M < M_2$, such a minimizers always exists.

We have another example.

Theorem 5.3. Let $f(x, s)$ be a measurable function such that $f(x, \cdot)\in C^2({\mathbb R})$ for every $x\in\Omega$. Let $G(x, \eta)$ be a measurable function such that $G(x, \cdot)\in C^1({\mathbb R}^N)\cap C^2({\mathbb R}^N\setminus\{0\})$ for every $x\in\Omega$. Let $p$ and $\lambda$ as in Section 2 and assume that either $f$ satisfies conditions $\rm f1, \cdots, f4$ in Theorem 5.1 or $f$ satisfies $\rm f1, f3$ in Theorem 5.1 and

f4' $|f_s(x, s)|\le \Lambda_0\big(1+|s|^{p(x)-1}\big)$.

On the other hand, $G$ satisfies

G1 $\mu\big(|\eta|^{p(x)}- 1\big)\le G(x, \eta)\le \mu^{-1}\big(|\eta|^{p(x)}+ 1\big)$ with $\mu > 0$.

G2 $\nabla_\eta G(x, 0)\equiv0$ in $\Omega$.

G3 $\sum_{i, j}\frac{\partial^2 G}{\partial\eta_i\partial\eta_j}\xi_i\xi_j\ge \lambda_0 |\eta|^{p(x)-2}|\xi|^2$.

G4 $\sum_{i, j}\Big|\frac{\partial^2 G}{\partial\eta_i\partial\eta_j}\Big|\le \Lambda_0 |\eta|^{p(x)-2}$.

G5 $|\nabla_\eta G(x_1, \eta)-\nabla_\eta G(x_2, \eta)|\le \Lambda_0 |x_1-x_2|^\beta \big(|\eta|^{p(x_1)-1}+|\eta|^{p(x_2)-1}\big)\big(1+\big|\log|\eta|\big|\big)$ for some $0 < \beta\le1$.

Let

and let $u\in W^{1, p(\cdot)}(\Omega)\cap L^\infty(\Omega)$ be a nonnegative, local minimizer of (1.1). Then, $u$ is locally Lipschitz continuous in $\Omega$.

Proof. There holds that

And it is clear that $F, A$ and $B$ satisfy the assumptions in Theorem 4.5.

Example 5.2. A possible example of function $G$ satisfying the assumptions of Theorem 5.3 is

with $p(x)$ as in Section 2, $a(x)$ a Hölder continuous function such that $a_0\le a(x)\le a_1$, with $a_0, a_1$ positive constants and $\widetilde G\in C^2\big([0, \infty)\big)$ a function satisfying:

In fact, since $c_0\le\widetilde G'(t)\le C_0$, condition G1 in Theorem 5.3 holds. We have $\nabla_\eta G(x, \eta) = a(x)\widetilde G'\big(|\eta|^{p(x)}\big)p(x)|\eta|^{p(x)-2}\eta$, so we get condition G2. We obtain condition G3 by reasoning as in (5.1), using that in the present case we have $\widetilde G''(t)\ge 0$ and $\widetilde G'(t)\ge c_0$.

We get condition G4 by using in our computations that $\widetilde G'(t)\le C_0$ and $\widetilde G''(t)t\le C_0$.

Finally, applying again that $\widetilde G''(t)t\le C_0$, we can obtain the estimate

which combined with computations similar as those in (4) in Theorem 5.1 leads to condition G5.

A possible example of function $f$ satisfying the assumptions of Theorem 5.3 is

In fact, it is immediate that $f$ satisfies conditions f1, f3 and f4'.

On the other hand, $f(x, s) = b(x)|s|^{\tau(x)}$ with $b$ and $\tau$ as in Example 5.1 and $f(x, s)$ as in (5.3) are other possible choices.

Let us present another example

Example 5.3. Another possible example of function $G$ satisfying the assumptions of Theorem 5.3 is

with $p(x)$ as in Section 2 and $\widetilde A(x)\in {{\mathbb R}}^{N\times N}$, symmetric, Hölder continuous in $\Omega$ and such that

Here $\lambda_0\le \lambda(x)\le \Lambda (x)\le \Lambda_0$ with $\lambda_0, \Lambda_0$ positive constants and $\Lambda(x)-\lambda(x)\le c_0$, with $c_0$ a suitable positive constant depending only on $N$, $p_{\min}$, $p_{\max}$ and $\lambda_0$.

In fact, conditions G1 and G2 in Theorem 5.3 are easy to verify. The computations leading to G4 and G5 are similar to the computations in Theorem 5.1.

In order to verify G3, we observe that, denoting $a(x)$ the smaller eigenvalue of $\widetilde A(x)$, there holds that

Then we can write

Now, proceeding as in Theorem 5.1, we get

It is not hard to see that

with $C$ depending only on $N$, $p_{\min}$ and $p_{\max}$. Then, combining (5.4) and (5.5) we deduce that $G(x, \eta)$ satisfies condition G3, if we take $||\Lambda(x)-\lambda(x)||_{L^\infty(\Omega)}\le c_0$, with $c_0$ depending only on $\lambda_0$, $N$, $p_{\min}$ and $p_{\max}$.

For choices of suitable functions $f(x, s)$ for this $G(x, \eta)$ we refer to Example 5.2.

Remark 5.1. We can present further examples of functions satisfying our assumptions. Let $p$ and $\lambda$ be as in Section 2. Let $F_1$ and $F_2$ satisfy the assumptions on Theorem 4.5, with $B_i = \partial_s F_i$ satisfying $B_i(x, 0, 0)\equiv 0$ for $x\in\Omega$, $i = 1, 2$. Then Theorem 4.5 also applies to the function

for any choice of Hölder continuous functions $a_1(x), a_2(x)$, which are bounded from above and below by positive constants.

The same result holds if $F_1$ and $F_2$ satisfy the assumptions on Theorem 4.5, with $B_i = \partial_s F_i$ satisfying (3.32) for $x\in \Omega$, $i = 1, 2$.

Similar consideration applies to functions $F_1$ and $F_2$ under the assumptions of Theorem 5.2.

Acknowledgments

Supported by the Argentine Council of Research CONICET under the project PIP 11220150100032CO 2016-2019, UBACYT 20020150100154BA and ANPCyT PICT 2016-1022.

Conflict of interest

The authors declare no conflict of interest.

A.   Appendix

In Section 1 we included some preliminaries on Lebesgue and Sobolev spaces with variable exponent. For the sake of completeness we collect here some additional results on these spaces.

Proposition A.1. There holds

Some important results for these spaces are

Theorem A.1. Let $p'(x)$ such that

Then $L^{p'(\cdot)}(\Omega)$ is the dual of $L^{p(\cdot)}(\Omega)$. Moreover, if $p_{\min} > 1$, $L^{p(\cdot)}(\Omega)$ and $W^{1, p(\cdot)}(\Omega)$ are reflexive.

Theorem A.2. Let $q(x)\leq p(x)$. If $\Omega$ has finite measure, then $L^{p(\cdot)}(\Omega)\hookrightarrow L^{q(\cdot)}(\Omega)$ continuously.

We also have the following Hölder's inequality

Theorem A.3. Let $p'(x)$ be as in Theorem A.1. Then there holds

for all $f\in L^{p(\cdot)}(\Omega)$ and $g\in L^{p'(\cdot)}(\Omega)$.

The following version of Poincare's inequality holds

Theorem A.4. Let $\Omega$ be bounded. Assume that $p(x)$ is log-Hölder continuous in $\Omega$ (that is, $p$ has a modulus of continuity $\omega(r) = C(\log \frac{1}{r})^{-1}$). For every $u\in W_0^{1, p(\cdot)}(\Omega)$, the inequality

holds with a constant $C$ depending only on N, $\rm{diam}(\Omega)$ and the log-Hölder modulus of continuity of $p(x)$.

The following Sobolev embedding holds. We assume for simplicity that the domain is $C^1$, but the result holds with weaker assumptions on the smoothness of the boundary.

Theorem A.5. Let $\Omega$ be a $C^1$ bounded domain. Assume that $p(x)$ is log-Hölder continuous in $\Omega$ and $1 < p_{\min}\le p(x)\le p_{\max} < \infty$. Let $\tau$ be such that $\tau(x)\ge p(x)$ and $\tau\in C(\overline\Omega)$. Assume moreover that $\tau(x)\le p^*(x) = \frac{Np(x)}{N-p(x)}$ if $p_{\max} < N$, $\tau(x)$ is arbitrary if $p_{\min} > N$, $\tau(x) = p(x)$ if $p_{\min}\le N \le p_{\max}$.

Then, $W^{1, p(\cdot)}(\Omega)\hookrightarrow L^{\tau(\cdot)}(\Omega)$ continuously. The embedding constant depends only on $N$, $|\Omega|$, the log-Hölder modulus of continuity of $p(x)$, $p_{\min}$, $p_{\max}$, $||\tau||_{L^{\infty}}$ and the $C^1$ norm of $\partial\Omega$.

For the proof of these results and more about these spaces, see ^[9,13,14,20] and the references therein.