High-order numerical method for the fractional Korteweg-de Vries equation using the discontinuous Galerkin method

1.   Introduction

We consider the general second order elliptic equation in divergence form

where $\Omega$ is an open set of $\mathbb{R}^{n}$, $n\geq 2$, the vector field $\left(a^{i}\left(x, u, \xi \right) \right) _{i = 1, \ldots, n}$ and the right hand side $b\left(x, u, \xi \right)$ are Carathéodory applications defined in $\Omega \times \mathbb{R}\times \mathbb{R}^{n}$. We study the elliptic equations (1.1) under some general growth conditions on the gradient variable $\xi = Du$, named $p, q-$ conditions, which we are going to state in the next Section 3.2. Under these assumptions we will obtain the local boundedness of the weak solutions, as stated in Theorem 3.2.

A strong motivation to study the local boundedness of solutions to (1.1) relies on the recent research in ^[53], where the local Lipschitz continuity of the weak solutions of the Eq (1.1) has been obtained under general growth conditions, precisely some $p, q-$growth assumptions, with the explicit dependence of the differential equation on $u$, other than on its gradient $Du$ and on the $x$ variable. In ^[53] the Sobolev class of functions where to start in order to get more regularity of the weak solutions was pointed out, precisely $u\in W_{\mathrm{loc}}^{1, q}\left(\Omega \right) \cap L_{\mathrm{loc}}^{\infty }\left(\Omega \right)$. That is, in particular the local boundedness $u\in L_{\mathrm{loc}}^{\infty }\left(\Omega \right)$ of weak solutions is a starting assumption for more interior regularity; i.e., for obtaining $u\in W_{\mathrm{loc}}^{1, \infty }\left(\Omega \right)$ and more. When we refer to the classical cases this is a well known aspect which appears in the mathematical literature on a-priori regularity: in fact, for instance, under the so-called natural growth conditions, i.e., when $q = p$, then the a-priori boundedness of $u$ often is a natural assumption to obtain the boundedness of its gradient $Du$ too; see for instance the classical reference book by Ladyzhenskaya-Ural'tseva [45,Chapter 4,Section 3] and the $C^{1, \alpha }-$regularity result by Tolksdorf ^[60].

The aim of this paper is to derive the local boundedness of solutions to (1.1); i.e., to deduce the local boundedness of $u$ only from the growth assumptions on the vector field $\left(a^{i}\left(x, u, \xi \right) \right) _{i = 1, \ldots, n}$ and the right hand side $b\left(x, u, \xi \right)$ in (1.1). The precise conditions and the related results are stated in Section 3.

We start with a relevant aspect to remark in our context, which is different from what happens in minimization problems and it is peculiar for equations: although under $p, q-$growth conditions (with $p < q$) the Eq (1.1) is elliptic and coercive in $W_{\mathrm{loc} }^{1, p}\left(\Omega \right)$, it is not possible a-priori to look for weak solutions only in the Sobolev class $W_{\mathrm{loc}}^{1, p}\left(\Omega \right)$, but it is necessary to emphasize that the notion of weak solution is consistent if a-priori we assume $u\in W_{\mathrm{loc} }^{1, q}\left(\Omega \right)$. This is detailed in Section 2.

Going into more detail, in this article we study the local boundedness of weak solutions to the $p-$elliptic equation (1.1) with $q-$growth, $1 < p\leq q < p+1$, as in (3.2), (3.3) and (3.7)–(3.10). Starting from the integrability condition $u\in W_{\mathrm{loc}}^{1, q}\left(\Omega\right)$ on the weak solution, under the bound on the ratio $\tfrac{q}{p}$

we obtain $u\in L^{\infty}_{\mathrm{loc}}\left(\Omega \right)$. The proof is based on the powerful De Giorgi technique ^[29], by showing first a Caccioppoli-type inequality and then applying an iteration procedure. The result is obtained via a Sobolev embedding theorem on spheres, a procedure introduced by Bella and Schäffner in ^[3], that allows a dimensional gain in the gap between $p$ and $q$. This idea has been later used by the same authors in ^[4], by Schäffner ^[58] and, particularly close to the topic of our paper, by Hirsch and Schäffner ^[43] and De Rosa and Grimaldi ^[30], where the local boundedness of scalar minimizers of a class of convex energy integrals with $p, q-$growth was obtained with the bound $\frac{q}{p} < 1+\frac{q}{n-1}$.

Some references about the local boundedness of solutions to elliptic equations and systems, with general and $p, q-$growth conditions, start by Kolodīĭ ^[44] in 1970 in the specific case of some anisotropic elliptic equations. The local boundedness of solution to classes of anisotropic elliptic equations or systems have been investigated by the authors ^{[18,19,20,21,22,23,24]} and by Di Benedetto, Gianazza and Vespri ^[31]. Other results on the boundedness of solutions of PDEs or of minimizers of integral functionals can be found in Boccardo, Marcellini and Sbordone ^[7], Fusco and Sbordone ^[37,38], Stroffolini ^[59], Cianchi ^[14], Pucci and Servadei ^[57], Cupini, Leonetti and Mascolo ^[17], Carozza, Gao, Giova and Leonetti ^[12], Granucci and Randolfi ^[42], Biagi, Cupini and Mascolo ^[5].

Interior $L^{\infty }-$gradient bound, i.e., the local Lipschitz continuity, of weak solutions to nonlinear elliptic equations and systems under non standard growth conditions have been obtained since 1989 in ^{[46,47,48,49,50]}. See also the following recent references for other Lipschitz regularity results: Colombo and Mingione ^[16], Baroni, Colombo and Mingione ^[1], Eleuteri, Marcellini and Mascolo ^[34,35], Di Marco and Marcellini ^[32], Beck and Mingione ^[2], Bousquet and Brasco ^[9], De Filippis and Mingione ^[26,27], Caselli, Eleuteri and Passarelli di Napoli ^[13], Gentile ^[39], the authors and Passarelli di Napoli ^[25], Eleuteri, Marcellini, Mascolo and Perrotta ^[36]; see also ^[53]. For other related results see also Byun and Oh ^[10] and Mingione and Palatucci ^[55]. The local boundedness of the solution $u$ can be used to achieve further regularity properties, as the Hölder continuity of $u$ or of its gradient $Du$; we limit here to cite Bildhauer and Fuchs ^[6], Düzgun, Marcellini and Vespri ^[33], Di Benedetto, Gianazza and Vespri ^[31], Byun and Oh ^[11] as examples of this approach. For recent boundary regularity results in the context considered in this manuscript we mention Cianchi and Maz'ya ^[15], Bögelein, Duzaar, Marcellini and Scheven ^[8], De Filippis and Piccinini ^[28]. A well known reference about the regularity theory is the article ^[54] by Giuseppe Mingione. We also refer to ^[51,52,53] and to De Filippis and Mingione ^[27], Mingione and Rădulescu ^[56], who have outlined the recent trends and advances in the regularity theory for variational problems with non-standard growths and non-uniform ellipticity.

2.   On the definition of weak solution

In order to investigate the consistency of the notion of weak solution, we anticipate the ellipticity and growth conditions of Section 3, in particular the growth in (3.3), (3.4),

As well known the integral form of the equation, for a smooth test function $\varphi$ with compact support in $\Omega$, is

Let us discuss the summability conditions for the pairings above to be well defined. Since each $a^{i}$ in the gradient variable $\xi$ grows at most as $\left\vert \xi \right\vert ^{q-1}$, more generally we can consider test functions $\varphi \in W_{0}^{1, q}\left(\Omega \right)$. In fact, starting with the first addendum and applying the Young inequality with conjugate exponents $\frac{q}{q-1}$ and $q$, we obtain the $L^{1}$ local summability

if $u\in W_{\mathrm{loc}}^{1, q}\left(\Omega \right)$ and if $\frac{q}{q-1} \gamma _{1}\leq q^{\ast }$, where $q^{\ast }$ is the Sobolev conjugate exponent of $q$, and $b_{1}\in L_{\mathrm{loc}}^{\frac{q}{q-1}}\left(\Omega \right)$. On $\gamma _{1}$ equivalently we require (if $q < n$) $\gamma _{1}\leq q^{\ast }\frac{q-1}{q} = \frac{nq}{n-q}\frac{q-1}{q} = \frac{n\left(q-1\right) }{n-q}$, which essentially corresponds to our assumption (3.8) below (the difference being the strict sign "$<$" for compactness reasons). We also observe that the summability condition $b_{1}\in L_{\mathrm{loc}}^{\frac{q}{q-1}}\left(\Omega \right)$ is satisfied if $b_{1}\in L_{\mathrm{loc}}^{s_{1}}\left(\Omega \right)$, with $s_{1} > \frac{n}{q-1}$, as in (3.10).

Similar computations apply to $\left\vert b\left(x, u, \xi \right) \varphi \right\vert$, again if $q < n$ and with conjugate exponents $\frac{q^{\ast }}{ q^{\ast }-1}$ and $q^{\ast }$,

and we obtain $b_{2}\in L_{\mathrm{loc}}^{\frac{q^{\ast }}{q^{\ast }-1} }\left(\Omega \right)$ (compare with (3.10), where $b_{2}\in L_{\mathrm{loc}}^{s_{2}}\left(\Omega \right)$ with $s_{2} > \frac{n}{p}$, since $\frac{q^{\ast }}{q^{\ast }-1}\le \frac{p^{\ast }}{p^{\ast }-1}\le \frac{p^{\ast }}{p^{\ast }-p} = \frac{n}{p}$) and the conditions for $r$ and $\gamma _{2}$ expressed by $r\frac{q^{\ast }}{q^{\ast }-1}\leq q$ and $\gamma _{2}\frac{q^{\ast }}{q^{\ast }-1}\leq q^{\ast }$; i.e., for the first one,

which correspond to the more strict assumption (3.9), with $r < p+ \frac{p}{n}-1$, with the sign "$<$" and where $q$ is replaced by $p$. Finally for $\gamma _{2}$ we obtain $\gamma _{2}\leq q^{\ast }-1$, which again corresponds to our assumption (3.8) with the strict sign.

Therefore our assumptions for Theorem 3.2 are more strict than that ones considered in this section and they are consistent with a correct definition of weak solution to the elliptic equation (1.1).

3.   Statement of the main result

Let $a^i:\Omega \times \mathbb{R}\times\mathbb{R}^{n}\rightarrow \mathbb{R}$, $i = 1, ..., n$, and $b:\Omega \times \mathbb{R} \times\mathbb{R}^{ n} \rightarrow \mathbb{R}$ be Carathéodory functions, $\Omega$ be an open set in $\mathbb{R}^n$, $n\geq 2$. Consider the nonlinear partial differential equation

For the sake of simplicity we use the following notation: $a(x, u, \xi) = (a^{i}(x, u, \xi))_{i = 1, ..., n}$, for all $i = 1, \ldots, n$.

We assume the following properties:

● $p-$ellipticity condition at infinity:

there exist an exponent $p > 1$ and a positive constant $\lambda$ such that

for a.e. $x\in \Omega$, for every $u\in \mathbb{R}$ and for all $\xi \in \mathbb{R}^{n}$ such that $\left\vert \xi \right\vert \geq 1$.

● $q-$growth condition:

there exist exponents $q\geq p$, $\gamma _{1}\geq 0$, $s_{1} > 1$, a positive constant $\Lambda$ and a positive function $b_{1}\in L_{\text{loc} }^{s_{1}}(\Omega)$ such that, for a.e. $x\in \Omega$, for every $u\in \mathbb{R}$ and for all $\xi \in \mathbb{R}^{n}$,

● growth conditions for the right hand side $b\left(x, u, \xi \right)$:

there exist further exponents $r\geq 0$, $\gamma _{2}\geq 0$, $s_{2} > 1$ and a positive function $b_{2}\in L_{\text{loc}}^{s_{2}}(\Omega)$ such that

for a.e. $x\in \Omega$, for every $u\in \mathbb{R}$ and for all $\xi \in \mathbb{R}^{n}$.

Without loss of generality we can assume $\Lambda \geq 1$ and $b_{1}, b_{2}\geq 1$ a.e. in $\Omega$. We recall the definition of weak solution to (3.1).

Definition 3.1. A function $u\in W_{\mathrm{loc}}^{1, q}(\Omega)$ is a weak solution to (3.1) if

for all $\varphi \in W^{1, q}(\Omega)$, $\mathrm{supp}\, \varphi\Subset \Omega$.

3.1.   Assumptions on the exponents

Our aim is to study the local boundedness of weak solutions to (3.1). Since this regularity property is trivially satisfied for functions in $W_{\mathrm{loc}}^{1, q}(\Omega)$ with $q > n$, from now on we only consider the case $q\leq n$; more precisely

since if $q > n$ then weak solutions are Hölder continuous as an application of the Sobolev-Morrey embedding theorem, see Remark 3.3.

Other assumptions on the exponents are

3.2.   The statement of the boundedness result

Under the conditions described above the following local boundedness result holds.

Theorem 3.2 (Boundedness result). Let $u\in W_{\mathrm{loc}}^{1, q}(\Omega)$, $1 < q\le n$, be a weak solution to the elliptic equation (3.1). If (3.2)–(3.4) and (3.6)–(3.10) hold true, then $u$ is locally bounded. Precisely, for every open set $\Omega ^{\prime }\Subset \Omega$ there exist constants $R_0, c > 0$ depending on the data $n, p, q, r, \gamma _{1}, \gamma _{2}, s_{1}, s_{2}$ and on the norm $\Vert u\Vert _{W^{1, q}(\Omega ^{\prime })}$ such that $\Vert u\Vert _{L^{\infty }(B_{R/2}(x_{0}))}\leq c$ for every $R\le R_0$, with $B_{R_0}(x_0)\subseteq \Omega'$.

Remark 3.3. We already observed that if $q > n$ then the weak solutions to (3.1) are locally Hölder continuous. Let us now discuss why in (3.6) we do not consider the case $p = q = n$. If $p = q$ ($\le n$), the same computations in the proof of Theorem 3.2 work with the set of assumptions (3.8)–(3.10). They can be written, coherently with the previous ones, as

Here $p^*$ denotes the Sobolev exponent appearing in the Sobolev embedding theorem for functions in $W^{1, p}(\Omega)$ with $\Omega$ bounded open set in $\mathbb{R}^n$; i.e.,

Following the computations in [40,Theorem 2.1] and [41,Chapter 6] it can be proved that the weak solutions to (3.1) are quasi-minima of the functional

with $\tau: = \max\{\gamma_1\frac{p}{p-1}, \gamma_2\frac{p^*}{p^*-1}\}$. It is known that if

then the gradient of quasi-minima of the functional (3.15) satisfies a higher integrability property; i.e., they belong to $W^{1, p+\epsilon}$, for some $\epsilon > 0$.

Under our assumptions, (3.16) is satisfied; indeed, taking into account that we are considering $p = q$, by (3.10)

and, by (3.13)

Analogously, by (3.11),

In particular, if $p = q = n$ the quasi-minima of (3.15) are in $W_{\rm{loc}}^{1, n+\epsilon}(\Omega)$ for some $\epsilon > 0$, therefore the weak solutions to (3.1) are Hölder continuous. We refer to ^[41] Chapter 6 for more details.

4.   Notation and remarks

If $p\ge 1$ and $d\in \mathbb{N}$, $d\ge 2$, we define

The Sobolev exponent appearing in the Sobolev embedding theorem for functions in $W^{1, p}(\Omega)$, $p\ge 1$, with $\Omega$ bounded open set in $\mathbb{R}^n$, is $(p_n)^*$ and will be denoted, as usual, $p^*$.

Let $t\in \mathbb{R}$, $t > 0$. We define $t_*$ as follows:

We have, if $n\ge 3$,

and, if $n = 2$, $t_* = 1$ for every $t$.

We notice that, if $n\ge 3$,

and, if $n = 2$, for every $t$, $((t_*)_{n-1})^*$ stands for any real number greater than $1$.

Remark 4.1. Let us consider the exponents $p, q$ satisfying (3.6) and (3.7) in Section 3. We notice that

Due to assumption (3.7), if $n = 2$, then $\left(\frac{p}{p-q+1}\right)_{*} = 1$.

Moreover, if we denote $t: = \left(\frac{p}{p-q+1}\right)_{*}$ then, if $n\ge 3$,

if instead $n = 2$ than $(t_{n-1})^*$ is any real number greater than $1$.

Let $p, q$ satisfy (3.6) and (3.7). It is easy to prove that

In the following it will be useful to introduce the following notation:

or, more explicitly,

Remark 4.2. Assume $1 < p\le q$. Then easy computations give

To get the sharp bound for $q$, we use a result proved in ^[43], see also ^[3,4,30,58]. Here we denote $S_{\sigma}(x_0)$ the boundary of the ball $B_{\sigma}(x_0)$ in $\mathbb{R}^n$.

Lemma 4.3. Let $n\in \mathbb{N}$, $n\ge 2$. Consider $B_{\sigma}(x_0)$ ball in $\mathbb{R}^n$ and $u\in L^1(B_{\sigma}(x_0))$ and $s > 1$. For any $0 < \rho < \sigma < +\infty$, define

Then for every $\delta\in]0, 1]$,

The following result is the Sobolev inequality on spheres.

Lemma 4.4. Let $n\in \mathbb{N}$, $n\ge 3$, and $\gamma\in [1, n-1[$. Then there exists $c$ depending on $n$ and $\gamma$ such that for every $u\in W^{1, p}(S_1(x_0), d\mathcal{H}^{n-1})$

Lemma 4.5. Let $n = 2$. Then there exists $c$ such that for every $u\in W^{1, 1}(S_1(x_0), d\mathcal{H}^{1})$ and every $r > 1$,

Proof. By the one-dimensional Sobolev inequality

Then, for every $r > 1$,

We conclude this section, by stating a classical result; see, e.g., ^[41]. that will be useful to prove Theorem 3.2.

Lemma 4.6. Let $\alpha > 0$ and $(J_h)$ a sequence of real positive numbers, such that

with $A > 0$ and $\lambda > 1$.

If $J_0 \leq A^{-\frac{1}{\alpha}}\lambda^{-\frac{1}{\alpha^2}}$, then $J_h \le \lambda^{-\frac{h}{\alpha}}J_0$ and $\lim_{h\to \infty} J_h = 0$.

5.   Caccioppoli's inequality

Under the assumptions in Section 3 we have the following Caccioppoli-type inequality.

Given a measurable function $u: \Omega\to \mathbb{R}$, with $\Omega$ open set in $\mathbb{R}^n$, and fixed $x_0\in \mathbb{R}^n$, $k\in \mathbb{R}$ and $\tau > 0$, we denote the super-level set of $u$ as follows:

usually dropping the dependence on $x_0$. We denote $|A_{k, \tau}|$ its Lebesgue measure.

Proposition 5.1 (Caccioppoli's inequality). Let $u\in W^{1, q}_{\rm{loc}}(\Omega)$ be a weak solution to (3.1). If (3.6)–(3.10) hold true, then there exists a constant $c > 0$, such that for any $B_{R_0}(x_0) \Subset \Omega$, $0 < \rho < R\le R_0$

with $\nu$ as in (4.4) and $c$ is a constant depending on $n, p, q, r, R_0$, the $L^{s_1}$-norm of $b_1$ and the $L^{s_2}$-norm of $b_2$ in $B_{R_0}$.

Proof. Without loss of generality we assume that the functions $b_1, b_2$ in (3.3) are a.e. greater than or equal to $1$ in $\Omega$. We split the proof into steps.

Step 1. Consider $B_{R_0}(x_0) \Subset \Omega$, $0 < \frac{R_0}{2}\le \rho < R\le R_0\le 1$.

We set

For every $\eta \in \mathcal{A}(\rho, R)$ and fixed $k > 1$ we define the test function $\varphi_k$ as follows

with

that is greater than $1$ because $q > 1$.

Notice that $\varphi_{k}\in W_0^{1, q}(B_{R_0}(x_0))$, $\operatorname{supp\, }\varphi_k\Subset B_R(x_0)$.

Step 2. Let us consider the super-level sets:

In this step we prove that

for some constant $c$ independent of $u$ and $\eta$.

Using $\varphi_k$ as a test function in (3.5) we get

Now, we separately consider and estimate $I_i$, $i = 1, 2, 3$.

ESTIMATE OF $I_3$

Using (3.4) we obtain

We estimate the right-hand side using the Young inequality, with exponents $\frac{p}{r}$ and $\frac{p}{p-r}$, and (3.2). There exists $c$, depending on $\lambda$, $\Lambda$, $n$, $p$, $r$, such that

and, recalling that $b_2\ge 1$,

Therefore,

Collecting (5.5)–(5.7) we get

Using (3.2) and (3.3) we get

ESTIMATE OF $I_2$. For a.e. $x\in A_{k, R}\cap \{\eta\ne 0\}$ we have

For a.e. $x\in \{|Du|\ge 1\} \cap A_{k, R}\cap \{\eta\ne 0\}$, by $q < p+1$ and the Young inequality with exponents $\frac{p}{q-1}$ and $\frac{p}{p-q+1}$, and noting that $\mu-1 = \mu\frac{q-1}{p}$, we get

On the other hand we have

a.e. in $\{|Du| < 1\} \cap A_{k, R}\cap \{\eta\ne 0\}$.

Therefore,

By (5.8) and the inequality above, we get

Taking into account that $b_1\ge 1$

therefore

and we obtain

We have

By Hölder inequality with exponents $\frac{p}{q-1}$ and $\frac{p}{p-q+1}$, we get

Analogously,

and

obtaining

Therefore,

Since $b_1\ge 1$ and $q < p+1$, then

and we get (5.4).

Step 3. In this step we prove that

We obtain this estimate starting by (5.4).

Consider $\tau\in (\rho, R)$ and define the function

where

This function $w$ is in $W^{1, \left(\frac{p}{p-q+1}\right)_{*}}(S_1, d\mathcal{H}^{n-1})$, with

Let us consider the case

By (4.1) in Remark 4.1, we get

By (4.2) and the Sobolev embedding theorem, see Lemma 4.4, we get

When

we distinguish among two cases: $n\ge 3$ and $n = 2$. If $n\ge 3$, by using Hölder's inequality, we get

by (4.2) and the Sobolev embedding theorem, see Lemma 4.4, we obtain the inequality (5.16).

If $n = 2$, then $\left(\frac{p}{p-q+1}\right)_{*} = 1$, then we obtain the inequality (5.16) by applying Lemma 4.5 with $r = \frac{p}{p-q+1}$.

Let $\mathcal{A}(\rho, R)$ be as in (5.2). We apply Lemma 4.3, with

that is a function in $L^{1}(B_R(x_0))$. Using (5.16) and recalling that $\frac{R_0}{2}\le \rho < R\le R_0$, reasoning as in ^[30], we get

By coarea formula, inequality (5.17) implies

and, taking into account (3.7), Remark 4.1 and (4.5)

by Hölder's inequality we get

By (5.4) we get

Since

we get (5.13).

Step 4. In this step we estimate the integrals at the right hand side of (5.13).

Consider

ESTIMATE OF $J_1$.

By assumptions (3.8) and (3.9),

Therefore, by using Hölder inequality with exponent $\frac{p^*(q-1)}{p\gamma_1}$ we get

Hölder inequality with exponent $p^*\frac{p-r}{p}$ implies

Moreover, by using Hölder inequality with exponent $\frac{p^*}{\gamma_2+1}$ we get

by using Hölder inequality with exponent $\frac{p^*}{\gamma_2}$ we get

Therefore, by using the Sobolev embedding theorem

Let us consider now the following integral in (5.13):

Trivially,

By assumption $b_2\in L^{s_2}$, $s_2 > \frac{n}{p} = \frac{p^*}{p^*-p}$. Since $\frac{p^*}{p^*-p} > \frac{p^*}{p^*-1}$, then $\frac{s_2}{s_2-1} < p^*$. Therefore, by Hölder inequality

which implies

Now, $b_1\in L^{s_1}$ with $s_1 > \frac{p}{q-1}$; by using Hölder inequality with exponent $\frac{s_1(q-1)}{p}$ we get

We obtain

Step 5. By Steps 3, 4 we get

and the inequality (5.1) follows.

6.   Proof of the boundedness result

Let $u\in W_{\mathrm{loc}}^{1, q}(\Omega)$, $1 < q\le n$, be weak solution to (3.1). Consider $\Omega' \Subset \Omega$ an open set.

I case $q > p$. Let $B_{R_0}(x_0)\subseteq \Omega'$.

For every $k\ge 0$

In particular, chosen $R_0$ such that

we get

II case $q = p$. By a well known result by Giaquinta and Giusti ^[40], the gradient of the weak solution satisfies a higher integrability property: its gradient is in $L^{p+\varepsilon }(B_{R_{0}}(x_{0}))$, for some $\varepsilon > 0$ sufficiently small. Moreover, $u\in L^{p^{\ast }}(B_{R_{0}}(x_{0}))$; because $p = q$, we can repeat the above argument with $q$ replaced by $p+\varepsilon$ so obtaining (6.1). $R_{0} > 0$ depends on the norm $\Vert u\Vert _{W^{1, p+\varepsilon }(B_{R_{0}}(x_{0}))}$. Again, by the Giaquinta and Giusti result, the norm $\Vert u\Vert _{W^{1, p+\varepsilon }(B_{R_{0}}(x_{0}))}$ can be estimated in terms of the $\Vert u\Vert _{W^{1, p}(\Omega ^{\prime })}$ for $B_{R_{0}}(x_{0})\subseteq\Omega ^{\prime }\Subset \Omega$.

Finally, we can summarize: in both cases, either if $q > p$ or if $q = p$, we can choose $R_{0}$ such that (6.2) holds with $R_{0} > 0$ depending on the norm $\Vert u\Vert _{W^{1, q}(\Omega ^{\prime })}$. We also assume $R_0 < 1$ such that $|B_{R_0}| < 1$, $0 < R \le R_0$.

Define the decreasing sequences

Fixed a positive constant $d\ge 2$, to be chosen later, define the increasing sequence of positive real numbers $(k_h)$

Define the decreasing sequence $(J_h)$,

Notice that

Moreover, by (6.2),

Let us introduce the following notation:

and

where $\nu$ is defined in (4.4).

Proposition 6.1 (Estimate of $J_{h+1}$). Let $u\in W^{1, q}_{\rm{loc}}(\Omega)$ be a weak solution to (3.1). Assume (3.2)–(3.4) with the exponents satisfying the inequalities listed in Section 3.1. Then for every $h\in \mathbb{N}$

where $c$ is a constant depending on $n, p, q, r, R_0$, the $L^{s_1}$-norm of $b_1$ and the $L^{s_2}$-norm of $b_2$ in $B_{R_0}$.

We precede the proof with the following remark.

Remark 6.2. We remark that, by assumptions (3.6)–(3.10), then $\tau, \theta > 0$ and $\sigma > 1$. As far as these inequalities are concerned, we remark that

that is satisfied, because $p\le q$

that is the first assumption in (3.10); this assumption also implies

that is equivalent to

By the second assumption in (3.10),

Proof of Proposition 6.1. By (5.1), used with $k = k_{h+1}$, $\rho = \rho_{h+1}$, $R = \rho_{h}$, we have

Let us write the estimate above as

To estimate the sum at the right-hand side it is useful to remark that, for all $h$,

and

Since

by the Sobolev inequality we get

that, together with (6.9), gives

Moreover,

Inequalities (6.10) and (6.11) imply that

therefore, by (6.9),

This estimate, together with (6.10), implies:

and, analogously,

Moreover, taking into account that

Let us now estimate $H_1$.

Inequalities (6.10) and (6.11) imply

that gives

Taking into account that for every $h$

we conclude that

Collecting (6.13)–(6.21), by (6.8) we get

Let us now add to both sides of (6.22) the integral $\int_{B_{\rho_{h+1}}}|(u-k_{h+1})_+|^p\, dx$.

By Hölder inequality

Since

the Sobolev embedding theorem gives

Taking into account (6.10), we obtain

therefore, the inequality (6.23) implies

Inequalities (6.22) and (6.24) give

where $c$ is a constant depending on $n, p, q, r, R_0$, the $L^{s_1}$-norm of $b_1$ and the $L^{s_2}$-norm of $b_2$ in $B_{R_0}$.

By taking in account the notation in (6.3)–(6.5), we get, by (6.25), the inequality (6.6).

We are now ready to prove our regularity result.

Proof of Theorem 3.2. By Proposition 6.1, for every $h\in \mathbb{N}$,

where $c$ is a constant depending on $n, p, q, R_0$, the $L^{s_1}$-norm of $b_1$ and the $L^{s_2}$-norm of $b_2$ in $B_{R_0}$ and for every $d\ge 2$. Thus, the following inequality holds:

with

where $\theta$, $\tau$ and $\sigma$ are defined in (6.4), (6.3), (6.5). We recall that $\theta, \tau > 0,$ $\sigma-1 > 0$, see Remark 6.2.

To apply Lemma 4.6, we need

Since

if we choose $d\ge 2$ satisfying

we get $0 = \lim_{h\to +\infty}J_h = \|\big(u-d)_+\|^p_{W^{1, p}(B_{\frac{R}{2}})}$ and we conclude that

To prove that $u$ is locally bounded from below, we proceed as follows. The function $-u$ is a weak solution to

where

Notice that, by (3.2)–(3.4) the following properties hold:

● $p-$ellipticity condition at infinity:

for a.e. $x\in \Omega$ and for every $u\in \mathbb{R}$,

● $q-$growth condition:

for a.e. $x\in \Omega$ and every $u\in \mathbb{R}$ and $\xi\in \mathbb{R}^{n}$

● growth condition for the right hand side $b\left(x, u, \xi \right)$:

To prove the analogue of Proposition 5.1 we now consider the test function $\varphi_{k}(x): = (k-u(x))_+[\eta (x)]^{\mu}$ where $\eta$ is a cut-off function. Let us consider the sub-level sets:

Then we obtain, in place of (5.5),

The proof goes on with no significant changes with respect the previous case, arriving to the conclusion that there exists $d'$ such that we obtain that $B_{\frac{R}{2}}\subseteq \{u\ge d'\}$, and

Collecting the estimates from below and from above for $u$, we conclude.

Acknowledgments

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Conflict of interest

The authors declare no conflict of interest.