On renormalized solutions to elliptic inclusions with nonstandard growth


We study the elliptic inclusion given in the following divergence form

$$\begin{aligned}&-\mathrm {div}\,A(x,\nabla u) \ni f\quad \mathrm {in}\quad \Omega ,\\&u=0\quad \mathrm {on}\quad \partial \Omega . \end{aligned}$$

As we assume that \(f\in L^1(\Omega )\), the solutions to the above problem are understood in the renormalized sense. We also assume nonstandard, possibly nonpolynomial, heterogeneous and anisotropic growth and coercivity conditions on the maximally monotone multifunction A which necessitates the use of the nonseparable and nonreflexive Musielak–Orlicz spaces. We prove the existence and uniqueness of the renormalized solution as well as, under additional assumptions on the problem data, its boundedness. The key difficulty, the lack of a Carathéodory selection of the maximally monotone multifunction is overcome with the use of the Minty transform.

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Corresponding author

Correspondence to Piotr Kalita.

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Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Work of AD has been supported by the Cracow University of Economics by the Project No.36/EIM/2020/POT. Work of PG has been supported by the National Science Center of the Republic of Poland by the Project No. UMO-2018/31/B/ST1/02289. Work of PK has been supported by the National Science Center of the Republic of Poland by the Project UMO-2016/22/A/ST1/00077. We thank Iwona Chlebicka for discussion and useful remarks. We also thank anonymous reviewers for valuable comments.

Communicated by Y. Giga.


Appendix A: N-functions and Musielak–Orlicz spaces

N-functions. We start from the definition of N-functions.

Definition A.1

The function \(M:\Omega \times {\mathbb {R}}^d \rightarrow [0,\infty )\) is an N-function if

  1. (N1)

    M is Carathéodory, that is, \(M(\cdot ,\xi )\) is measurable for every \(\xi \in {\mathbb {R}}^d\) and \(M(x,\cdot )\) is continuous for almost every \(x\in \Omega \),

  2. (N2)

    \(M(x,\xi ) = M(x,-\xi )\) for every \(\xi \in {\mathbb {R}}^d\) a.e. in \(\Omega \) and \(M(x,\xi ) = 0\) is and only if \(\xi =0\) a.e. in \(\Omega \),

  3. (N3)

    \(M(x,\cdot )\) is convex for almost every \(x\in \Omega \),

  4. (N4)

    M has superlinear growth in \(\xi \) at zero and infinity, that is,

    $$\begin{aligned} \lim _{|\xi |\rightarrow 0}\mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega }\frac{M(x,\xi )}{|\xi |} = 0\qquad \text {and}\qquad \lim _{|\xi |\rightarrow \infty }\mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\frac{M(x,\xi )}{|\xi |} = \infty . \end{aligned}$$
  5. (N5)

    \( \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\inf _{|\xi |=s}M(x,\xi ) > 0 \ \text {for every}\ s\in (0,\infty )\) and \({{\,\mathrm{ess\,sup}\,}}_{x\in \Omega }M(x,\xi ) < \infty \ \text {for every}\ \xi \ne 0. \)

If \(d=1\) we will use small letters, such as m, to denote N-functions, we will call such functions one dimensional N-functions. In such case by assumption (N2) there holds \(m(x,-\xi ) = m(x,\xi )\) for every \(\xi \in {\mathbb {R}}\) so it is enough to define one dimensional N-function for \(\xi \in [0,\infty )\). Indeed we will assume that one dimensional N-functions are defined only on \(\Omega \times [0,\infty )\), and with some abuse of notation we will define one dimensional N-functions sometimes on the whole line and sometimes on the half-line. If an N-function does not depend on x we will call it homogeneous. So, homogeneous N-function leads from \({\mathbb {R}}^d\) to \( [0,\infty )\) and homogeneous one dimensional N-function from \([0,\infty )\) to \([0,\infty )\). If \(M:\Omega \times {\mathbb {R}}^d\rightarrow {\mathbb {R}}\) then its complementary function \({\widetilde{M}}\) is defined by the Fenchel transform in the following way

$$\begin{aligned} {\widetilde{M}}(x,\eta ) = \sup _{\xi \in {\mathbb {R}}^d} \left\{ \xi \cdot \eta - M (x,\xi ) \right\} . \end{aligned}$$

In the following results we discuss the assumptions in the definition of an N-function and establish some of its properties. We remark that the behavior of an N-function close to the origin is not important for the main results of the present paper, as it does not influence the generated Musielak–Orlicz space nor the arguments of the proofs. In the discussion below, however, for the sake of the exposition completeness, we discuss both the behavior close to infinity as well as close to the origin.

Remark A.2

It is easy to verify that the complementary function of a function which satisfies (N1)–(N4) satisfies (N1)–(N3). It does not have to satisfy either the first, or the second assertion of (N4). Indeed, let \(\Omega =(0,1)\) and let

$$\begin{aligned} M(x,\xi ) = {\left\{ \begin{array}{ll} x\frac{|\xi |^2}{2}\ \text {when}\ |\xi |\le 1,\\ \frac{|\xi |^2}{2}-\frac{1}{2}+\frac{x}{2}\ \text {otherwise}. \end{array}\right. }\quad \text {Then}\quad {\widetilde{M}}(x,\eta )={\left\{ \begin{array}{ll}\frac{|\eta |^2}{2x}\ \text {when}\ |\eta |\le x,\\ |\eta |-\frac{x}{2}\ \text {when}\ |\eta |\in (x,1),\\ \frac{|\eta |^2}{2}-\frac{x}{2}+\frac{1}{2}\ \text {when}\ |\eta |>1. \end{array}\right. }. \end{aligned}$$

It is clear that

$$\begin{aligned} \lim _{|\xi |\rightarrow 0}\mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\eta )}{|\eta |} = \frac{1}{2}. \end{aligned}$$


$$\begin{aligned}&\text {if}\ \ M(x,\xi ) = {\left\{ \begin{array}{ll} \frac{|\xi |^2}{2}\quad \text {when}\quad |\xi |\le \sqrt{2},\\ \frac{|\xi |^2}{2x} + 1 - \frac{1}{x}\quad \text {otherwise}, \end{array}\right. }\ \ \text {then}\\&{\widetilde{M}}(x,\eta ) = {\left\{ \begin{array}{ll} \frac{|\eta |^2}{2}\quad \text {when}\quad |\eta |\le \sqrt{2},\\ \sqrt{2}|\eta | - 1\quad \text {when}\quad \eta \in (\sqrt{2},\sqrt{2}/x),\\ \frac{|\eta |^2x}{2} - 1 + \frac{1}{x}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

It is not hard to verify that

$$\begin{aligned} \lim _{|\eta |\rightarrow \infty }\mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\eta )}{|\eta |} = \sqrt{2}. \end{aligned}$$

Note that the two examples do not satisfy (N5). As we will later show, the complementary function of an N-function is also an N-function.

In the following Lemma A.3 and Remark A.4 we establish that N-functions always have a minorant and majorant being one dimensional homogeneous N-functions.

Lemma A.3

Let \(M:\Omega \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}\) be a function satisfying (N1)–(N4). Then \(M:\Omega \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}\) is an N-function (i.e. it satisfies (N5)) if and only if it is stable, i.e., if there exist homogeneous one dimensional N-functions \(m_1, m_2:[0,\infty )\rightarrow [0,\infty )\) such that for every \(\xi \in {\mathbb {R}}^d\) and almost every \(x\in \Omega \) there holds

$$\begin{aligned} m_1(|\xi |) \le M(x,\xi ) \le m_2(|\xi |). \end{aligned}$$

In particular every N-function is stable.


The fact that stability implies (N5) is straightforward. For the opposite implication define \(m_2:[0,\infty ) \rightarrow [0,\infty )\) by the formula

$$\begin{aligned} m_2(s) = \mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega } \sup _{|\xi |=s} M(x,\xi ). \end{aligned}$$

It is straightforward to check that this function is finite, nonzero for \(\xi \ne 0\), and satisfies (N1)–(N4). To get the lower bound, let us define

$$\begin{aligned} m_{inf}(s) = \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } \inf _{|\xi |=s} M(x,\xi ). \end{aligned}$$

The function \(m_{inf}\) is nondecreasing and hence it has at most countable number of discontinuities and each of these discontinuities is a jump point. Define

$$\begin{aligned} m_{lsc}(s) = {\left\{ \begin{array}{ll} m_{inf}(s)\ \mathrm {if}\ m_{inf}\ \text {is continuous at}\ s,\\ m_{inf}(s^-)\ \mathrm {otherwise}. \end{array}\right. } \end{aligned}$$

This function is in fact the lower semicontinuous envelope of \(m_{inf}\). Now define \(m_1:[0,\infty ) \rightarrow [0,\infty )\) as

$$\begin{aligned} m_1(s) = \widetilde{\widetilde{m_{lsc}}}(s). \end{aligned}$$

that is, the greatest convex minorant of \(m_{lsc}\). The fact that \(m_1\) satisfies (N1)–(N3), (N5), as well as the growth at zero in (N4) is clear. To prove the growth at infinity assume, for contradiction, that there exist constants \(c< \infty \), \(R > 0\) and \(\epsilon >0\) such that for every \(s\ge R\)

$$\begin{aligned} m_1(s) < cs +\epsilon . \end{aligned}$$

So, for a sequence \(s_n\rightarrow \infty \), there exist numbers \(s_n^1,s_n^2\) and \(\lambda _n^1, \lambda _n^2\ge 0\) such that \(\lambda _n^1 + \lambda _n^2 = 1\) and

$$\begin{aligned} (s_n,m_1(s_n)) = \left( \lambda _n^1 s_n^1 + \lambda _n^1 s_n^2, \lambda _n^1 m_{lsc}(s_n^1) + \lambda _n^1 m_{lsc}(s_n^2)\right) . \end{aligned}$$


$$\begin{aligned} s_n^1 \ge s_n, \quad m_{lsc}(s_n^1) = m_1(s_n^1)\quad \text {and}\quad \lambda _n^1 > 0. \end{aligned}$$

Such a choice of \(s_n^1, \lambda _n^1, s_n^2, \lambda _n^2\) is possible due to [38, Theorem 2.1, Remark 2.1]. This means that \(s_n^1\rightarrow \infty \) as \(n\rightarrow \infty \) and \(m_{lsc}(s_n^1) = m_1(s_n^1) < cs_n^1 +\epsilon \), whence

$$\begin{aligned} \frac{m_{lsc}(s_n^1)}{s_n^1} \le c + \frac{\epsilon }{s_n^1}. \end{aligned}$$

Thus we can construct a sequence \(r_n \rightarrow \infty \) such that

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{m_{inf}(r_n)}{r_n}\le c. \end{aligned}$$

This is a contradiction with superlinear growth at infinity of M. \(\square \)

Remark A.4

We are tempted to replace the lower bound in (N5) with its weaker version

$$\begin{aligned} \mathop {{\mathrm{\mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits }}}\limits _{x\in \Omega }M(x,\xi ) > 0\ \text {for every}\ \xi \ne 0. \end{aligned}$$

Unfortunately in such case Lemma A.3 does not hold anymore. To demonstrate this consider \(\Omega =(0,1)\), \(d=2\) and let \(\theta (\xi )\) denote the angular polar coordinate of \(\xi \). Define the function

$$\begin{aligned} P(x,\xi ) = {\left\{ \begin{array}{ll} |\xi |^2\ \ \text {if}\ \ |\xi |\ne 1,\\ 1\ \ \text {if}\ \ |\xi |= 1\ \text {and}\ \theta (\xi )\notin (0,x)\cup (\pi ,\pi +x),\\ x+(1-x)\cos ^2\left( \frac{\theta (\xi )}{x}\pi \right) \ \text {if}\ \ |\xi |= 1\ \text {and} \ \theta (\xi )\in (0,x),\\ x+(1-x)\cos ^2\left( \frac{\theta (\xi )-\pi }{x}\pi \right) \ \text {if}\ \ |\xi |= 1\ \text {and} \ \theta (\xi )\in (\pi ,\pi +x). \end{array}\right. } \end{aligned}$$

Note that P is lower semicontinuous with respect to \(\xi \). Define \(M(x,\xi ) = \widetilde{{\widetilde{P}}}(x,\xi )\), the convex envelope of P. Then if only \(|\xi |=1\) and \(\theta (\xi )\in (0,1)\cup (\pi ,\pi +1)\) we obtain

$$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } M(x,\xi ) \le \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } P(x,\xi ) = \frac{\theta (\xi )}{2}. \end{aligned}$$

This means that

$$\begin{aligned} \inf _{|\xi |=1}\mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } M(x,\xi ) = 0, \end{aligned}$$

and hence we can have only \(m_1(1) = 0\). Lower semicontinuity of P with respect to \(\xi \) and [38, Theorem 2.1, Remark 2.1] imply that \({{\,\mathrm{ess\,inf}\,}}_{x\in \Omega } M(x,\xi ) > 0\) for every nonzero \(\xi \). We will verify that M satisfies all remaining conditions in the definition on an N-function. Indeed, (N1)-(N3), growth at zero in (N4), and condition with \({{\,\mathrm{ess\,sup}\,}}\) in (N5) are clear. We will verify growth at infinity in (N4). Assume for contradiction that there exists a sequence \(\xi _n\) with \(|\xi _n|\rightarrow \infty \) and a constant \(C>0\) such that

$$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\frac{M(x,\xi _n)}{|\xi _n|} \le C. \end{aligned}$$

This means that there exists \(x_n \in (0,1)\) such that

$$\begin{aligned} \frac{M(x_n,\xi _n)}{|\xi _n|} \le C+1. \end{aligned}$$

Now [38, Theorem 2.1] implies that there exist nonnegative numbers \(\lambda ^1_n+\lambda ^2_n+\lambda ^3_n = 1\) and \(\xi ^1_n,\xi ^2_n,\xi ^3_n\in {\mathbb {R}}^2\) such that \(\xi _n = \lambda ^1_n\xi ^1_n + \lambda ^2_n\xi ^2_n+\lambda ^3_n\xi ^3_n\) and

$$\begin{aligned} M(x_n,\xi _n) = \lambda ^1_nP(x_n,\xi ^1_n)+\lambda ^2_nP(x_n,\xi ^2_n)+\lambda ^3_nP(x_n,\xi ^3_n). \end{aligned}$$

We can assume that \(|\xi ^1_n|\ge |\xi _n|\) and \(\lambda ^1_n > 0\). If, for a given n, neither of \(|\xi ^k_n|\) is on a unit circle then

$$\begin{aligned} |\xi _n|^2\le \lambda _n^1|\xi _n^1|^2+ \lambda _n^2|\xi _n^2|^2+ \lambda _n^3|\xi _n^3|^2 = M(x_n,\xi _n) \le P(x_n,\xi _n)= |\xi _n|^2. \end{aligned}$$

It follows that

$$\begin{aligned} \frac{M(x_n,\xi _n)}{|\xi _n|} = |\xi _n|\le C+1. \end{aligned}$$

If exactly one of \(\xi _n^k\) is on the unit circle, say \(\xi ^3_n\), then

$$\begin{aligned} |\xi _n|^2&\le \lambda _n^1|\xi _n^1|^2+ \lambda _n^2|\xi _n^2|^2+ \lambda _n^3|\xi _n^3|^2 = \lambda _n^1|\xi _n^1|^2+ \lambda _n^2|\xi _n^2|^2+ \lambda _n^3\\&\qquad + \lambda _n^3P(x_n,\xi _n^3) - \lambda ^3_nP(x_n,\xi _n^3)\\&\quad = M(x_n,\xi _n) + \lambda _n^3(1-P(x_n,\xi _n^3))\le M(x_n,\xi _n) + 1 \le (C+1)|\xi _n| + 1. \end{aligned}$$

It must be

$$\begin{aligned} |\xi _n| \le (C+1) + \sqrt{2}. \end{aligned}$$

In the final possibility two points \(\xi _n^2\) and \(\xi _n^3\) are on the unit circle. Then

$$\begin{aligned} |\xi _n|^2&\le \lambda _n^1|\xi _n^1|^2+ \lambda _n^2|\xi _n^2|^2+ \lambda _n^3|\xi _n^3|^2\\&= \lambda _n^1|\xi _n^1|^2+ \lambda _n^2+\lambda _n^2P(x_n,\xi _n^2) - \lambda ^2_nP(x_n,\xi _n^2) + \lambda _n^3 + \lambda _n^3P(x_n,\xi _n^3) - \lambda ^3_nP(x_n,\xi _n^3)\\&= M(x_n,\xi _n) + \lambda _n^2(1-P(x_n,\xi _n^2)) + \lambda _n^3(1-P(x_n,\xi _n^3))\le M(x_n,\xi _n) + 2\\&\le (C+1)|\xi _n| + 2. \end{aligned}$$

It follows that \(|\xi _n|\) is bounded, and the proof by contradiction is complete.

Remark A.5

Sometimes the assertion (N4) in the definition of an N-function is replaced with its weaker, nonuniform version

  1. (N4’)

    \( \lim _{|\xi |\rightarrow 0}\frac{M(x,\xi )}{|\xi |} = 0\qquad \text {and}\qquad \lim _{|\xi |\rightarrow \infty }\frac{M(x,\xi )}{|\xi |} = \infty \quad \text {for a.e.}\quad x\in \Omega \).

It is clear that (N4) implies (N4’). We demonstrate that if we replace (N4) with (N4’) in the definition of an N-function, then we lose the stability property. Indeed, consider \(\Omega = (0,1)\), \(d=1\) and

$$\begin{aligned} M(x,\xi ) = {\left\{ \begin{array}{ll} \frac{|\xi |^2}{x+|\xi |}\quad \text {for} \quad |\xi |\le 1,\\ \frac{|\xi |^3}{x+|\xi |}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

This function satisfies (N1)–(N3), as well as (N4’) and (N5), but it is not stable, as

$$\begin{aligned} m_2(s) = {\left\{ \begin{array}{ll} s\quad \text {for} \quad |\xi |\le 1,\\ s^2\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

which is not an N-function as it has linear growth at zero. Moreover, consider \(\Omega = (0,1)\), \(d=1\) and

$$\begin{aligned} M(x,\xi ) = {\left\{ \begin{array}{ll} \left( \frac{1}{2}+x\right) |\xi |^2\quad \text {for} \quad |\xi |\le 1,\\ |\xi |+x|\xi |^2\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

This function satisfies (N1)–(N3), (N4’), and (N5) but it does not satisfy (N4) and it is not stable, as

$$\begin{aligned} m_1(s) = {\left\{ \begin{array}{ll} \frac{1}{2}s^2\quad \text {for} \quad |\xi |\le 1,\\ s\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

and it has linear, and not superlinear growth at infinity.

Remark A.6

We will make use of the function \(M_1:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) defined as

$$\begin{aligned} M_1(\xi ) = \widetilde{\widetilde{\mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } M(x,\xi )}}, \end{aligned}$$

the greatest convex minorant of \({{\,\mathrm{ess\,inf}\,}}_{x\in \Omega } M(x,\xi )\). It is not hard to verify that if M is an N-function, then \(M_1\) is a homogeneous N-function and for every \(\xi \in {\mathbb {R}}^d\) and for almost every \(x\in \Omega \) there holds

$$\begin{aligned} m_1(|\xi |) \le M_1(\xi ) \le M(x,\xi ). \end{aligned}$$

The following result shows the integrability of an N-function

Lemma A.7

Every N-function is integrable, i.e. for every \(\xi \in {\mathbb {R}}^d\) there holds

$$\begin{aligned} \int _{\Omega }M(x,\xi )\, dx < \infty . \end{aligned}$$


The result readily follows from stability of M and the fact that \(\Omega \) is bounded. \(\square \)

We continue by reminding some properties of N-functions.

Lemma A.8

If LM are N-functions, then the following assertions hold

  1. 1.

    For almost every \(x\in \Omega \) and for every \(\xi ,\eta \in {\mathbb {R}}^d\) there holds the following Fenchel–Young inequality

    $$\begin{aligned} |\xi \cdot \eta | \le M(x,\xi ) + {\widetilde{M}}(x,\eta ). \end{aligned}$$
  2. 2.

    If, for some \(x\in \Omega \), there holds \(L(x,\xi )\le M(x,\xi )\) for all \(\xi \in {\mathbb {R}}^d\) then, for this x, \({\widetilde{M}}(x,\xi )\le {\widetilde{L}}(x,\xi )\) for every \(\xi \in {\mathbb {R}}^d\).

  3. 3.

    There holds

    $$\begin{aligned} \lim _{|\xi |\rightarrow \infty } \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\xi )}{|\xi |} = \infty \quad \text {and}\quad \lim _{|\xi |\rightarrow 0}\mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\xi )}{|\xi |}=0. \end{aligned}$$
  4. 4.

    There holds

    $$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega } \inf _{|\xi |=s} {\widetilde{M}}(x,\xi ) \ge \widetilde{m_2}(s) > 0\quad \text {for every nonzero}\quad \xi \in {\mathbb {R}}^d. \end{aligned}$$
  5. 5.

    There holds

    $$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega } {\widetilde{M}}(x,\xi ) \le \widetilde{m_1}(|\xi |) < \infty \quad \text {for every}\quad \xi \in {\mathbb {R}}^d. \end{aligned}$$

It follows that the complementary function of an N-function is also an N-function.


The assertions 1. and 2. of the above lemma are standard properties of the Fenchel conjugate valid for functions which satisfy (N1)–(N3) even without (N4) and (N5). To prove 3. take \(\xi = K \eta / |\eta |\) in the Fenchel–Young inequality which yields

$$\begin{aligned} K |\eta | \le {\widetilde{M}}(x,\eta ) + M(x, K\eta / |\eta |) \le {\widetilde{M}}(x,\eta ) + m_2(K). \end{aligned}$$

Dividing by \(|\eta |\) and taking \({{\,\mathrm{ess\,inf}\,}}_{x\in \Omega }\) we obtain

$$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,inf}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\eta )}{|\eta |} \ge K - \frac{m_2(K)}{|\eta |}, \end{aligned}$$

and the assertion follows by passing with \(|\eta |\) to \(\infty \). To prove the second assertion of 3. observe that

$$\begin{aligned} \mathop {{\mathrm{{{\,\mathrm{ess\,sup}\,}}}}}\limits _{x\in \Omega }\frac{{\widetilde{M}}(x,\xi )}{|\xi |} \le \frac{\widetilde{m_1}(|\xi |)}{|\xi |}. \end{aligned}$$

For any \(s\in (0,\infty )\) we can find \(t=t(s)\in (0,\infty )\) such that \(\widetilde{m_1}(s) + m_1(t) = st\). Such t always exists as the equality in the Fenchel–Young inequality is equivalent to the fact that \(t\in \partial m_1(s)\) (i.e. t belongs to the convex subdifferential of \(m_1\) at s), and for convex functions leading from \({\mathbb {R}}\) to \({\mathbb {R}}\) (which have to be continuous) the subdifferential is always nonempty. Let \(s_n\rightarrow 0^+\). The corresponding sequence \(t(s_n)\) must be bounded. Indeed, if this is not the case, then, for a subsequence

$$\begin{aligned} \frac{m_1(t(s_n))}{t(s_n)} \le \frac{m_1(t(s_n)) + \widetilde{m_1}(s_n)}{t(s_n)} = s_n, \end{aligned}$$

a contradiction. So, for a subsequence, \(t(s_n) \rightarrow t_0\). Passing to the limit with n to \(\infty \) in the expression \(\widetilde{m_1}(s_n) + m_1(t(s_n)) = s_n t(s_n)\) we deduce that \(t_0 = 0\) and the whole sequence converges. Now

$$\begin{aligned} \frac{\widetilde{m_1}(s_n)}{s_n} \le \frac{\widetilde{m_1}(s_n)+m_1(t(s_n))}{s_n} = t(s_n) \rightarrow 0\quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

The assertion 3. is proved. The first inequality in 4. follows from 2. The fact that \(\widetilde{m_2}(s)\ne 0\) for \(s\ne 0\) follows from the fact that \(m_2\) is an N-function. Indeed, assume the contrary, that is \( \widetilde{m_2}(s) = 0 \) for some nonzero \(s\in {\mathbb {R}}\). Taking \(t = s/K\) in the Fenchel–Young inequality we obtain

$$\begin{aligned} m_2(s/K) + \widetilde{m_2}(s) \ge s^2 / K. \end{aligned}$$

It follows that

$$\begin{aligned} \frac{m_2(s/K)}{s/K} \ge s, \end{aligned}$$

and we have the contradiction by passing with K to infinity, as \(m_2\), an N-function, in particular satisfies the assertion (N4) of Definition A.1. Let us prove 5. To this end assume, for contradiction, that there exists \(s\in {\mathbb {R}}\) and the sequence \(\{ t_n \}\subset {\mathbb {R}}\) such that \(t_n s - m_1(t_n) \ge n\). It is clear that \(t_n \rightarrow \infty \). Then we obtain

$$\begin{aligned} s\ge \frac{m_1(t_n)}{t_n} + \frac{n}{t_n} \ge \frac{m_1(t_n)}{t_n}, \end{aligned}$$

and we get the contradiction with superlinear growth of \(m_1\) at infinity by passing with n to infinity. \(\square \)

Sometimes we will use the so called \(\Delta _2\) condition which states that there exists a constant \(c>0\) and a nonnegative function \(h\in L^1(\Omega )\) such that for a.e. \(x\in \Omega \) and every \(\xi \in {\mathbb {R}}^d\)

$$\begin{aligned} M(x,2\xi ) \le c M(x,\xi ) + h(x). \end{aligned}$$

If M satisfies the above condition for \(\xi \in {\mathbb {R}}^d \setminus B(0,R)\) for some \(R>0\) than we write that \(M \in \Delta _2^\infty \) and we say that M satisfies \(\Delta _2\) far from origin.

Orlicz–Musielak spaces. We remind the definition of the Orlicz–Musielak class \({\mathcal {L}}_M(\Omega )\) and two spaces \(L_M(\Omega )\), and \(E_M(\Omega )\).

Definition A.9

Suppose that M is an N-function.

  1. 1.

    \({\mathcal {L}}_M(\Omega )\), the Orlicz–Musielak class, is the set of all measurable functions \(\xi :\Omega \rightarrow {\mathbb {R}}^d\) such that

    $$\begin{aligned} \int _{\Omega } M(x,\xi (x))\, dx < \infty . \end{aligned}$$
  2. 2.

    \(L_M(\Omega )\) is the generalized Orlicz–Musielak space, which is the smallest linear space containing \({\mathcal {L}}_M(\Omega )\), equipped with the Luxemburg norm

    $$\begin{aligned} \Vert \xi \Vert _{L_M} = \inf \left\{ \lambda > 0\, :\ \int _{\Omega } M\left( x,\frac{\xi (x)}{\lambda }\right) \, dx \le 1 \right\} . \end{aligned}$$
  3. 3.

    \(E_M(\Omega )\) is the closure of \(L^\infty (\Omega )^d\) in \(L_M\) norm.

We prove the following result

Theorem A.10

The space \(C^\infty _0(\Omega )^d\) is dense in \(E_M(\Omega )\) in \(L_M\) norm.


Since \(E_M(\Omega )\) is a closure of \(L^\infty (\Omega )^d\) in \(L_M\) norm, it is enough to prove that \(C^\infty _0(\Omega )^d\) is dense in \(L^\infty (\Omega )^d\) in \(L_M(\Omega )\) norm. We first prove that \(C_c(\Omega )^d\) is \(L_M\) dense in \(L^\infty (\Omega )^d\). Take \(v\in L^\infty (\Omega )^d\). By the Luzin theorem there exists a sequence of compact sets \(E_n\) such that \(|\Omega \setminus E_n| < 1/n\) and the functions \(v|_{E_n}:E_n\rightarrow {\mathbb {R}}\) are continuous. These functions, by the Tietze–Urysohn lemma can be extended to functions \(v_n\) in \(C_c(\Omega )^d\) such that \(\Vert v_n\Vert _{L^\infty } \le \Vert v\Vert _{L^\infty }\). Now

$$\begin{aligned}&\int _{\Omega }M\left( x,\frac{v_n(x)-v(x)}{\lambda }\right) \, dx = \int _{\Omega \setminus {E_n}}M\left( x,\frac{v_n(x)-v(x)}{\lambda }\right) \, dx \\&\quad \le \int _{\Omega \setminus {E_n}} m_2 \left( \frac{|v_n(x)|+|v(x)|}{\lambda }\right) \,dx\\&\quad \le m_2 \left( \frac{2\Vert v\Vert _{L^\infty (\Omega )^d}}{\lambda }\right) |\Omega \setminus {E_n}| \rightarrow 0 \quad \text {as}\quad n\rightarrow \infty \quad \text {for every}\ \ \lambda >0. \end{aligned}$$

We now prove that \(C^\infty _0(\Omega )^d\) is \(L_M\) dense in \(C_c(\Omega )^d\). To this end take \(v\in C_c(\Omega )^d\) and extend it to \({\mathbb {R}}^d\) by taking \(v=0\) outside \(\Omega \). Choose a standard mollifier kernel

$$\begin{aligned} \theta _\epsilon (x) = \frac{1}{\epsilon ^n}\theta \left( \frac{x}{\epsilon }\right) \quad \text {where}\quad \theta (x) = {\left\{ \begin{array}{ll} Ce^{\frac{1}{|x|^2}-1}\quad \text {for}\quad |x|\le 1,\\ 0\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

where the constant C is chosen such that \(\int _{{\mathbb {R}}^n}\theta (x)\, dx = 1\). Define \(v^\epsilon (x) = \int _{{\mathbb {R}}^d} \theta _\epsilon (x-y)v(y)\, dy\). Then, if only \(\epsilon \) is small enough, \(v^\epsilon \in C^\infty _0({\mathbb {R}}^d)\) and there holds the pointwise convergence \(\lim _{\epsilon \rightarrow 0}v^\epsilon (x) = v(x)\) for almost every \(x\in \Omega \). Moreover, \(|v^\epsilon (x)| \le \Vert v\Vert _{L^\infty (\Omega )^d}\) for almost every \(x\in \Omega \). There holds

$$\begin{aligned} M\left( x,\frac{v_\epsilon (x)-v(x)}{\lambda }\right) \le m_2 \left( \frac{|v_\epsilon (x)|+|v(x)|}{\lambda }\right) \le m_2 \left( \frac{2\Vert v\Vert _{L^\infty (\Omega )^d}}{\lambda }\right) \quad \text {for every}\ \ \lambda >0. \end{aligned}$$

We can use the Lebesgue dominated convergence theorem to deduce that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\int _{\Omega } M\left( x,\frac{v_\epsilon (x)-v(x)}{\lambda }\right) \, dx = 0\quad \text {for every}\ \ \lambda >0, \end{aligned}$$

whence \(\lim _{\epsilon \rightarrow 0}\Vert v_\epsilon -v\Vert _{L_M} = 0\), and the proof is complete. \(\square \)

It is not hard to verify, that for \(v\in L_M(\Omega )\) there holds

$$\begin{aligned}&\Vert v\Vert _{L_M} \le \int _{\Omega }M(x,v(x))\, dx+1, \end{aligned}$$
$$\begin{aligned}&\Vert v\Vert _{L_M} \le 1 \Rightarrow \int _{\Omega }M(x,v(x))\, dx \le \Vert v\Vert _{L_M}. \end{aligned}$$

It is clear that \(L^\infty (\Omega )^d \subset E_M(\Omega )\) and \({\mathcal {L}}_M(\Omega ) \subset L_M(\Omega )\). We also observe that \(E_M(\Omega )\subset {\mathcal {L}}_M(\Omega )\). Indeed, if \(v_k\) is a sequence in \(L^\infty (\Omega )^d\) such that \(\Vert v_k-v\Vert _{L_M} \rightarrow 0\), then

$$\begin{aligned} \int _{\Omega }M(x,v(x))\, dx \le \frac{1}{2}\left( \int _{\Omega }M(x,2v_k(x))\, dx+\int _{\Omega }M(x,2(v(x)-v_k(x)))\, dx\right) . \end{aligned}$$

Taking k large enough, such that \(2\Vert v-v_k\Vert _{L_M} \le 1\), it follows that

$$\begin{aligned} \int _{\Omega }M(x,v(x))\, dx&\le \frac{1}{2}\int _{\Omega }M(x,2v_k(x))\, dx+\Vert v-v_k\Vert _{L_M} \\&\quad \le \frac{1}{2}|\Omega | m_2(2\Vert v_k\Vert _{L^\infty }))+\Vert v-v_k\Vert _{L_M} < \infty . \end{aligned}$$

Some functional analytic properties of the defined spaces are summarized in the following lemmas [16, 41, 57, 58, 64].

Lemma A.11

If M is an N-function, \(\xi \in L_M(\Omega )\), and \(\eta \in L_{{\widetilde{M}}}(\Omega )\), then the following generalized Hölder inequality holds

$$\begin{aligned} \left| \int _{\Omega } \xi \cdot \eta \, dx\right| \le 2 \Vert \xi \Vert _{L_M(\Omega )}\Vert \eta \Vert _{L_{{\widetilde{M}}}(\Omega )} \end{aligned}$$

Lemma A.12

Let M be an N-function. Then

  • \(E_M(\Omega )\) is separable.

  • \(E_M(\Omega )^* = L_{{\widetilde{M}}}(\Omega )\).

  • \(E_M(\Omega ) = L_M(\Omega )\) if and only if \(M\in \Delta _2^\infty \).

  • \(L_M(\Omega )\) is separable if and only if \(M\in \Delta _2^\infty \).

  • \(L_M(\Omega )\) is reflexive if and only if both \(M\in \Delta _2^\infty \) and \({\widetilde{M}}\in \Delta _2^\infty \).

In this article we assume nowhere that both M and \({\widetilde{M}}\) satisfy the \(\Delta _2\) condition, so that we have to deal with the lack of reflexivity. We also deal with the case where M does not satisfy the \(\Delta _2\) condition, so we cannot use the separability of \(L_M\). Despite this difficulties we are still in position to obtain the existence results using the functional analytic tools developed in [16, 40, 44, 45].

If M is an N-function, we define the space

$$\begin{aligned} V_0^M = \{ v\in W^{1,1}_0(\Omega )\, :\ \nabla v \in L_M(\Omega ) \}. \end{aligned}$$

We will need the following version of the modular Poincaré inequality, cf. [13, Lemma 1], [61, Lemma 3], or more recent works [40, Theorem 2.2], [26, Corollary 4.2].

Theorem A.13

Let \(m:[0,\infty )\rightarrow [0,\infty )\) be an N-function. There exist constants \(\lambda >0\) and \(C>0\) such that for every \(u\in W^{1,1}_0(\Omega )\) satisfying \(\int _{\Omega }m(\lambda |\nabla u|)\, dx < \infty \) there holds

$$\begin{aligned} \int _{\Omega }m(|u|)\, dx \le C \int _{\Omega }m(\lambda |\nabla u|)\, dx. \end{aligned}$$

Remark A.14

In [40] it is proved that the above theorem holds with \(\lambda =1\) provided m satisfies the \(\Delta _2\) condition. A careful analysis of its proof, however, reveals that without the \(\Delta _2\) condition the result holds with a constant \(\lambda \) not necessary equal to one, but dependent only on \(\Omega \) and d.

We remind the definition of modular convergence, cf. [16, 40, 41, 43].

Definition A.15

A sequence \(\{ v_m \}_{m=1}^\infty \) of measurable \({\mathbb {R}}^d\) valued function on \(\Omega \) is said to converge modularly to a function v if there exists \(\lambda >0\) such that

$$\begin{aligned} \lim _{m\rightarrow \infty }\int _{\Omega } M\left( x,\frac{v_m-v}{\lambda }\right) \, \, dx = 0. \end{aligned}$$

We denote the modular convergence by \(v_m \xrightarrow {M} v\). Equivalently, cf. [41, Lemma 2.1], \(\{v_m\}_{m=1}^\infty \) converges modularly to v if \(v_m\rightarrow v\) in measure and the sequence \(\{ M(\cdot , \lambda v_m) \}_{m=1}^\infty \) is uniformly integrable for some \(\lambda >0\).

Lemma A.16

(cf. [41, Lemma 2.2]) Let M be an N-function. If, for constants \(c,\lambda >0\), we have \(\int _{\Omega }M(x,\lambda v_m)\, dx \le c\) for all \(m\in {\mathbb {N}}\) then the sequence \(\{v_m\}_{m=1}^\infty \) is uniformly integrable.


For every \(m\in {\mathbb {N}}\) it holds

$$\begin{aligned} c\ge \int _{\{x\in \Omega \, :\ |v_m(x)|\ge R\}} M(x,\lambda v_m(x))\, dx \ge \lambda \int _{\{x\in \Omega \, :\ |v_m(x)|\ge R\}} \frac{m_1(\lambda |v_m(x)|)}{\lambda |v_m(x)|} |v_m(x)|\, dx. \end{aligned}$$

As \(m_1\) is an N-function, for any \(D>0\) there exists \(R_0(D)>0\) such that for any \(s\ge R_0\) there holds \(\frac{m_1(\lambda s)}{\lambda s} \ge D\). Hence

$$\begin{aligned} \frac{c}{\lambda D} \ge \int _{\{x\in \Omega \, :\ |v_m(x)|\ge R_0(D)\}} |v_m(x)|\, dx, \end{aligned}$$

and the assertion follows easily. \(\square \)

The following approximation theorem which has been proved in [27, Theorem 3.1] is valid in nonreflexive and nonseparable Musielak–Orlicz spaces.

Theorem A.17

Let \(\Omega \) be a Lipschitz domain and let an N-function M satisfy (C2). Then for any \(u\in L^{\infty }(\Omega )\cap V_0^M\) there exists a sequence \(\{u_m\}_{m=1}^\infty \) of functions belonging to \(C_0^\infty (\Omega )\) such that \(u_m\rightarrow u\) strongly in \(L^1(\Omega )\) and \(\nabla u_m \xrightarrow {M} \nabla u\) in \(L_M(\Omega ).\) Moreover, there exists a constant \(c=c(\Omega ) > 0\), such that \(\Vert u_m\Vert _{L^{\infty }(\Omega )}\le c\Vert u\Vert _{L^{\infty }(\Omega )}.\)

We remind an important property of the modular convergence

Lemma A.18

(cf. [41, Proposition 2.2]) Suppose that the sequences \(\{v_k\}_{k=1}^\infty \) and \(\{w_k\}_{k=1}^\infty \) are uniformly bounded in \(L_M(\Omega )\) and \(L_{{\widetilde{M}}}(\Omega )\), respectively. If \(v_k\xrightarrow {M} v\) and \(w_k\xrightarrow {{\widetilde{M}}} w\) then \(v_k\cdot w_k \rightarrow v\cdot w\) in \(L^1(\Omega )\).

Appendix B: Some useful tools of nonlinear analysis

We recall some tools useful in the arguments of this article.

Definition B.1

A sequence \(\{f_m\}_{m=1}^\infty \) of measurable functions \(f_m:\Omega \rightarrow {\mathbb {R}}\) is uniformly integrable if, equivalently, one of the following conditions holds:

  1. (i)
    $$\begin{aligned} \lim _{R\rightarrow \infty }\left( \sup _{m\in {\mathbb {N}}}\int _{\{ x\in \Omega \, :\ |f_m(x)| \ge R \}}|f_m(x)|\, dx\right) = 0. \end{aligned}$$
  2. (ii)
    $$\begin{aligned} \forall \, \epsilon>0\ \ \exists \, \delta >0\quad \sup _{m\in {\mathbb {N}}} \int _{\Omega }\left( |f_m(x)|-\frac{1}{\sqrt{\delta }}\right) _+\, dx \le \epsilon . \end{aligned}$$
  3. (iii)

    There exists a continuous, concave, and nondecreasing function \(\omega :[0,\infty ) \rightarrow [0,\infty )\) such that \(\omega (0)=0\) and for every measurable set \(E\subset \Omega \) and for every \(m\in {\mathbb {N}}\) there holds

    $$\begin{aligned} \int _{E}|f_m(x)|\, dx \le \omega (|E|). \end{aligned}$$
  4. (iv)

    There exists a function \(\Phi :[0,\infty )\rightarrow [0,\infty )\) which is convex, \(\Phi (0) = 0\), \(\lim _{s\rightarrow \infty }\frac{\Phi (s)}{s} = \infty \), and

    $$\begin{aligned} \sup _{n\in {\mathbb {N}}} \int _{\Omega }\Phi (|f_n(x)|)\, dx < \infty . \end{aligned}$$
  5. (v)

    The set \(\{ f_n\}_{n=1}^\infty \) is relatively compact (or, equivalently, relatively sequentially compact) in the weak topology of \(L^1(\Omega )\).

Remark B.2

The fact that condition (iv) of the above definition is equivalent to the other three is known as the de la Vallée Poussin theorem. The equivalence of assertion (v) to the remaining ones is known as the Dunford–Pettis theorem.

Young measures are now a standard tool of nonlinear analysis, we refer for example to [55] for a comprehensive exposition of their theory. We will need the following version of the generalized fundamental theorem on Young measures from [47], where by \({\mathcal {M}}({\mathbb {R}}^N)\) we denote the space of bounded Radon measures.

Proposition B.3

(cf. [47, Theorem 4.1]) Let \(\Omega \) be an open and bounded subset of \({\mathbb {R}}^d.\) Assume that the sequence \(\{ \nu ^j \}_{j=1}^\infty \subset L_{w}^{\infty }(\Omega , {\mathcal {M}}({\mathbb {R}}^N))\) of weakly-* measurable mappings is such that \(\nu ^j(x)=\nu ^j_x\) is a probability measure for almost every \(x\in \Omega \). If the sequence \(\nu ^j\) satisfies the tightness condition:

$$\begin{aligned} \lim _{M\rightarrow \infty } \sup _j |\{x\in \Omega \,\ \mathrm {supp}(\nu ^j_x)\setminus B(0,M)\ne \emptyset \}| \rightarrow 0, \end{aligned}$$

then the following assertions are true.

  1. (1)

    There exists a weakly-* measurable mapping \(\nu \in L^\infty _w(\Omega ,{\mathcal {M}}({\mathbb {R}}^N))\) such that, for a subsequence still denoted by j, there holds

    $$\begin{aligned} \nu ^j\rightarrow \nu \quad \mathrm {weakly}-*\ \mathrm {in} \quad L_w^{\infty }(\Omega , {\mathcal {M}}({\mathbb {R}}^N)), \end{aligned}$$
  2. (2)

    \(\Vert \nu _x\Vert _{{\mathcal {M}}({\mathbb {R}}^N)}=1\) a.e in \(\Omega .\) Moreover, for every \(f\in L^{\infty }(\Omega ,C_b({\mathbb {R}}^N))\) there holds

    $$\begin{aligned} \int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x^j(\lambda )\rightarrow \int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x(\lambda ) ~~\mathrm {weakly}-*\ \mathrm {in}~~L^\infty (\Omega ). \end{aligned}$$
  3. (3)

    For every measurable subset \(A\subset \Omega \) and for every Carathéodory function f (measurable in the first, and continuous in the second variable) such that

    $$\begin{aligned} \lim _{R\rightarrow 0}\sup _{j\in {\mathbb {N}}}\int _A\int _{\{\lambda \in {\mathbb {R}}^N\, :\ |f(x,\lambda )| > R\}} |f(x,\lambda )|d\nu ^j_x(\lambda )dx=0, \end{aligned}$$

    there holds

    $$\begin{aligned} \int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x^j(\lambda )\rightarrow \int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x(\lambda ) ~~\mathrm {weakly}\ \ \mathrm {in}~~L^1(A) \end{aligned}$$

The following corollary is the generalization of the result on the lower-semicontinuity of Young measure generated by a sequence of functions, cf. [54, Corollary 3.3], to the case when the measure is generated by a sequence of measures.

Corollary B.4

Let \( \Omega \subset {\mathbb {R}}^d\) be an open and bounded subset of \({\mathbb {R}}^d\). Suppose that

$$\begin{aligned} \{ \nu ^j \}_{j=1}^\infty \subset L_{w}^{\infty }(\Omega , {\mathcal {M}}({\mathbb {R}}^N))\quad \mathrm {and}\quad \nu \in L^\infty _w(\Omega , {\mathcal {M}}({\mathbb {R}}^N)) \end{aligned}$$

are weakly-* measurable mappings such that \(\nu ^j_x\) and \(\nu _x\) are probability measures for a.e. \(x\in \Omega \). Moreover assume that

$$\begin{aligned} \nu ^{j} \rightarrow \nu \quad \mathrm {weakly}-*\ \ \mathrm {in}\quad L^\infty _w(\Omega ,{\mathcal {M}}({\mathbb {R}}^N)). \end{aligned}$$

Then for any measurable set \( E\subset \Omega \) and every Carathéodory function such that there exists \(m\in L^1(\Omega ), \) with \(m\ge 0\) and \(f(x,\lambda )> - m(x)\) for almost every \(x\in \Omega \) and every \(\lambda \in \bigcup _{j=1}^\infty \mathrm {supp}\, \nu _x^j\) , there holds

$$\begin{aligned} \int _{E}\int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x(\lambda )\le \liminf _{j\rightarrow \infty } \int _{E}\int _{{\mathbb {R}}^N}f(x,\lambda )d\nu _x^j(\lambda ) \end{aligned}$$


The proof follows the lines of the proof of Corollary 3.3 in [54]. First assume that there exist \(R>0\) such that \(f(x,\lambda )=0\) for \( |\lambda |\ge R\). By the Scorza–Dragoni theorem there exists an increasing sequence of compact sets \(E_k\) such that \(|E \setminus E_k| \rightarrow 0\) as \(k\rightarrow \infty \) and \(f|_{E_k\times \mathbb {R^N}}\) is continuous. Define \(F_k: E\rightarrow C_0({\mathbb {R}}^N)\) as \(F_k(x)= \chi _{E_k}(x)f(x,\cdot ).\) We observe that \(F_k \in L^1(E,C_0({\mathbb {R}}^N))\). Indeed,

$$\begin{aligned}&\int _{E}\Vert F_k(x)\Vert _{C_0({\mathbb {R}}^N)}dx=\int _{E}\sup _{\lambda \in {\mathbb {R}}^N}|F_k(x,\lambda )|dx=\int _{E_k}\sup _{\lambda \in {\mathbb {R}}^N}|f(x,\lambda )|dx \\&\qquad \le |\Omega | \sup _{(x,\lambda )\in E_k\times {\mathbb {R}}^N}|f(x,\lambda )|= |\Omega | \sup _{(x,\lambda )\in E_k\times \overline{B(-R,R)}}|f(x,\lambda )| < \infty . \end{aligned}$$

Now, as \((L^1(E,C_0({\mathbb {R}}^N)))' = L^\infty _w(\Omega ,{\mathcal {M}}({\mathbb {R}}^N))\), there holds

$$\begin{aligned}&\lim _{j\rightarrow \infty }\int _{E}\int _{{\mathbb {R}}^N}F_k(x,\lambda )d\nu _x^{j}(\lambda )dx= \int _{E}\int _{\mathbb {R^N}}F_k(x,\lambda )d\nu _x(\lambda )dx, \\&\lim _{j\rightarrow \infty } \int _{E_k}\int _{{\mathbb {R}}^N}(f(x,\lambda )+m(x)-m(x))d\nu _x^{j}(\lambda )dx\\&\quad =\int _{E_k}\int _{\mathbb {R^N}}(f(x,\lambda )+m(x)-m(x))d\nu _x(\lambda )dx. \end{aligned}$$

As \(\nu ^j_x\) and \(\nu \) are probability measures, it follows that

$$\begin{aligned} \lim _{j\rightarrow \infty } \int _{E_k}\int _{{\mathbb {R}}^N}(f(x,\lambda )+m(x))d\nu _x^{j}(\lambda )dx=\int _{E_k}\int _{\mathbb {R^N}}(f(x,\lambda )+m(x))d\nu _x(\lambda )dx. \end{aligned}$$

It follows that

$$\begin{aligned} \int _{E}\int _{\mathbb {R^N}}\chi _{E_k}(x)(f(x,\lambda )+m(x))d\nu _x(\lambda )dx \le \liminf _{j\rightarrow \infty } \int _{E}\int _{{\mathbb {R}}^N}(f(x,\lambda )+m(x))d\nu _x^{j}(\lambda )dx. \end{aligned}$$

Letting \(k\rightarrow \infty \) we obtain the assertion by the monotone convergence theorem. To remove the assumption that \(f(x,\lambda ) = 0\) \(|\lambda |\ge R\) consider an increasing sequence of nonnegative functions \(\eta _l\subset C_0^{\infty }({\mathbb {R}}^N)\) that converges pointwise to 1. We use the above result for \(f(x,\lambda )\eta _l(\lambda )\)

$$\begin{aligned}&\int _{E}\int _{\mathbb {R^N}}f(x,\lambda )\eta _l(\lambda ) d\nu _x(\lambda )dx\le \liminf _{j\rightarrow \infty }\int _{E}\int _{\mathbb {R^N}}f(x,\lambda )\eta _l(\lambda )d\nu _x^{j}(\lambda )dx\\&\qquad \le \liminf _{j\rightarrow \infty }\left( \int _{E}\int _{\mathbb {R^N}}f(x,\lambda )+m(x) d \nu _x^{\epsilon }(\lambda )dx - \int _{E}\int _{\mathbb {R^N}}m(x)\eta ^l(\lambda ) d\nu _x^{j}(\lambda )dx\right) . \end{aligned}$$

But \(m(x)\eta ^l(\lambda ) \in L^1(E,C_0({\mathbb {R}}^N))\) and hence

$$\begin{aligned} \int _{E}\int _{\mathbb {R^N}}(f(x,\lambda )+m(x))\eta _l(\lambda ) d\nu _x(\lambda )dx \le \liminf _{j\rightarrow \infty }\int _{E}\int _{\mathbb {R^N}}f(x,\lambda )+m(x) d \nu _x^{j}(\lambda )dx. \end{aligned}$$

We can pass to the limit \(l\rightarrow \infty \) in the left-hand side by the monotone convergence theorem

$$\begin{aligned} \int _{E}\int _{\mathbb {R^N}}f(x,\lambda )+m(x) d\nu _x(\lambda )dx \le \liminf _{j\rightarrow \infty }\int _{E}\int _{\mathbb {R^N}}f(x,\lambda )+m(x) d \nu _x^{j}(\lambda )dx, \end{aligned}$$

and the assertion follows. \(\square \)

We recall the definition of the biting convergence and the statement of the Chacon biting Lemma.

Definition B.5

Let \(\Omega \subset {\mathbb {R}}^d\) be a measurable set. We say that a sequence \(\{ f_j \}_{j=1}^\infty \subset L^1(\Omega )\) converges to a function \(f\in L^1(\Omega )\) in a biting sense (and we write \(f_j {\mathop {\rightarrow }\limits ^{b}} f\)) if there exists a sequence of measurable sets \(E_l \subset \Omega \) with \(|\Omega \setminus E_l| \rightarrow 0\) as \(l\rightarrow \infty \) and \(E_1\subseteq E_2 \subseteq \cdots \subseteq E_l \subseteq \cdots \subseteq \Omega \) such that

$$\begin{aligned} f_j \rightarrow f\quad \mathrm {weakly}\ \mathrm {in}\ L^1(E_l)\quad \mathrm {for}\ \mathrm {every} \ l\in {\mathbb {N}}. \end{aligned}$$

The proof of the following proposition (known as the Chacon biting lemma) can be found for example in [7].

Proposition B.6

Let \(\Omega \subset {\mathbb {R}}^d\) be a measurable set and let the sequence \(\{ f_j \}_{j=1}^\infty \subset L^1(\Omega )\) be bounded in \(L^1(\Omega )\). There exists a subsequence of indices, still denoted by j, and a function \(f\in L^1(\Omega )\) such that \(f_j {\mathop {\rightarrow }\limits ^{b}} f\).

We will need the following result which states when the sequence which converges in the biting sense is convergent weakly in \(L^1(\Omega )\), cf. [46, Lemma 4.6].

Proposition B.7

Let the sequence \(\{ a_j \}_{j=1}^\infty \subset L^1(\Omega )\) and let \(0\le a_0\in L^1(\Omega )\) be such that \(a_j(x) \ge -a_0(x)\) for almost every \(x\in \Omega \). If

$$\begin{aligned} a_n {\mathop {\rightarrow }\limits ^{b}} a \quad \mathrm {and} \quad \limsup _{j\rightarrow \infty }\int _{\Omega }a_j\, dx \le \int _{\Omega } a\, dx, \end{aligned}$$


$$\begin{aligned} a_j \rightarrow a\quad \mathrm {weakly} \ \mathrm {in}\quad L^1(\Omega ). \end{aligned}$$

We also make use of the following well known result

Proposition B.8

Assume that \(\Omega \subset {\mathbb {R}}^d\) is a bounded set. Let the sequence \(f_j\rightarrow f\) weakly in \(L^1(\Omega )\), and let \(g_j, g\in L^\infty (\Omega )\) be such that \(\Vert g_j\Vert _{L^\infty (\Omega )} \le C\) and \(\Vert g\Vert _{L^\infty (\Omega )} \le C\), where the constant C is independent of j and for almost every \(x\in \Omega \) there holds the pointwise convergence \(g_j(x)\rightarrow g(x)\). Then

$$\begin{aligned} \lim _{j\rightarrow \infty } \int _{\Omega }f_j g_j\, dx = \int _{\Omega }f g\, dx \end{aligned}$$

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Denkowska, A., Gwiazda, P. & Kalita, P. On renormalized solutions to elliptic inclusions with nonstandard growth. Calc. Var. 60, 21 (2021). https://doi.org/10.1007/s00526-020-01893-4

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Mathematics Subject Classification

  • 35J60
  • 35D30