# Doctoral Paper Functionals

Christoph Tietz

###### Abstract

Abstract. We present an order-theoretical fixed point theorem for increasing
multivalued operators suitable for the method of sub-supersolutions and its
application to the following multivalued quasi-variational inclusion: Let
$\Omega\subset\mathbb{R}^{N}$ be a bounded Lipschitz domain and
$W=W_{0}^{1,p}(\Omega)$. Find $u\in W$ such that for some measurable selection
$\eta$ of $f(\cdot,u,u)$ it holds

$\langle Eu,w-u\rangle+\int_{\Omega}\eta(w-u)+K(w,u)-K(u,u)\geq
0\quad\text{for all }w\in W,$

where $E\colon W\to W^{\ast}$ is an elliptic Leray-Lions operator of
divergence form,
$f\colon\Omega\times\mathbb{R}\times\mathbb{R}\to\mathcal{P}(\mathbb{R})$ is a
multivalued bifunction being upper semicontinuous in the second and decreasing
in the third argument, and $K(\cdot,u)$ is a convex functional for each $u\in
W$. Under weak assumptions on the data we will prove that there are smallest
and greatest solutions between each pair of appropriately defined sub-
supersolutions.

main [][][][][][0 ] subtext0

## An Order-Theoretical Multi-Valued Fixed Point Approach to Quasi-Variational
Inclusions with Bifunctions

Christoph Tietz, Technische Universität Dresden

##  1 Introduction

In order to solve variational inequalities and inclusions in various
analytical settings lacking coercivity, the method of sub-supersolutions has
been applied successfully, see, e.g., [2, 3, 5, 7, 8] and the recent monograph
[4]. This method relies on a combination of sophisticated analytical existence
results and truncation-techniques with respect to appropriately defined sub-
supersolutions. In [13, 14] we proposed a framework for this method in order
to make the interplay of analysis and order-theory as lucid as possible. This
opens the possibility to replace functionals appearing in variational
inequalities and inclusions by bifunctionals, which considerably extends the
class of solvable problems while maintaining qualitative properties of the
solution set. It is the aim of this paper to refurnish this framework, give an
easier proof, and present a sophisticated application.

To be a little bit more concrete, let $W$ be a partially ordered space and,
for each $u\in W$, $A_{u}\colon W\to\mathbb{R}\cup\\{\infty\\}$ a functional.
Then the variational problem is to find $u\in W$ such that

$A_{u}(w)-A_{u}(u)\geq 0\quad\text{for all }w\in W.$ (1)

In order to ensure the existence of a solution $u\in W$ of (1), one clearly
needs more information. In particular, in order to apply analytical methods,
one presumes that the mapping $u\mapsto A_{u}$ is continuous in some way.
Then, one incorporates a subsolution $\underline{\smash{u}}\in W$, which is a
solution of the appropriately defined variational inequality

$\underline{\smash{A}}_{\underline{\smash{u}}}(w)-\underline{\smash{A}}_{\underline{\smash{u}}}(\underline{\smash{u}})\geq
0\quad\text{for all }w\in\underline{\smash{T}}_{\underline{\smash{u}}}\subset
W,$

and a supersolution $\overline{u}$, which is similarly defined. At this point,
order-theory comes into play. For instance, it is preferable that the supremum
of increasing sequences of subsolutions is again a subsolution, and that there
is a solution above each subsolution. To obtain those results, sophisticated
constructions and proofs are needed. However, in doing so, one realizes that
it is often possible to extend the scope of the developed theory by allowing
that the functional $A_{u}$ depends on $u$ not only continuously, but order-
continuously. This leads to bifunctionals $A_{u,v}$, where $u\mapsto A_{u,v}$
preserves convergence in some sense, and $v\mapsto A_{u,v}$ preserves order in
some sense. Thus, if one can solve variational inequalities of the form (1) by
means of sub-supersolutions, there is good hope that one can also solve the
variational inequality

$A_{u,u}(w)-A_{u,u}(u)\geq 0\quad\text{for all }w\in W.$

This procedure extends in a natural way to multivalued quasi-variational
inclusions as we will see in this paper. In the second section, we will
present the related fixed point theory, where we incorporate ideas from [9] to
obtain a more general form of the framework presented in [13] and a simpler
proof that relies on a multivalued Tarski fixed point theorem and not anymore
on an abstract chain generating principle. The third section contains some
abstract order-theoretical results about quasi-variational inclusions with
bifunctions of the general form

$\text{find }u\in K(u,u)\quad\text{s.t.}\quad a(w)-a(u)\geq 0\quad\text{for
some $a\in A(u,u)$ and all }w\in T(u,u),$

where the elements of $A(u,u)$ are functionals $a\colon
W\to\mathbb{R}\cup\\{\infty\\}$ and $K(u,u),T(u,u)\subset W$. To conclude the
paper, we will apply the developed theory to give a concise proof without the
need for technical approximation considerations for the existence of solutions
of the following multivalued quasi-variational inclusion problem considered in
[7]: Find $u\in W_{0}^{1,p}(\Omega)$ such that there is a measurable selection
$\eta$ of $f(\cdot,u,u)$ such that

$\langle Eu,w-u\rangle+\int_{\Omega}\eta(w-u)+K(w,u)-K(u,u)\geq
0\quad\text{for all }w\in W^{1,p}(\Omega).$

There, $W=W_{0}^{1,p}(\Omega)$ is a Sobolev space over a bounded Lipschitz
domain $\Omega\subset\mathbb{R}^{N}$, $E\colon W\to W^{\ast}$ is an elliptic
differential operator of Leray-Lions type,
$f\colon\Omega\times\mathbb{R}\times\mathbb{R}\to\mathcal{P}(\mathbb{R})$ is a
multivalued bifunction, and $K(\cdot,u)\colon W\to\mathbb{R}\cup\\{\infty\\}$
is a convex functional for each $u\in W$. The key features in this problem are
the weak assumptions on $K$ and $f$ such that a wide range of quasi-
variational inclusions is covered. For example, $K(\cdot,u)$ allows for the
characteristic functional $I_{K(u)}\colon W\to\mathbb{R}\cup\\{\infty\\}$ of a
convex set $K(u)\subset W$, and $f$ is assumed to be upper semicontinuous in
the second and increasing in the third argument, so that the mapping $s\mapsto
f(x,s,s)$ is in general neither upper semicontinuous nor lower semicontinuous
nor monotone. The lack of continuity allows the solution set to be not
compact, but still one can show that there are extremal (i.e. smallest and
greatest) solutions between every ordered pair of sub-supersolutions.

We would like to mention that the method developed here can be used to treat
the following even more general problem (cf. [3]): Find $u\in W^{1,p}(\Omega)$
such that there are measurable selections $\eta$ of $f(\cdot,u,u)$ and $\xi$
of $\hat{f}(\cdot,\gamma u,\gamma u)$ such that

$\begin{split}\langle
Eu,w-u\rangle&+\int_{\Omega}\eta(w-u)dx+\int_{\partial\Omega}\xi(\gamma
w-\gamma u)d\sigma\\\ &+K(w,u)-K(u,u)+\hat{K}(\gamma w,\gamma
u)-\hat{K}(\gamma u,\gamma u)\geq 0\quad\text{for all }w\in
W^{1,p}(\Omega),\end{split}$

where $\gamma u$ denotes the trace of $u$, $\sigma$ the boundary measure on
the boundary $\partial\Omega$ of $\Omega$, and where
$\hat{f}\colon\partial\Omega\times\mathbb{R}\times\mathbb{R}\to\mathcal{P}(\mathbb{R})$
and $\hat{K}\colon L^{p}(\partial\Omega)\times
L^{p}(\partial\Omega)\to\mathbb{R}\cup\\{\infty\\}$ have similar properties as
$f$ and $K$.

##  2 Abstract Subpoint Method

Let $V=(V,\leq)$ be a poset, i.e. a set $V$ equipped with a partial order
$\leq$. We will equip each subset $W\subset V$ with the induced partial order.
Further, for $A,B\in\mathcal{P}(V)$, the power set of $V$, we write

$A\leq^{\ast}B\quad:\Longleftrightarrow\quad\text{for all }a\in A\text{ there
is }b\in B\text{ such that }a\leq b.$

Obviously, $\leq^{\ast}$ is a preorder on $\mathcal{P}(V)$ which extends
$\leq$ in the sense that $a\leq b$ if and only if $\\{a\\}\leq^{\ast}\\{b\\}$.
We use $\leq^{\ast}$ to extend the notion of increasing functions to the case
of multifunctions:

######  2.1 Definition.

Let $V$ be a poset and $S\colon V\to\mathcal{P}(V)$ a multifunction. Then $S$
is called increasing upward if $v\leq w$ in $V$ implies $S(v)\leq^{\ast}S(w)$.

Obviously, a single-valued function $s\colon V\to V$ is increasing if and only
if the corresponding multifunction $S\colon V\to\mathcal{P}(V)$ with
$S(v)=\\{s(v)\\}$ is increasing upward. In the following, we will identify $s$
with $S$ and each $v\in V$ with the singleton $\\{v\\}$.

Furthermore, we extend the notion of fixed points to multifunctions as
follows:

######  2.2 Definition.

Let $V$ be a poset, $W\subset V$, and $S\colon V\to\mathcal{P}(W)$ a
multifunction. Then the set $\operatorname{Sub}S$ of subpoints of $S$ and the
set $\operatorname{Fix}S$ of fixed points of $S$ are given by

$\operatorname{Sub}S:=\\{v\in
W\,:\,v\leq^{\ast}S(v)\\},\qquad\operatorname{Fix}S:=\\{v\in W\,:\,v\in
S(v)\\}.$

Note that subpoints of $S$ belong by definition to the codomain $W$. This will
be important when assigning different properties to the domain $V$ and its
subset $W$. In the following, we will first provide results concerning maximal
subpoints and fixed points and then concerning greatest ones. Recall that
$v^{\ast}$ is maximal in some poset $V$ if for each $w\in V$ with
$v^{\ast}\leq w$ it follows $v^{\ast}=w$, and that $v^{\ast}$ is the greatest
element of $V$ if $w\leq v^{\ast}$ for all $w\in V$. Minimal and smallest
elements are defined by duality.

The interplay of increasing upward multifunctions and subpoints as well as
fixed points is illustrated in the following proposition:

######  2.3 Proposition.

Let $V$ be a poset, $W\subset V$, and let $S\colon V\to\mathcal{P}(W)$ be
increasing upward. Then every maximal subpoint of $S$ is a maximal fixed point
of $S$.

###### Proof.

Let $v^{\ast}$ be a maximal element of $\operatorname{Sub}S$. Then there is
$w\in S(v^{\ast})$ such that $v^{\ast}\leq w$. Since $S$ is increasing upward,
it follows $S(v^{\ast})\leq^{\ast}S(w)$ and especially $w\leq^{\ast}S(w)$,
i.e. $w\in\operatorname{Sub}S$. By maximality of $v^{\ast}$, we obtain
$v^{\ast}=w$ and thus $v^{\ast}\in S(v^{\ast})$, i.e.
$v^{\ast}\in\operatorname{Fix}S$.

Now, let $w\in\operatorname{Fix}S$ with $v^{\ast}\leq w$. Then from
$\operatorname{Fix}S\subset\operatorname{Sub}S$ and the maximality of
$v^{\ast}$ we obtain $v^{\ast}=w$, i.e. $v^{\ast}$ is maximal in
$\operatorname{Fix}S$. ∎

This result is useful because order-theoretical existence results can be
applied with more ease to $\operatorname{Sub}S$ than to $\operatorname{Fix}S$.
In [13, 14], e.g., we have applied an abstract chain generating principle from
[6]. Here, let us present a simpler approach which is essentially given in
[9], where the notion of universally inductive subsets (which we will simply
call inductive in their supersets, see below) was introduced. The key idea is
to extend the famous Fixed Point Theorem of Tarski (whose proof works via
inspection of subpoints) to the multi-valued case.

In order to prove the existence of a maximal subpoint, we will use two notions
concerning the existence of special upper bounds:

######  2.4 Definition.

Let $V$ be a poset and $C,D\subset V$.

  1. (i)

$d\in D$ is called an upper bound of $C$ if $C\leq^{\ast}d$.

  2. (ii)

$C$ is called a chain if it is totally ordered. $D$ is called inductive in
$V$, if for each chain $C\subset V$ from $C\leq^{\ast}D$ it follows
$C\leq^{\ast}d$ for some $d\in D$.

  3. (iii)

If $C$ has a smallest upper bound in $V$, this upper bound is called supremum
of $C$ and denoted by $\sup C$. $V$ is called chain-complete if $\sup C$
exists in $V$ for each chain $C\subset V$.

As usual, a poset $V$ is called inductive if it is inductive in itself, i.e.
if each chain $C\subset V$ is bounded by some $v\in V$. Then, the famous Lemma
of Zorn states that each inductive poset has a maximal element $v^{\ast}$. To
ensure the existence of a maximal subpoint it thus suffices to provide
conditions under which $\operatorname{Sub}S$ is inductive. The following
purely order-theoretical result in combination with Proposition 2.3 is a
slight generalization of [9, Theorem 3.4]:

######  2.5 Proposition.

Let $V$ be a chain-complete poset, $W\subset V$, and let $S\colon
V\to\mathcal{P}(W)$ be increasing upward and such that all its values $S(v)$
are inductive in $W$. Then $\operatorname{Sub}S$ is inductive.

###### Proof.

Let $C\subset\operatorname{Sub}S$ be a chain. Then $C\subset V$ and thus
$c:=\sup C$ exists in $V$. Now, for every $v\in C$ we have
$v\leq^{\ast}S(v)\leq^{\ast}S(c)$, since $S$ is increasing upward. Since
$\leq^{\ast}$ is transitive, we obtain $C\leq^{\ast}S(c)$, and since $S(c)$ is
inductive in $W$, there is some $c^{\ast}\in S(c)$ such that
$C\leq^{\ast}c^{\ast}$. By definition of $c$, we have $c\leq c^{\ast}$ and
thus $c\leq c^{\ast}\in S(c)\leq^{\ast}S(c^{\ast})$, which implies that
$c^{\ast}\in\operatorname{Sub}S$ is an upper bound of $C$. Consequently,
$\operatorname{Sub}S$ is inductive. ∎

######  2.6 Remark.

Since $\emptyset$ is a chain, it follows that every inductive set is non-empty
and that each chain-complete poset has a smallest element
$\underline{\smash{u}}$. So, let $V$ be a poset with smallest element
$\underline{\smash{u}}$, let $W\subset V$ and let $S\colon V\to\mathcal{P}(W)$
be increasing upward and such that all its values are non-empty. Take
$u^{\ast}\in S(\underline{\smash{u}})$, then $\underline{\smash{u}}\leq
u^{\ast}$ and thus $u^{\ast}\in
S(\underline{\smash{u}})\leq^{\ast}S(u^{\ast})$, meaning that
$u^{\ast}\in\operatorname{Sub}S$, i.e. $\operatorname{Sub}S\neq\emptyset$.

There are simple criteria for sets to be chain-complete or inductive if one
considers ordered topological spaces, i.e. posets $V$ equipped with a topology
$\tau$ such that for all $a,b\in V$ the order intervals

$a^{\uparrow}:=\\{c\in V:a\leq c\\},\qquad b^{\downarrow}:=\\{c\in V:c\leq
b\\},\qquad[a,b]:=a^{\uparrow}\cap b^{\downarrow}$

are closed. We recall them from [9], but give a much simpler proof for the
second part, which is inspired by [6, Proposition 1.1.4].

######  2.7 Proposition.

Let $V$ be an ordered topological space and let $A\subset V$ be compact.

  1. (i)

If $A$ is closed and non-empty, then $A$ is inductive in $V$.

  2. (ii)

If $A$ has a smallest element, then $A$ is chain-complete.

###### Proof.

Ad (i), let $C\subset V$ be a chain such that $C\leq^{\ast}A$. If
$C=\emptyset$, then any $a\in A$ satisfies $C\leq^{\ast}a$. If
$C\neq\emptyset$, then the family $\mathcal{C}:=\\{c^{\uparrow}\cap A\,:\,c\in
C\\}$ consists of closed subsets of the compact set $A$ and has the finite
intersection property. Thus there is some $a\in\bigcap\mathcal{C}$, such that
$C\leq^{\ast}a$ and $a\in A$. Consequently, $A$ is inductive in $V$.

Ad (ii), let $C\subset A$ be a chain. If $C=\emptyset$, then $\sup C$ is the
smallest element of $A$. If $C\neq\emptyset$, then the closure $\overline{C}$
of $C$ is non-empty. Then the family
$\mathcal{C}:=\\{c^{\uparrow}\cap\overline{C}\,:\,c\in C\\}$ consists of
closed subsets of the compact set $A$ and has the finite intersection
property. Thus there is some $a\in\bigcap\mathcal{C}$, such that
$C\leq^{\ast}a$ and $a\in\overline{C}$. For any upper bound $b$ of $C$ we have
$\overline{C}\subset b^{\downarrow}$, and so it follows $a\leq b$ and $\sup
C=a\in A$. Consequently, $A$ is chain-complete. ∎

It is straightforward to combine the results so far to obtain the following
fixed point theorem, from which [13, Theorem 2.11] readily follows:

######  2.8 Theorem.

Let $V$ be a poset, $W\subset V$, and let $V$ and $W$ be equipped with
topologies such that they become ordered topological spaces. Further, let
$D\subset V$ be a compact set with smallest element $\underline{\smash{u}}$,
and let $S\colon D\to\mathcal{P}(D\cap W)$ be an increasing upward
multifunction whose values are non-empty and compact and closed in $W$. Then
$S$ has a maximal fixed point.

######  2.9 Remark.

If $V$ is an ordered topological space then $W$ becomes an ordered topological
space if equipped with the induced topology as a subset of $V$ (and, as
always, with the induced partial order). However, there is no need to use the
induced topology on $W$.

In order to obtain even greatest fixed points, we use the following elementary
results from [13, Section 2.3.2]:

######  2.10 Proposition.

Let $V$ be an upper semilattice (i.e. a poset in which $\sup\\{a,b\\}$ exists
for all $a,b\in V$) and let $\underline{\smash{S}},S\colon V\to\mathcal{P}(V)$
be multifunctions.

  1. (i)

If $S$ has a maximal fixed point, then $S$ has a greatest fixed point if and
only if $\operatorname{Fix}S$ is directed upward (i.e. for all
$a,b\in\operatorname{Fix}S$ there is $c\in\operatorname{Fix}S$ such that
$\\{a,b\\}\leq^{\ast}c$).

  2. (ii)

If all values of $S$ are directed upward and if $S$ is permanent upward (i.e.
from $a\leq b$ it follows $S(a)\subset S(b)$), then $\operatorname{Fix}S$ is
directed upward.

  3. (iii)

If $\underline{\smash{S}}$ and $S$ are equivalent in the sense that
$S(v)\leq^{\ast}\underline{\smash{S}}(v)\leq^{\ast}S(v)$ for all $v\in V$,
then $S$ is increasing upward if and only if $\underline{\smash{S}}$ is
increasing upward.

  4. (iv)

If $\underline{\smash{S}}$ and $S$ are equivalent in the sense that
$\operatorname{Fix}S\leq^{\ast}\operatorname{Fix}\underline{\smash{S}}\leq^{\ast}\operatorname{Fix}S$,
then $S$ and $\underline{\smash{S}}$ have the same maximal and greatest fixed
points.

Now, we have all the ingredients to provide the general framework for the
abstract subpoint method. To this end, the so called suboperator
$\underline{\smash{S}}$ has to satisfy various order-theoretical assumptions,
while the main operator $S$ is asked to have good topological properties.

######  2.11 Theorem.

Let $V$ be a poset, $W\subset V$, and let $V$ and $W$ be equipped with
topologies such that they become ordered topological spaces. Further, let
$D\subset V$ and let $\underline{\smash{S}}\colon D\to\mathcal{P}(D)$ and
$S\colon D\to\mathcal{P}(D\cap W)$ be multifunctions such that the following
conditions are satisfied:

  1. (i)

$D$ is an upper semilattice and compact in $V$,

  2. (ii)

all values of $S$ are compact and closed in $W$,

  3. (iii)

$\underline{\smash{S}}$ is permanent upward, all its values are directed
upward, and for all $v\in D$ it holds
$S(v)\subset\underline{\smash{S}}(v)\leq^{\ast}S(v)$,

  4. (iv)

there is $\underline{\smash{u}}\in D$ such that
$\underline{\smash{u}}\leq^{\ast}\underline{\smash{S}}(\underline{\smash{u}})$.

Then $S$ has a greatest fixed point $u^{\ast}$ and it holds
$\underline{\smash{u}}\leq u^{\ast}$.

###### Proof.

Without loss of generality, we can assume
$\underline{\smash{u}}\in\operatorname{Sub}\underline{\smash{S}}$, for
otherwise from
$\underline{\smash{u}}\leq^{\ast}\underline{\smash{S}}(\underline{\smash{u}})\leq^{\ast}S(\underline{\smash{u}})$
there is some $\underline{\smash{w}}$ such that
$\underline{\smash{u}}\leq\underline{\smash{w}}\in
S(\underline{\smash{u}})\subset\underline{\smash{S}}(\underline{\smash{u}})\leq^{\ast}\underline{\smash{S}}(\underline{\smash{w}})$,
implying $\underline{\smash{w}}\in\operatorname{Sub}\underline{\smash{S}}$, so
that we can replace $\underline{\smash{u}}$ by $\underline{\smash{w}}$.

Further, we can assume that $\underline{\smash{u}}$ is the smallest element of
$D$, for otherwise we restrict our considerations to
$\underline{\smash{u}}^{\uparrow}$ and check that the restrictions of $D$,
$\underline{\smash{S}}$ and $S$ to $\underline{\smash{u}}^{\uparrow}$ satisfy
the same properties as $D$, $\underline{\smash{S}}$ and $S$ (for this we need
$\underline{\smash{u}}\in W$ such that $\underline{\smash{u}}^{\uparrow}\cap
W$ is closed in $W$). We conclude that the values of $S$ are non-empty, since
for all $v\in D$ we have $\underline{\smash{u}}\leq v$ and thus
$\emptyset\neq\underline{\smash{S}}(\underline{\smash{u}})\leq^{\ast}\underline{\smash{S}}(v)\leq^{\ast}S(v)$.

By Proposition 2.10(iii) we have that $S$ is increasing upward, and thus
Theorem 2.8 ensures that $S$ has a maximal fixed point. Especially, by
replacing $\underline{\smash{u}}$ with any
$\underline{\smash{w}}\in\operatorname{Sub}\underline{\smash{S}}$, the
arguments so far imply
$\operatorname{Fix}\underline{\smash{S}}\subset\operatorname{Sub}\underline{\smash{S}}\leq^{\ast}\operatorname{Fix}S$,
whereas the inclusion
$\operatorname{Fix}S\subset\operatorname{Fix}\underline{\smash{S}}$ follows
from $S(v)\subset\underline{\smash{S}}(v)$. Thus, by repeated application of
Proposition 2.10 we obtain that $S$ has a greatest fixed point. ∎

######  2.12 Remark.

In applications, the fixed points of $S$ will be the solutions of a problem,
while the fixed points or even the subpoints of $\underline{\smash{S}}$ will
be so called subsolutions. Since it is natural to define subsolutions in a way
that a solution is always a subsolution, the seemingly strong condition
$S(v)\subset\underline{\smash{S}}(v)$ is natural. However, it is often non-
trivial to provide the counterpart $\underline{\smash{S}}(v)\leq^{\ast}S(v)$
and the directedness of $\underline{\smash{S}}(v)$. In Section 4 we will
present some analytic tools for this task which, however, depend on the
existence of
$\underline{\smash{u}}\in\operatorname{Fix}\underline{\smash{S}}$.

######  2.13 Remark.

All results so far remain true if we replace the partial order $\leq$ in all
statements by its dual order $\geq$. Thus, a multifunction $\overline{S}\colon
D\to\mathcal{P}(D)$ and the existence of some
$\overline{u}\in\operatorname{Fix}\overline{S}$ such that
$\underline{u}\leq\overline{u}$ can be used to find smallest fixed points of
$S$ in $D=[\underline{u},\overline{u}]$.

##  3 Quasi-Variational Inclusions with Bifunctions

Let us consider quasi-variational inclusions with bifunctions, i.e.
variational inequalities with multi-valued functions in which the constraint
set for the solution, the function to be minimized, as well as the set of test
functions depend on the unknown solution itself, and this in a twofold way. To
this end, let again $V$ be a poset and assume that $W\subset V$ is a lattice
(i.e. for all $u,w\in W$ the supremum $u\vee w:=\sup\\{u,v\\}$ and the infimum
$u\wedge w$ (the largest lower bound of $u$ and $w$) exist in $W$). Further,
let $\Gamma(W)$ be the set of functions $a\colon
W\to\mathbb{R}\cup\\{\infty\\}$ for which $a(u)\in\mathbb{R}$ for at least one
$u\in W$, i.e. the effective domain $\mathcal{D}(a):=\\{u\in
W:a(u)\in\mathbb{R}\\}$ is non-empty. Then we consider the following general
problem:

######  3.1 Definition.

Let $A\colon W\times V\to\mathcal{P}(\Gamma(W))$ and $K,T\colon W\times
V\to\mathcal{P}(W)$ be multi-valued bifunctions. Then we say that $u\in W$ is
a solution of the associated quasi-variational inclusion problem (QVIP) if and
only if

$u\in K(u,u)\quad\text{and}\quad a(w)-a(u)\geq 0\quad\text{for some $a\in
A(u,u)$ and all }w\in T(u,u)$

(which implicitly implies $u\in\mathcal{D}(a)$). In this context, a function
$u$ is admissible if and only if $u\in K(u,u)$, and the set $T(u,u)$ contains
the corresponding test functions.

Naturally, we can interpret (QVIP) as a fixed point problem by fixing one
argument of the bifunctions. This leads to the following simpler problem:

######  3.2 Definition.

Let $A\colon W\times V\to\mathcal{P}(\Gamma(W))$ and $K,T\colon W\times
V\to\mathcal{P}(W)$ be multi-valued bifunctions. Then for each $v\in V$ we say
that $u\in W$ is a solution of the associated parameterized quasi-variational
inclusion problem (QVIPv) if and only if

$u\in K(u,v)\quad\text{and}\quad a(w)-a(u)\geq 0\quad\text{for some $a\in
A(u,v)$ and all }w\in T(u,v).$ (2)

The associated multi-valued solution operator is defined as

$S\colon V\to\mathcal{P}(W),\quad S(v):=\\{u\in W\,:\,u\text{ is a solution of
(QVIP${}_{v}$)}\\}.$

Clearly, $u$ is a solution of (QVIP) if and only if $u$ is a solution of
(QVIPu), i.e. $u\in\operatorname{Fix}S$. Thus, by introducing subsolutions, we
may be able to apply the abstract framework. It will come in useful to define
subsolutions as solutions of certain appropriate defined quasi-variational
inclusions of the form (2) with slightly modified multifunctions
$\underline{A}$, $\underline{K}$ and $\underline{T}$. Thus, we next provide
some results about the monotone behavior of the solution operator $S$. To this
end, we use the following relation for $a,b\in\Gamma(W)$, which goes well with
variational problems on lattices:

$a\preccurlyeq b\quad:\Longleftrightarrow\quad a(u\wedge w)+b(u\vee w)\leq
a(u)+b(w)\quad\text{for all }u,w\in W.$

This relation was introduced in [8] as a natural extension of the so called
strong set order on sets $A,B\in\mathcal{P}(W)$, i.e.

$A\preccurlyeq B\quad:\Longleftrightarrow\quad u\wedge w\in
A\quad\text{and}\quad u\vee w\in B\quad\text{for all }u\in A,w\in B$

(we use the same relation symbol since $A\preccurlyeq B$ holds for non-empty
$A,B\in\mathcal{P}(W)$ if and only if $I_{A}\preccurlyeq I_{B}$ holds for
their characteristic functionals). Let us recall a few elementary properties:
If $W$ contains two non-comparable elements then $\preccurlyeq$ is not
reflexive. Thus, functions $a\in\Gamma(W)$ with $a\preccurlyeq a$ are
meaningful; they are commonly called submodular (and a set
$A\in\mathcal{P}(W)$ with $A\preccurlyeq A$ is called sublattice of $W$).
Furthermore, $\preccurlyeq$ is in general neither symmetric nor anti-symmetric
nor transitive. However, the following modified transitivity holds true:

######  3.3 Proposition.

Let $a,b,c\in\Gamma(W)$ such that $a\preccurlyeq b$ and $b\preccurlyeq c$.

  1. (i)

Then $\mathcal{D}(a)\preccurlyeq\mathcal{D}(b)\preccurlyeq\mathcal{D}(c)$ and
from this it follows $\mathcal{D}(a)\preccurlyeq\mathcal{D}(c)$.

  2. (ii

If $b\preccurlyeq b$ and if $W$ is a distributive lattice (i.e. $\wedge$ and
$\vee$ distribute over each other), then $a\preccurlyeq c$.

###### Proof.

Assertion (i) follows readily. The proof of assertion (ii) is a little bit
more complicated since $a$, $b$ and $c$ may attain infinite values. Thus, let
$u,w\in W$ be arbitrary. We claim that

$a(u\wedge w)+c(u\vee w)\leq a(u)+c(w).$ (3)

If $a(u)=\infty$ or $c(w)=\infty$, then this holds true trivially. Otherwise,
let $v\in W$ such that $b(v)\in\mathbb{R}$. Then from $a\preccurlyeq b$ and
$u\wedge(w\wedge(u\vee v))=u\wedge w$ we obtain

$a(u\wedge w)+b(u\vee(w\wedge(u\vee v)))\leq a(u)+b(w\wedge(u\vee v)).$ (4)

By dual reasoning, we also obtain

$c(w\vee u)+b(w\wedge(u\vee(w\wedge v)))\leq c(w)+b(u\vee(w\wedge v)).$ (5)

All terms in (4) and (5) are finite (recall that
$\mathcal{D}(a)\preccurlyeq\mathcal{D}(b)\preccurlyeq\mathcal{D}(c)$) and thus
(3) follows from $b\preccurlyeq b$ and the identity

$\displaystyle[u\vee(w\wedge(u\vee v))]\wedge[w\wedge(u\vee(w\wedge v))]$
$\displaystyle=(u\vee w)\wedge(u\vee v)\wedge w\wedge(u\vee w)\wedge(u\vee v)$
$\displaystyle=w\wedge(u\vee v)$

together with its dual counterpart. ∎

Despite $\preccurlyeq$ being no partial order on $\Gamma(W)$, we use the same
terminology as for partial orders. Especially,
$a^{\uparrow}=\\{b\in\Gamma(W):a\preccurlyeq b\\}$ and we have the following
notion:

######  3.4 Definition.

A multifunction $A\colon V\to\mathcal{P}(\Gamma(W))$ is called increasing
upward if $v\leq v^{\prime}$ in $V$ implies
$A(v)\preccurlyeq^{\ast}A(v^{\prime})$, i.e. for all $a\in A(v)$ there is
$b\in A(v^{\prime})$ such that

$a(u\wedge w)+b(u\vee w)\leq a(u)+b(w)\quad\text{for all }u,w\in W.$

######  3.5 Remark.

Note that parameterized problems like (QVIPv) where considered for a long time
in the case where $A$ is single-valued and constant, see, e.g., [15, Lemma
2.8.1]. There, parameterized functions $a_{v}\colon W\to\mathbb{R}$ with the
following monotone compatibility property were considered:

$a_{v}(u\wedge w)+a_{v^{\prime}}(u\vee w)\leq
a_{v}(u)+a_{v^{\prime}}(w)\quad\text{for all }u,w\in W\text{ and for $v\leq
v^{\prime}$ in $V$}.$

Obviously, this is a special case of Definition 3.4 and consequently, our next
lemma generalizes the result of [15].

Now, let us provide the monotonic dependence of the solution operator on the
data, which gives us a guideline for the construction of the multifunctions
$A$, $K$ and $T$. To this end, we consider a second quasi-variational
inclusion problem (QVIP’) with data $A^{\prime}$, $K^{\prime}$ and
$T^{\prime}$ and solution operator $S^{\prime}$.

######  3.6 Lemma.

Let for all $u\in W$ and $v,v^{\prime}\in V$ such that $u\in S(v)$ and $v\leq
v^{\prime}$ the following relations hold true:

$A(u,v)\preccurlyeq^{\ast}A^{\prime}(u,v^{\prime}),\quad K(u,v)\subset
K^{\prime}(u,v^{\prime}),\quad T^{\prime}(u,v^{\prime})\subset T(u,v),\quad
T^{\prime}(u,v^{\prime})\leq^{\ast}u.$

Then $S(v)\subset S^{\prime}(v^{\prime})$.

###### Proof.

Let $v\leq v^{\prime}\in V$ and $u\in S(v)$. Then it holds $u\in K(u,v)\subset
K^{\prime}(u,v^{\prime})$ and there is $a\in A(u,v)$ such that

$a(w)-a(u)\geq 0\quad\text{for all }w\in T(u,v).$ (6)

Since $A(u,v)\preccurlyeq^{\ast}A^{\prime}(u,v^{\prime})$, there is $b\in
A^{\prime}(u,v^{\prime})$ such that $a\preccurlyeq b$, that is

$a(u\wedge w)+b(u\vee w)\leq a(u)+b(w)\quad\text{for all }w\in W.$

Taking $w\in T^{\prime}(u,v^{\prime})\subset T(u,v)$, we have $w\leq u$ and
thus $u\wedge w=w$ and $u\vee w=u$, resulting in

$a(w)+b(u)\leq a(u)+b(w).$ (7)

Using (6) and (7), we conclude

$b(u)\leq b(u)+a(w)-a(u)\leq a(u)+b(w)-a(u)=b(w),$

and since $b(w)<\infty$ for at least one $w\in W$, we obtain
$u\in\mathcal{D}(b)$ and $u\in S^{\prime}(v^{\prime})$.

∎

As an immediate consequence of Lemma 3.6 for the case (QVIP) $=$ (QVIP’) we
obtain conditions on $A$, $K$ and $T$ under which $S$ is permanent upward;
especially, $v\mapsto A(u,v)$ should be asked to be increasing upward.
Furthermore, one can utilize the fact that the sum of two increasing upward
multifunctions is again increasing upward, and there is even a connection to
positive linear functionals. To this end, we have to assume that $W$ is an
ordered linear space (i.e. $W$ is a linear space over $\mathbb{R}$ such that
$u\leq w$ is equivalent to $0\leq w-u$, and $0\leq\lambda$ in $\mathbb{R}$ and
$0\leq u\in W$ imply $0\leq\lambda w$). Then for $u\in W$ we define
$u^{+}:=u\wedge 0$ and obtain the identities

$u-(u\wedge w)=(u-w)^{+}=(u\vee w)-w.$ (8)

By $W^{\prime}$ we denote the linear space of all linear functionals $a\colon
W\to\mathbb{R}$, and for $a\in W^{\prime}$ and $u\in W$ we write $\langle
a,u\rangle:=a(u)$ and say that $a$ is positive if $\langle a,u^{+}\rangle\geq
0$ for all $u\in W$.

######  3.7 Proposition.

For $a,b\in W^{\prime}$ it holds $a\preccurlyeq b$ if and only if $a-b$ is
positive.

###### Proof.

Let $u,w\in W$. Then from the identities in (8) and the linearity of $a$ and
$b$ we obtain

$\displaystyle a(u\wedge w)+b(u\vee w)\leq a(u)+b(w)\quad$
$\displaystyle\Longleftrightarrow\quad\langle b,(u\vee w)-w\rangle\leq\langle
a,u-(u\wedge w)\rangle$ $\displaystyle\Longleftrightarrow\quad\langle
a-b,(u-w)^{+}\rangle\geq 0.$

The claim follows readily.∎

The calculations above suggest the following notions:

  * (i)

For $a,b\in\Gamma(W)$ and $u,w\in W$ we say that $a\preccurlyeq b$ w.r.t.
$(u,w)$ if

$a(u\wedge w)+b(u\vee w)\leq a(u)+b(w).$

  * (ii)

A multifunction $A\colon W\to\mathcal{P}(W^{\prime})$ is called T-monotone if
for all $u,w\in W$ and all $a\in A(u)$ and $b\in A(w)$ it holds

$\langle a-b,(u-w)^{+}\rangle\geq 0.$

######  3.8 Corollary.

Let $A\colon W\to\mathcal{P}(W^{\prime})$ be T-monotone and $a\in A(u)$, $b\in
A(w)$. Then $a\preccurlyeq b$ w.r.t. $(u,w)$.

######  3.9 Remark.

For the use of T-monotone functions $A\colon W\to W^{\prime}$ see, e.g., [11,
Section 4:5]. The use of T-monotone multifunctions in the context of
variational inclusions is standard, see, e.g., [7].

######  3.10 Remark.

We have seen that our definition of increasing upward mappings is productive.
In the special case $v=v^{\prime}$ we obtain the following notion: A set
$A\subset\Gamma(W)$ is called upper submodular if $A\preccurlyeq^{\ast}A$,
that is, if for all $a\in A$ there is $b\in B$ such that $a(w\wedge
w^{\prime})+b(w\vee w^{\prime})\leq a(w)+b(w^{\prime})$ for all
$w,w^{\prime}\in W$.

##  4 Elliptic Quasi-Variational Inclusions with Bifunctions

Finally, let us apply the abstract results in order to find solutions of an
involved elliptic quasi-variational inclusion problem whose lower order terms
are given by bifunctions. To this end, we will presume some knowledge of
functional analysis (for more information, we refer to, e.g., [4, 16]) and use
the following setting throughout this section:

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 1$, be a bounded domain with
Lipschitz boundary. We equip $\Omega$ with the Lebesgue measure and denote by
$L^{0}(\Omega)$ the linear space of all (equivalence classes of) measurable
functions $u\colon\Omega\to\mathbb{R}$ (where $\mathbb{R}$ is equipped with
the Borel measure). We fix $p\in(1,\infty)$ and $q\in(1,p^{\ast})$, where
$p^{\ast}$ denotes the critical Sobolev exponent associated with $p$ and the
dimension $N$, given by $p^{\ast}=Np/(N-p)$ for $p<N$, and $p^{\ast}$
arbitrarily large for $p\geq N$. Furthermore, $q^{\prime}$ denotes the Hölder
conjugate of $q$. In dependence of these constants, we introduce the following
subspaces of $L^{0}(\Omega)$:

$L:=L^{q^{\prime}}(\Omega),\qquad V:=L^{p^{\ast}}(\Omega),\qquad
W:=W^{1,p}_{0}(\Omega),$

where $L^{r}(\Omega)$ is the usual Lebesgue space of $r$-th power integrable
functions, and $W^{1,p}_{0}(\Omega)$ the usual Sobolev space with zero
boundary values in the sense of traces. We will denote the norm of
$L^{r}(\Omega)$ by $\|\cdot\|_{r}$, and we note that $\|\nabla u\|_{p}$ gives
an equivalent norm on $W$ (where $\nabla u$ is the weak gradient of $u$). The
spaces are chosen in such a way that the embedding $W\hookrightarrow V$ is
continuous and that the embedding $i_{q}:W\hookrightarrow L^{\ast}$ and its
adjoint $i_{q}^{\ast}\colon L\hookrightarrow W^{\ast}$ are compact (with
respect to the norm topology). There, $L^{\ast}$ and $W^{\ast}$ denote the
topological dual of $L$ and $W$, respectively, and $i_{q}^{\ast}$ is the
natural embedding, i.e. for $\eta\in L$ we have

$\langle i_{q}^{\ast}\eta,u\rangle:=\int_{\Omega}\eta
u:=\int_{\Omega}\eta(x)u(x)\,\mathrm{d}x\quad\text{for all }u\in W.$

N ote that we suppress arguments in the notation of integrals.

Concerning order, we equip $\mathbb{R}$ with the usual partial order $\leq$
and $L^{0}(\Omega)$ (or even the space of (equivalence classes of) functions
$u\colon\Omega\to\mathbb{R}$) and all its subspaces with the natural partial
order $\leq$, defined by $u\leq v$ if $u(x)\leq v(x)$ for a.e. $x\in\Omega$,
by which $L^{0}(\Omega)$ is known to be an ordered linear space with lattice
structure and distributive sub-lattices $L$, $V$ and $W$.

Concerning topology, besides the norm topology we will often use the weak
topology on the reflexive Banach spaces $L$, $V$ and $W$. If a sequence
$(u_{n})$ converges weakly to some function $u$, we write
$u_{n}\rightharpoonup u$. Recall that each interval $u^{\uparrow}$ is closed
and convex and thus weakly closed, such that especially $V$ and $W$ are
ordered topological spaces with respect to the weak topology. Recall further,
that a subset $K$ of a reflexive Banach space is weakly compact and weakly
closed if and only if it is norm bounded and sequentially closed (via Banach-
Alaoglu and Eberlein-Šmulian), such that (by monotonicity of the integral)
each order interval $[\underline{\smash{u}},\overline{u}]$ in $V$ is weakly
compact and weakly closed. Note, however, that such intervals in $W$ are not
bounded and thus not weakly compact since our partial order is not suitable to
control the derivatives of a Sobolev function.

In order to formulate the (QVIP) of this section, let $E\colon W\to W^{\ast}$
be an elliptic differential operator of Leray-Lions type (for a precise
description see Assumption 4.4 below),
$f\colon\Omega\times\mathbb{R}\times\mathbb{R}\to\mathcal{P}(\mathbb{R})$ a
multi-valued bifunction, and $K\colon W\times V\to\mathbb{R}\cup\\{\infty\\}$
a bifunctional. Then the (QVIP) reads as follows:

$\begin{split}\text{Find }u\in W\text{ }&\text{s.t. there is }\eta\in L\text{
with }\eta(x)\in f(x,u(x),u(x))\text{ for a.e. $x\in\Omega$}\\\ &\text{s.t.
}\langle Eu,w-u\rangle+\int_{\Omega}\eta(w-u)+K(w,u)-K(u,u)\geq
0\quad\text{for all }w\in W.\end{split}$ (9)

######  4.1 Remark.

This problem was considered in [7], where a rather involved proof of the
existence result was given. By using the abstract framework developed above we
are able to reduce the analytic work to be done to a minimum. Nevertheless,
not all difficulties can be skipped. For the sake of completeness, we are
going to present full arguments in our notation although most of the ideas are
given in [3, 5, 7, 8, 13] and the papers cited there.

To shorten the notation and to bring this problem in the form (QVIP) presented
in Section 3, we introduce the mapping

$A\colon W\times V\to\mathcal{P}(\Gamma(W)),\quad
A(u,v):=E(u)+i^{\ast}_{q}F(u,v)+K(\cdot,v),$

where $E$ is the elliptic differential operator from above and

$F\colon L^{0}(\Omega)\times L^{0}(\Omega)\to\mathcal{P}(L),\quad
F(u,v):=\\{\eta\in L:\eta\subset f(\cdot,u,v)\\},$

where $\eta\subset f(\cdot,u,v)$ means $\eta(x)\in f(x,u(x),v(x)$ for a.e.
$x\in\Omega$. Then $u\in W$ is a solution of (9) if and only if

$a(w)-a(u)\geq 0\quad\text{for some }a\in A(u,u)\text{ and all }w\in W.$

There, implicitly, only functions $u\in W$ with $u\in\mathcal{D}(K(\cdot,v))$
are admissible, and every function $w\in W$ is a test function.

In order to define subsolutions, we modify this setting by expanding $A(u,u)$
and shrinking the space of test functions so that it is as easy as possible
for subsolutions to exist. In accordance with [7], we define

$\underline{A}\colon W\times
V\to\mathcal{P}(\Gamma(W)),\quad\underline{\smash{A}}(u,v):=E(u)+i^{\ast}_{q}F(u,v)+K(\cdot,v)^{\downarrow}$

(where $K(\cdot,v)^{\downarrow}=\\{k\in\Gamma(W):k\preccurlyeq K(\cdot,v)\\}$)
and

$\underline{\smash{T}}\colon W\times
V\to\mathcal{P}(W),\quad\underline{\smash{T}}(u,v):=u\wedge\mathcal{D}(K_{v}).$

Then, $\underline{\smash{u}}\in W$ is called subsolution of (9) if and only if

$a(w)-a(\underline{\smash{u}})\geq 0\quad\text{for some
}a\in\underline{\smash{A}}(\underline{\smash{u}},\underline{\smash{u}})\text{
and all
}w\in\underline{\smash{T}}(\underline{\smash{u}},\underline{\smash{u}}).$

Dually, a supersolution $\overline{u}\in W$ of (9) is defined as solution of
the quasi-variational inclusion with $K_{v}^{\uparrow}$ instead of
$K_{v}^{\downarrow}$ and $\overline{T}(u,v):=u\vee\mathcal{D}(K_{v})$ instead
of $\underline{\smash{T}}(u,v)$.

Next, let us give precise assumptions on the data which guarantee that all
conditions of Theorem 2.11 are fulfilled, so that Problem (9) has greatest and
smallest solutions between each ordered pair of sub-supersolutions.

######  4.2 Assumption.

Since we do not provide sub-supersolutions, we have to assume:

  * (S)

There is a subsolution $\underline{\smash{u}}$ and a supersolution
$\overline{u}$ such that $\underline{\smash{u}}\leq\overline{u}$.

######  4.3 Remark.

See, e.g., [7] for the construction of sub-supersolutions under appropriate
conditions by use of the monotone dependence provided in Lemma 3.6. The key
idea is to use solutions of simpler variational inclusions.

In the following, we will often restrict our consideration to the interval
$[\underline{\smash{u}},\overline{u}]$. Although
$\underline{\smash{u}},\overline{u}\in W$ by definition, this interval of sub-
supersolutions is always to be considered in $V$ such that it is non-empty,
bounded, closed and weakly compact. Especially, if we merely assume
$v\in[\underline{\smash{u}},\overline{u}]$, then $v$ may not have weak
derivatives.

######  4.4 Assumption.

Let $e\colon\Omega\times\mathbb{R}^{N}\to\mathbb{R}^{N}$ be a function
defining the (single-valued) differential operator $E\colon W\to W^{\ast}$ of
Leray-Lions type via

$Eu:=-\operatorname{div}e(\cdot,\nabla u)\quad\text{resp.}\quad\langle
Eu,w\rangle:=\int_{\Omega}e(\cdot,\nabla u)\nabla w\quad\text{for all }w\in
W.$

The following standard assumptions on $e$ are meant to hold for a.e.
$x\in\Omega$ and all $\xi\in\mathbb{R}^{N}$.

  * (E1)

The function $e$ is a Carathéodory function, i.e. $x\mapsto e(x,\xi)$ is
measurable on $\Omega$ and $\xi\mapsto e(x,\xi)$ is continuous on
$\mathbb{R}^{N}$.

  * (E2)

The function $e$ is $\boldsymbol{p}$-coercive in the second argument, i.e.
there is a constant $\alpha_{2}>0$ and a function $e_{2}\in L^{1}(\Omega)$
such that $e(x,\xi)\xi\geq\alpha_{2}|\xi|^{p}-e_{2}(x)$.

  * (E3)

There exists a constant $\alpha_{3}>0$ and a function $e_{3}\in
L^{p^{\prime}}(\Omega)$ such that $e$ satisfies the growth condition
$|e(x,\xi)|\leq\alpha_{3}|\xi|^{p-1}+e_{3}(x)$.

  * (E4)

The function $e$ is monotone in the second argument, i.e. it holds, for all
$\xi,\xi^{\prime}\in\mathbb{R}^{N}$,
$(e(x,\xi)-e(x,\xi^{\prime}))(\xi-\xi^{\prime})\geq 0$.

Under conditions (E1)—(E4) the operator $E\colon W\to W^{\ast}$ is known to be
well-defined, bounded, coercive, monotone and continuous, and thus
pseudomonotone (i.e. $u_{n}\rightharpoonup u$ in $W$ and $\limsup_{n}\langle
Eu_{n},u_{n}-u\rangle\leq 0$ imply $\langle
Eu,w-u\rangle\geq\limsup_{n}\langle Eu_{n},w-u_{n}\rangle$ for all $w\in W$).
An example is given by the $p$-Laplacian defined by $e(x,\xi)=|\xi|^{p-2}\xi$.
See, e.g., [12, Section 2.4] for an exhaustive treatment.

######  4.5 Assumption.

Let $f\colon\Omega\times\mathbb{R}\times\mathbb{R}\to\mathcal{P}(\mathbb{R})$
be a multifunction whose values are compact intervals. If not specified
otherwise, the following conditions are meant to hold for a.e. $x\in\Omega$
and all $s,t\in\mathbb{R}$.

  * (F1)

The function $(x,t)\mapsto f(x,s,t)$ is superpositionally measurable, i.e. for
all $v\in L^{0}(\Omega)$ the multivalued function $x\mapsto f(x,s,v(x))$ is
measurable, i.e. the set $\\{x\in\Omega:f(x,s,v(x))\subset U\\}$ is measurable
for each open $U\subset\mathbb{R}$.

  * (F2)

The function $s\mapsto f(x,s,t)$ is upper semicontinuous on $\mathbb{R}$, i.e.
the set $\\{s\in\mathbb{R}:f(x,s,t)\subset U\\}$ is open for each open
$U\subset\mathbb{R}$.

  * (F3)

The function $t\mapsto f(x,s,t)$ is decreasing on $\mathbb{R}$, i.e.
$f(x,s,t)\leq^{\ast}f(x,s,t^{\prime})$ and
$f(x,s,t^{\prime})\geq^{\ast}f(x,s,t)$ for all $t^{\prime}\in\mathbb{R}$ with
$t^{\prime}\leq t$.

  * (F4)

There is some $b_{4}\in L$ such that $f$ satisfies the growth condition

$|y|\leq b_{4}(x)\quad\text{for all $y\in f(x,s,t)$ and all
$s,t\in[\underline{u}(x),\overline{u}(x)]$.}$

######  4.6 Remark.

Since $f$ may not depend continuously on $t$, it may be a tedious task to
ensure (F1). However, if one only assumes (F4) and that $f$ is measurable in
$x$ (which follows at once from (F1) and is a standard assumption if $f$ does
not depend on $t$), then there may be a fairly simple criterion which ensures
(F1).

From (F4) it follows that all measurable selections $\eta\subset f(\cdot,u,v)$
with $u,v\in[\underline{\smash{u}},\overline{u}]$ belong to $L$, which means
that $F(u,v)$ is non-empty if $f$ is weakly superpositionally measurable, i.e.
if $f(\cdot,u,v)$ has at least one measurable selection whenever $u,v\in
L^{0}(\Omega)$. However, this follows readily from (F1) and (F2) and [1, Lemma
7.1]. Thus, $F$ has non-empty values if restricted to
$[\underline{\smash{u}},\overline{u}]$.

Further, we will need the following weak closure property of $f$, which
follows at once from (F1) and [5, Proposition 2]:

######  4.7 Proposition.

Let $v\in L^{0}(\Omega)$, $\eta_{n}\rightharpoonup\eta$ in $L^{1}(\Omega)$ and
$u_{n}\to u$ in $L^{0}(\Omega)$ pointwise a.e. If $\eta_{n}\subset
f(\cdot,u_{n},v)$, then $\eta\subset f(\cdot,u,v)$.

(Note that the convergence in Proposition 4.7 holds true for at least a
subsequence if $\eta_{n}\rightharpoonup\eta$ in $L^{r}(\Omega)$ and $u_{n}\to
u$ in $L^{s}(\Omega)$ for some $r,s\geq 1$ since $\Omega$ is bounded.)

######  4.8 Assumption.

Let $K\colon W\times V\to\mathbb{R}\cup\\{\infty\\}$ be a bifunctional
satisfying the following conditions:

  * (K1)

For all $v\in[\underline{\smash{u}},\overline{u}]$ the function
$K_{v}:=K(\cdot,v)\colon W\to\mathbb{R}\cup\\{\infty\\}$ is proper (i.e.
$\mathcal{D}(K_{v})\neq\emptyset$, thus $K_{v}\in\Gamma$), convex and lower
semicontinuous.

  * (K2)

The mapping $v\mapsto K_{v}$ is increasing, i.e. $v\leq v^{\prime}$ in
$[\underline{\smash{u}},\overline{u}]$ implies $K_{v}\preccurlyeq
K_{v^{\prime}}$.

  * (K3)

For each $v\in[\underline{\smash{u}},\overline{u}]$ there is some constant
$c_{3}>0$ such that for all $u\in[\underline{\smash{u}},\overline{u}]$ it
holds $K_{v}(u)\geq-c_{3}(\|\nabla u\|_{p}^{p-1}+1)$.

It is well-known that (K1) implies that $K_{v}$ is weakly sequentially lower
semicontinuous and that $\partial K_{v}$ (the subdifferential of $K_{v}$) is
maximal monotone. As an example, we have $K_{v}=I_{K(v)}$, the characteristic
functional of a non-empty closed convex set $K(v)\subset W$. In this case,
$\mathcal{D}(K_{v})=K(v)$ and (K2) becomes $K({v})\preccurlyeq
K({v^{\prime}})$ for $v\leq v^{\prime}$, while (K3) holds trivially.

######  4.9 Remark.

Note that we slightly changed the assumptions on the data given in [7]. In
particular, we do not assume that $(x,s)\mapsto f(x,s,t)$ is superpositionally
measurable, but only (by (F1) and (F2)) that $(x,s)\mapsto f(x,s,t)$ is weakly
superpositionally measurable. Furthermore, our assumption (K3) replaces the
technical assumptions (J3) and (J4) of [7] which is possible by separating
topological and order-theoretical properties. Furthermore, in order to present
the main ideas, we restrict our considerations to the space
$W^{1,p}_{0}(\Omega)$. At the cost of some slightly more involved calculation
we could treat also quasi-variational inclusions with non-zero boundary
conditions.

We are now in a position to check if all the conditions of Theorem 2.11 are
fulfilled with respect to $D:=[\underline{\smash{u}},\overline{u}]$, $V$ and
$W$ as given in this section, $S$ being the solution operator associated with
$A$, $K\equiv W$ and $T\equiv W$, and $\underline{\smash{S}}$ being the
solution operator associated with $\underline{A}$,
$\underline{\smash{K}}\equiv W$ and $\underline{T}$ as described in Definition
3.2. To be precise, in what follows we restrict both the domain and the values
of $S$ and $\underline{\smash{S}}$ to $D$, since we are only interested in
solutions between $\underline{\smash{u}}$ and $\overline{u}$.

First of all, $D$, $V$ and $W$ satisfy all required conditions, and we have
not only
$\underline{\smash{u}}\leq^{\ast}\underline{\smash{S}}(\underline{\smash{u}})$,
but even $\underline{\smash{u}}\in\operatorname{Fix}\underline{\smash{S}}$
(which will be relevant in the proof of Theorem 4.16). It thus remains to
provide the topological properties of $S$ and the order-theoretical properties
of $\underline{\smash{S}}$ and its connection to $S$. Let us start with the
properties of $S$ which are proved in a standard way as if we would have no
bifunctions:

######  4.10 Proposition.

All values of $S\colon D\to\mathcal{P}(D\cap W)$ are weakly compact (and thus
weakly closed) in $W$.

###### Proof.

Let $v\in D$ be fixed. We only have to prove that $S(v)$ is bounded and weakly
sequentially closed. So, let $u\in S(v)$ be arbitrary and fix some
$w\in\mathcal{D}(K_{v})$ (not depending on $u$). Then it holds, for some
$\eta\in F(u,v)$,

$\langle Eu,w\rangle+\int_{\Omega}\eta(w-u)+K_{v}(w)-K_{v}(u)\geq\langle
Eu,u\rangle.$ (10)

By (K3) we have $-K_{v}(u)\leq c_{3}(\|\nabla u\|_{p}^{p-1}+1)$ and we can
assume that also $K_{v}(w)\leq c_{3}$. Combining these estimates with those
from (A2)—(A3) and using Hölder’s inequality, we obtain from (10)

$\begin{split}(\alpha_{3}\|\nabla
u\|_{p}^{p-1}+\|a_{3}\|_{p^{\prime}})\|\nabla w\|_{p}\leavevmode\nobreak\
+\leavevmode\nobreak\
\|\eta\|_{{q^{\prime}}}(\|w\|_{{q}}+\|u\|_{{q}})\leavevmode\nobreak\
&+\leavevmode\nobreak\ c_{3}(\|\nabla u\|_{p}^{p-1}+2)\\\
&\geq\alpha_{2}\|\nabla u\|_{p}^{p}-\|e_{2}\|_{1}.\end{split}$ (11)

Now, $\|\nabla w\|_{p}$ and $\|w\|_{q}$ are constant, $\|\eta\|_{q^{\prime}}$
is bounded due to (F4), and we have
$\|u\|_{q}\leq\|\underline{u}\|_{q}+\|\overline{u}\|_{{q}}$, since $u\in D$.
Thus, (11) can not hold if $\|\nabla u\|_{p}$ is arbitrarily large, whence
$S(v)$ is bounded in $W$.

It remains to prove that $S(v)$ is weakly sequentially closed. To this end,
let $(u_{n})\subset S(v)$ and assume $u_{n}\rightharpoonup u$ for some $u\in
W$. From the continuous embedding $W\hookrightarrow V$ we obtain
$u_{n}\rightharpoonup u$ in $V$ (and thus $u\in D$) and from the compact
embedding $W\hookrightarrow L^{\ast}$ we obtain $u_{n}\to u$ in $L^{\ast}$.
Further, since $K_{v}$ is weakly sequentially lower semicontinuous due to
(K1), from $(u_{n})\subset\mathcal{D}(K_{v})$ we have
$u\in\mathcal{D}(K_{v})$. Now, let $(\eta_{n})\subset L$ be a sequence such
that $\eta_{n}\in F(u_{n},v)$ and

$\langle
Au_{n},w-u_{n}\rangle+\int_{\Omega}\eta_{n}(w-u_{n})+K_{v}(w)-K_{v}(u_{n})\geq
0\quad\text{for all }w\in W.$ (12)

We can assume $\eta_{n}\rightharpoonup\eta$ in $L$ for some $\eta\in V$ since
$(\eta_{n})$ is bounded in $L$ due to (F4), and thanks to Proposition 4.7 we
have $\eta\in F(u,v)$. Letting $w=u$ in (12) and passing to the limit yields,
using again that $K_{v}$ is weakly sequentially lower semicontinuous,

${\limsup}_{n}\langle Au_{n},u_{n}-u\rangle\leq 0.$

Since $A$ is pseudomonotone, we infer $\langle
Au,w-u\rangle\geq\limsup_{n}\langle Au_{n},w-u_{n}\rangle$ for all $w\in W$.
Using again (12), it follows, for all $w\in W$,

$\begin{split}\langle
Au,w-u\rangle&+\int_{\Omega}\eta(w-u)+K_{v}(w)-K_{v}(u)\\\
&\geq{\limsup}_{n}\langle
Au_{n},w-u_{n}\rangle+{\lim}_{n}\int_{\Omega}\eta_{n}(w-u_{n})+K_{v}(w)-{\liminf}_{n}K_{v}(u_{n})\\\
&\geq{\limsup}_{n}\left(\langle
Au_{n},w-u_{n}\rangle+\int_{\Omega}\eta_{n}(w-u_{n})+K_{v}(w)-K_{v}(u_{n})\right)\geq
0,\end{split}$

which proves $u\in S(v)$, thus $S(v)$ is weakly sequentially closed. ∎

Next, let us check that $\underline{\smash{S}}$ is permanent upward. This is
almost a purely order-theoretical proof, but note that it is vital for the
values of $F$ to be non-empty.

######  4.11 Proposition.

The operator $\underline{\smash{S}}\colon D\to\mathcal{P}(D\cap W)$ is
permanent upward.

###### Proof.

We apply Lemma 3.6. So, let $v\leq v^{\prime}$ in $D$ and $u\in S(v)$.

First, we have to prove
$\underline{\smash{A}}(u,v)\preccurlyeq^{\ast}\underline{\smash{A}}(u,v^{\prime})$,
where $\underline{\smash{A}}(u,v)=E(u)+i_{q}^{\ast}F(u,v)+K_{v}^{\downarrow}$.
We consider the summands separately:

  * (i)

Since $E(u)$ does not depend on $v$, there is nothing to do.

  * (ii)

Let $\eta\in F(u,v)$, let $\eta^{\prime}\in F(u,v^{\prime})$ be arbitrary and
set $\eta^{\prime\prime}:=\eta\wedge\eta^{\prime}\in L$. Further, let
$\alpha\colon\Omega\to\mathbb{R}$ be such that $\alpha\subset
f(\cdot,u,v^{\prime})$ and $\alpha\leq\eta$ (which exists due to (F3)).
$\alpha$ may not be measurable, but we have
$\alpha\wedge\eta^{\prime}\leq\eta^{\prime\prime}\leq\eta^{\prime}$. Now,
recall that all values of $f$ are closed. It thus follows
$\alpha\wedge\eta^{\prime}\subset f(\cdot,u,v^{\prime})$ and
$\eta^{\prime\prime}\subset f(\cdot,u,v^{\prime})$, i.e.
$\eta^{\prime\prime}\in F(u,v^{\prime})$. Further, from
$\eta^{\prime\prime}\leq\eta$ we obtain that
$i_{q}^{\ast}(\eta-\eta^{\prime\prime})$ is a positive linear functional such
that Proposition 3.7 implies $i_{q}^{\ast}\eta\preccurlyeq
i_{q}^{\ast}\eta^{\prime\prime}$.

  * (iii)

Let $k\in K_{v}^{\downarrow}$, i.e. $k\preccurlyeq K_{v}$. Then from (K2) we
have $K_{v}\preccurlyeq K_{v}\preccurlyeq K_{v^{\prime}}$ and Proposition 3.3
gives $k\preccurlyeq K_{v^{\prime}}$, i.e. $k\in K_{v^{\prime}}^{\downarrow}$.
That is, we even have $K_{v}^{\downarrow}\subset K_{v^{\prime}}^{\downarrow}$.

Second, $\underline{\smash{K}}(u,v)\subset\underline{\smash{K}}(u,v^{\prime})$
holds trivially since $\underline{K}(u,v)\equiv W$.

Third, for $u\in\underline{\smash{S}}(v)$ we have to prove that
$\underline{\smash{T}}(u,v^{\prime})\subset\underline{\smash{T}}(u,v)$, where
$\underline{\smash{T}}(u,v)=u\wedge\mathcal{D}(K_{v})$. To this end, let $k\in
K_{v}^{\downarrow}$ be such that $u\in\mathcal{D}(k)$, let
$w^{\prime}\in\mathcal{D}(K_{v^{\prime}})$ and let $w\in\mathcal{D}(K_{v})$ be
arbitrary. Then $u\wedge w^{\prime}=u\wedge((u\vee w)\wedge w^{\prime})\in
u\wedge\mathcal{D}(K_{v})$ since
$\mathcal{D}(k)\preccurlyeq\mathcal{D}(K_{v})\preccurlyeq\mathcal{D}(K_{v^{\prime}})$,
see Proposition 3.3.

Fourth, $\underline{\smash{T}}(u,v^{\prime})\leq^{\ast}u$ holds true by
definition of $\wedge$. ∎

It now remains to prove that all values of $\underline{\smash{S}}$ are
directed upward, and that it holds
$S(v)\subset\underline{\smash{S}}(v)\leq^{\ast}S(v)$. Of these properties, the
inclusion $S(v)\subset\underline{\smash{S}}(v)$ is easy to show due to the
variational definition of $\underline{\smash{S}}$:

######  4.12 Proposition.

For all $v\in D$ it holds $S(v)\subset\underline{\smash{S}}(v)$.

###### Proof.

We once again apply Lemma 3.6. To this end, let $v\in D$ and $u\in S(v)$. Then
from $K_{v}\preccurlyeq^{\ast}K_{v}^{\downarrow}$ it follows
$A(u,v)\preccurlyeq^{\ast}\underline{\smash{A}}(u,v)$. Further,
$\underline{\smash{T}}(u,v)\subset W$ as well as
$\underline{\smash{T}}(u,v)\leq^{\ast}u$ hold trivially. ∎

The remaining properties rely on sophisticated truncation arguments which
allow us to apply the following existence theorem for perturbed coercive
pseudomonotone multifunctions:

######  4.13 Theorem.

Let $X$ be a real reflexive Banach space, let $M\colon X\to\mathcal{P}(X)$ be
a maximal monotone mapping, suppose $M(u_{0})\neq\emptyset$, and let $T\colon
X\to\mathcal{P}(X^{\ast})$ be a bounded, pseudomonotone mapping having only
non-empty, closed and convex values. If $T$ is coercive with respect to
$u_{0}$, i.e. there exists a real-valued function $c$ on $\mathbb{R}^{+}$ with
$\lim_{s\to\infty}c(s)=\infty$ such that for all $u\in X$, $u^{\ast}\in T(u)$
one has $\langle u^{\ast},u-u_{0}\rangle\geq c(\|u\|)\|u\|$, then $M+T$ is
surjective, i.e. for all $y\in X^{\ast}$ there is $x\in X$ such that
$y\in(M+T)(x)$.

The proof of Theorem 4.13 together with further information is contained in,
e.g., [10]. We will skip the rather technical definition of multi-valued
pseudomonotonicity (but note that each single-valued pseudomonotone mapping is
also pseudomonotone as a multi-valued mapping, and that sums of pseudomonotone
mappings are pseudomonotone, too) and invoke [10, Proposition 2.2] to
formulate the following useful result:

######  4.14 Proposition.

Let $X$ be a real reflexive Banach space and let $T\colon
X\to\mathcal{P}(X^{\ast})$ be a multi-valued mapping satisfying the following
properties:

  * (i)

All values of $T$ are non-empty, closed and convex,

  * (ii)

$T$ is bounded, i.e. $T$ maps bounded sets to bounded ones,

  * (iii)

the graph $Gr(T)$ of $T$ is sequentially weakly closed, i.e. for all weakly
convergent sequences $u_{n}\rightharpoonup u$ in $X$ and
$u_{n}^{\ast}\rightharpoonup u^{\ast}$ in $X^{\ast}$ with $u_{n}^{\ast}\in
T(u_{n})$ one has $u^{\ast}\in T(u)$,

  * (iv)

the duality pairing is ($\boldsymbol{w\times w}$)-continuous on $Gr(T)$, i.e.
$u_{n}\rightharpoonup u$ in $X$ and $u_{n}^{\ast}\rightharpoonup u^{\ast}$ in
$X^{\ast}$ with $u_{n}^{\ast}\in T(u_{n})$ imply $\langle
u_{n}^{\ast},u_{n}\rangle\to\langle u^{\ast},u\rangle$.

Then $T$ is pseudomonotone.

Thanks to Proposition 4.14 we are able to show that upper Carathéodory
functions induce a pseudomonotone operator, which we will need to prove the
existence of solutions. The proof of the next lemma is contained in [5], thus
we keep it short. Note, however, that we do not need any knowledge about
Hausdorff upper semicontinuous multifunctions as in [8, Lemma 2.5] which
allows for a rather elementary approach.

######  4.15 Lemma.

Let $g\colon\Omega\times\mathbb{R}\to\mathcal{P}(\mathbb{R})$ be a
multifunction whose values are non-empty, closed and convex. Suppose further
that $g$ is upper Carathéodory, i.e. $x\mapsto g(x,s)$ is measurable for all
$s\in\mathbb{R}$ and $s\mapsto g(x,s)$ is upper semicontinuous for a.e.
$x\in\Omega$, and let there be some $b\in L$ such that $g$ satisfies, for a.e.
$x\in\Omega$,

$|y|\leq b(x)\quad\text{for all }y\in g(x,s)\text{, a.e }x\in\Omega\text{ and
all }s\in\mathbb{R}.$ (13)

Then the selection mapping

$G\colon W\to\mathcal{P}(W^{\ast}),\quad
G(u):=\\{i_{q}^{\ast}\eta:\eta\textrm{ is a measurable selection of
}g(\cdot,u)\\}$

is well-defined, pseudomonotone, bounded and has only non-empty, closed and
convex values.

###### Proof.

Let $u\in W$ be given. Due to the growth condition (13) we see that the
measurable selections of $g(\cdot,u)$ are uniformly bounded in $L$ independent
of $u$. Since $i_{q}^{\ast}\colon L\hookrightarrow W^{\ast}$ is a compact
embedding, we conclude that $G(u)$ is well-defined and uniformly bounded in
$W^{\ast}$, again independent of $u$. Further, $G(u)$ is convex and closed in
$W^{\ast}$ since $g$ has convex and closed values. Indeed, let
$(\eta_{n})\subset g(\cdot,u)$ be some sequence such that the sequence
$(i_{q}^{\ast}\eta_{n})$ converges in $W^{\ast}$. Since $g(\cdot,u)$ is
bounded in the reflexive space $L$, we conclude, up to a subsequence,
$\eta_{n}\rightharpoonup\eta$ in $L$ for some $\eta$, and due to Proposition
4.7 we have $\eta\subset g(\cdot,u)$. Because $i_{q}^{\ast}$ is linear and
bounded, it follows $i_{q}^{\ast}\eta_{n}\rightharpoonup i_{q}^{\ast}\eta$ in
$W^{\ast}$ and we infer that $G(u)$ is closed.

In order to apply Proposition 4.14, it remains to verify that $Gr(G)$ is
sequentially weakly closed and that the duality pairing is
(w$\times$w)-continuous on $Gr(G)$. To this end, let $(u_{n})\subset W$ and
$(i^{\ast}\eta_{n})\subset W^{\ast}$ be sequences such that $\eta_{n}\subset
g(\cdot,u_{n})$ for all $n$, and, for some $u\in W$, $\eta^{\ast}\in
W^{\ast}$, $u_{n}\rightharpoonup u$ in $W$ and
$i_{q}^{\ast}\eta_{n}\rightharpoonup\eta^{\ast}$ in $W^{\ast}$. By the compact
embedding $i_{q}$ and since $(\eta_{n})$ is bounded in $L$, we have, up to a
subsequence, $u_{n}\to u$ in $L^{\ast}$, $u_{n}\to u$ a.e. and
$\eta_{n}\rightharpoonup\eta$ in $L$ for some $\eta\in L$. We apply once more
Proposition 4.7 to conclude $\eta\subset g(\cdot,u)$. Since $i_{q}^{\ast}$ is
linear and bounded, we infer $i_{q}^{\ast}\eta_{n}\rightharpoonup
i_{q}^{\ast}\eta$, and thus $\eta^{\ast}=i_{q}^{\ast}\eta\in G(u)$. Further,
we apply the identity $\langle
i_{q}^{\ast}\theta,v\rangle=\langle\theta,v\rangle$ (for $\theta\in L$, $v\in
W$) and the triangle inequality to conclude

$0\leq|\langle\eta_{n}^{\ast},u_{n}\rangle-\langle\eta^{\ast},u\rangle|\leq\|\eta_{n}\|_{L}\|u_{n}-u\|_{L^{\ast}}+|\langle\eta_{n}-\eta,u\rangle|\to
0.$

Thus, $G$ fulfills the requirements of Proposition 4.14 and is thus
pseudomonotone. ∎

Note that (13) is a global growth condition while (F4) is only a local growth
condition. This is why we need some truncation procedure in the proof of the
next theorem.

######  4.16 Theorem.

Let $v\in D$ be arbitrary, and let
$\underline{\smash{v}}_{i}\in\underline{\smash{S}}(v)$ and
$\overline{v}_{i}\in\overline{S}(v)$, $i=1,2$, be such that

$\underline{\smash{u}}\leq\underline{\smash{v}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\underline{\smash{v}}_{1}\vee\underline{\smash{v}}_{2}\leq\overline{v}_{1}\wedge\overline{v}_{2}=\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}\overline{v}\leq\overline{u}.$

Then there is $u\in S(v)$ such that $\underline{\smash{v}}\leq
u\leq\overline{v}$. Especially, $S(v)$ is directed upward and it holds
$\underline{\smash{S}}(v)\leq^{\ast}S(v)$.

###### Proof.

Step 1: Auxiliary functions We are going to define an auxiliary problem whose
solutions will be the desired elements of $S(v)$. To this end, recall that, by
definition, for $i=1,2$ there are $\underline{\smash{\eta}}_{i}\in
F(\underline{\smash{v}}_{i},v)$ and $\underline{\smash{k}}_{i}\in\Gamma$ as
well as $\overline{\eta}_{i}\in F(\overline{v}_{i},v)$ and
$\overline{k}_{i}\in\Gamma$ such that we have the relations

$\underline{\smash{v}}_{i}\in\mathcal{D}(\underline{\smash{k}}_{i}),\quad\underline{\smash{k}}_{i}\preccurlyeq
K_{v}\preccurlyeq\overline{k}_{i},\quad\mathcal{D}(\overline{k}_{i})\ni\overline{v}_{i}$

and such that the following inequalities hold:

$\displaystyle\langle
E\underline{\smash{v}}_{i},w-\underline{\smash{v}}_{i}\rangle+\int_{\Omega}\underline{\smash{\eta}}_{i}(w-\underline{\smash{v}}_{i})+\underline{\smash{k}}_{i}(w)-\underline{\smash{k}}_{i}(\underline{\smash{v}}_{i})\geq
0,\quad\text{for all }w\in\underline{\smash{v}}_{i}\wedge\mathcal{D}(K_{v}),$
(14) $\displaystyle\langle
E\overline{v}_{i},w-\overline{v}_{i}\rangle+\int_{\Omega}\overline{\eta}_{i}(w-\overline{v}_{i})+\overline{k}_{i}(w)-\overline{k}_{i}(\overline{v}_{i})\geq
0,\quad\text{for all }w\in\overline{v}_{i}\wedge\mathcal{D}(K_{v}).$ (15)

Define the functions
$\underline{\smash{\eta}},\overline{\eta}\colon\Omega\to\mathbb{R}$ pointwise
a.e. by

$\underline{\smash{\eta}}(x):=\begin{cases}\underline{\smash{\eta}}_{1}(x)&\text{
if }\underline{\smash{v}}_{1}(x)\geq\underline{\smash{v}}_{2}(x),\\\
\underline{\smash{\eta}}_{2}(x)&\text{ if
}\underline{\smash{v}}_{1}(x)<\underline{\smash{v}}_{2}(x),\end{cases}\qquad\overline{\eta}(x):=\begin{cases}\overline{\eta}_{1}(x)&\text{
if }\overline{v}_{1}(x)\leq\overline{v}_{2}(x),\\\
\overline{\eta}_{2}(x)&\text{ if
}\overline{v}_{1}(x)>\overline{v}_{2}(x),\end{cases}$

and note that $\underline{\smash{\eta}}\in F(\underline{\smash{v}},v)$ and
$\overline{\eta}\in F(\overline{v},v)$. Further, we need three auxiliary
functions $d$, $g$ and $h$, which will be introduced subsequently.

First, let us introduce the cut-off function

$d\colon\Omega\times\mathbb{R}\to\mathbb{R},\quad
d(x,s):=\begin{cases}-(\underline{\smash{v}}(x)-s)^{p-1}&\text{ if
}s<\underline{\smash{v}}(x),\\\ 0&\text{ if }\underline{\smash{v}}(x)\leq
s\leq\overline{v}(x),\\\ (s-\overline{v}(x))^{p-1}&\text{ if
}\overline{v}(x)<s.\end{cases}$

Obviously, $d$ is a Carathéodory function and satisfies the growth condition

$|d(x,s)|\leq
d_{0}\left(|\underline{\smash{v}}(x)|^{p-1}+|s|^{p-1}+|\overline{v}(x)|^{p-1}\right)$
(16)

for some constant $d_{0}>0$. Hence, the Nemytskij operator $v\mapsto
d(\cdot,v)$ is known to be continuous and bounded from $L^{p}(\Omega)$ to its
dual space. Thanks to the compact embedding $W\hookrightarrow L^{p}(\Omega)$
we conclude that the mapping $D\colon W\to W^{\ast}$, defined by

$\langle Dv,w\rangle:=\int_{\Omega}d(\cdot,v)w\quad\text{for all }w\in W,$

is bounded and completely continuous, thus pseudomonotone. For further use,
let us note that there are constants $d_{1},d_{2}>0$ such that both
$-(t-s)^{p-1}s$ for $s\leq t$ and $(s-t)^{p-1}s$ for $s\geq t$ are bounded
from below by $d_{1}|s|^{p}-d_{2}|t|^{p-1}|s|$. One may take $d_{1}=1$ and
$d_{2}=2^{2-p}$ if $p\leq 2$ and $d_{1}=2^{2-p}$ and $d_{2}=1$ if $p\geq 2$.
By these estimates and by use of Young’s inequality with epsilon, we obtain
for all $v\in W$

$\langle Dv,v\rangle\geq
d_{3}\|v\|_{p}^{p}-d_{3}(\|\underline{\smash{v}}\|_{p}^{p}+\|\overline{v}\|_{p}^{p}),$
(17)

where $d_{3}>0$ is some constant not depending on $v$.

Second, let us truncate $f$ to define the multifunction

$g\colon\Omega\times\mathbb{R}\to\mathcal{P}(\mathbb{R}),\quad
g(x,s):=\begin{cases}\\{\underline{\smash{\eta}}(x)\,&\text{ if
}s<\underline{\smash{v}}(x),\\\ f(x,s,v(x))&\text{ if
}\underline{\smash{v}}(x)\leq s\leq\overline{v}(x),\\\
\\{\overline{\eta}(x)\\}&\text{ if }\overline{v}(x)<s.\end{cases}$

From (F1) and (F2) we have that $(x,s)\mapsto f(x,s,v(x))$ is upper
Carathéodory and it is readily seen (by invoking
$\underline{\smash{\eta}}\subset f(\cdot,\underline{\smash{v}},v)$ and
$\overline{\eta}\subset f(\cdot,\overline{v},v)$) that $g$ is upper
Carathéodory, too. Since $g$ has only non-empty, closed and convex values, and
due to (F4) and Lemma 4.15, the mapping

$G\colon W\to\mathcal{P}(W^{\ast}),\quad
G(u):=\\{i_{q}^{\ast}\eta:\eta\textrm{ is a measurable selection of
}g(\cdot,u)\\}$

is well-defined, pseudomonotone, bounded and its values are non-empty, closed
and convex.

Third, in order to define $h$, let us introduce the following notation: For
any real numbers $x_{1}<x_{2}$ and $y_{1},y_{2}$, denote by

$l=\big{[}(x_{1},y_{1})\rightsquigarrow(x_{2},y_{2})\big{]}$

the continuous piecewise linear function $l\colon\mathbb{R}\to\mathbb{R}$ that
satisfies $l(x)=y_{1}$ for $x\leq x_{1}$,
$l^{\prime}(x)=(y_{2}-y_{2})/(x_{2}-x_{1})$ for $x_{1}<x<x_{2}$, and
$l(x)=y_{2}$ for $x\geq x_{2}$.

This said, for $i=1,2$ we introduce the functions

$\displaystyle\underline{\smash{\theta}}_{i}\colon\Omega\times\mathbb{R}\to\mathbb{R},\quad\underline{\smash{\theta}}_{i}(x,\cdot):=\big{[}\big{(}\underline{\smash{v}}_{i}(x),\underline{\smash{\eta}}(x)-\underline{\smash{\eta}}_{i}(x)\big{)}\rightsquigarrow\big{(}\underline{\smash{v}}(x),0\big{)}\big{]},$
$\displaystyle\overline{\theta}_{i}\colon\Omega\times\mathbb{R}\to\mathbb{R},\quad\overline{\theta}_{i}(x,\cdot):=\big{[}\big{(}\overline{v}(x),0\big{)}\rightsquigarrow\big{(}\overline{v}_{i}(x),\overline{\eta}_{i}(x)-\overline{\eta}(x)\big{)}\big{]}.$

It is easy to check that $\underline{\smash{\theta}}_{i}$ and
$\overline{\theta}_{i}$ are measurable in the first argument and, of course,
continuous in the second. Let us combine them to form the Carathéodory
function

$h\colon\Omega\times\mathbb{R}\to\mathbb{R},\quad
h(x,s):=|\underline{\smash{\theta}}_{1}(x,s)|+|\underline{\smash{\theta}}_{2}(x,s)|-|\overline{\theta}_{1}(x,s)|-|\overline{\theta}_{2}(x,s)|.$

We have constructed $h$ in such a way that, for all $u\in L^{0}(\Omega)$,
$i=1,2$, the inequalities

$\underline{\smash{\eta}}-\underline{\smash{\eta}}_{i}-h(\cdot,u)\leq
0\quad\text{on
}\\{u<\underline{\smash{v}}_{i}\\},\qquad\overline{\eta}_{i}-\overline{\eta}+h(\cdot,u)\leq
0\quad\text{on }\\{\overline{v}_{i}<u\\}$ (18)

hold true. Indeed, if, e.g.,
$v(x)<\underline{\smash{v}}_{i}(x)\leq\underline{\smash{v}}(x)$ for some
$x\in\Omega$, then we obtain per definition
$\underline{\smash{\theta}}_{i}(x,v(x))=\underline{\smash{\eta}}(x)-\underline{\smash{\eta}}_{i}(x)$
and $\overline{\theta}_{i}(x,v(x))=0$, $i=1,2$, which implies the first
inequality in (18). Moreover, $h(x,s)=0$ if $\underline{\smash{v}}(x)\leq
s\leq\overline{v}(x)$.

Further, $h(x,\cdot)$ is obviously bounded by some function $\eta\in L$, so
that the Nemytskij operator $v\mapsto h(\cdot,v)$ is known to be a continuous
and bounded mapping from $L$ to $L^{\ast}$. Thanks to the compact embedding
$W\hookrightarrow L$ we conclude that the composed mapping $H\colon W\to
W^{\ast}$, defined by

$\langle Hv,w\rangle:=\int_{\Omega}h(\cdot,v)w\quad\text{for all }w\in W,$

is bounded and completely continuous, thus pseudomonotone.

Step 2: Solutions of auxiliary problem Let us consider the multi-valued
mapping

$E+D+G-H+\partial K_{v}\colon W\to\mathcal{P}(W^{\ast}).$

It is the sum of the bounded, pseudomonotone single-valued mappings $E$, $D$
and $-H$, the bounded, pseudomonotone mapping $G$ with non-empty, closed and
convex values, and the maximal monotone mapping $\partial K_{v}$. Further, $E$
is coercive with respect to any $u_{0}\in\mathcal{D}(\partial K_{v})$, since
by (E2)—(E3) we have for all $u,u_{0}\in W$

$\langle Eu,u-u_{0}\rangle\geq\alpha_{2}\|\nabla
u\|_{p}^{p}-\|a_{2}\|_{1}-(\alpha_{3}\|\nabla
u\|_{p}^{p-1}+\|a_{3}\|_{p^{\prime}})\|\nabla u_{0}\|_{p},$ (19)

which implies

$\frac{\langle
Eu,u-u_{0}\rangle}{\|u\|_{W}}\to\infty\quad\text{as}\quad\|u\|_{W}\to\infty$

(note again that $\|\nabla u\|_{p}$ defines an equivalent norm on $W$).
Further, for all $u,u_{0}\in W$ and any $\eta\subset g(\cdot,u)$ we have,
since the selections of $g(\cdot,u)$ and $Hu$ are uniformly bounded,

$\langle i^{\ast}\eta-Hu,u-u_{0}\rangle\geq-\|\eta-
Hu\|_{L}\|u-u_{0}\|_{L^{\prime}}\geq-c_{0}\|u\|_{W}-c_{1}$ (20)

for some constants $c_{0},c_{1}>0$. In addition, estimates (16) and (17) imply

$\langle Du,u-u_{0}\rangle\geq-
d_{3}(\|\underline{\smash{v}}\|_{p}^{p}+\|\overline{v}\|_{p}^{p})-d_{0}(\|\underline{\smash{v}}\|_{p}^{p-1}+\|u\|_{p}^{p-1}+\|\overline{v}\|_{p}^{p-1})\geq
d_{4}-d_{5}\|u\|_{W}^{p-1}$ (21)

for constants $d_{4},d_{5}>0$. From (19), (20) and (21) it follows that
$E+D+G-H\colon W\to\mathcal{P}(W^{\ast})$ is coercive with respect to $u_{0}$.

Taken together, all conditions of Theorem 4.13 are fulfilled, and thus there
is some $u\in W$ such that $(E+D+G-H+\partial K_{v})(u)\ni 0$. By definition
of the subdifferential this implies $u\in\mathcal{D}(K_{v})$ and that there is
$\eta\subset g(\cdot,u)$ such that, for all $w\in W$,

$\langle
Eu,w-u\rangle+\int_{\Omega}d(\cdot,u)(w-u)+\int_{\Omega}\left(\eta-h(\cdot,u)\right)(w-u)+K_{v}(w)-K_{v}(u)\geq
0.$ (22)

Step 3: Solution of the QVIP In this last step, let us check that for any
$u\in W$ and $\eta\subset g(\cdot,u)$ with (22) we have
$\underline{\smash{v}}\leq u\leq\overline{v}$. To this end, for $i=1,2$, take

$w=\underline{\smash{v}}_{i}-(\underline{\smash{v}}_{i}-u)^{+}=\underline{\smash{v}}_{i}\wedge
u\in\underline{\smash{v}}_{i}\wedge\mathcal{D}(K_{v})$

as test function in (14), take

$w=u+(\underline{\smash{v}}_{i}-u)^{+}=\underline{\smash{v}}_{i}\vee
u\in\mathcal{D}(\underline{\smash{k}}_{i})\vee\mathcal{D}(K_{v})\subset\mathcal{D}(K_{v})$

as test function in (22), and add the resulting inequalities to obtain

$\begin{split}\langle
Eu-E\underline{\smash{v}}_{i},(\underline{\smash{v}}_{i}-u)^{+}\rangle+\int_{\Omega}d(\cdot,u)(\underline{\smash{v}}_{i}-u)^{+}+\int_{\Omega}\left(\eta-\underline{\smash{\eta}}_{i}-h(\cdot,u)\right)(\underline{\smash{v}}_{i}-u)^{+}&\\\
+K_{v}(\underline{\smash{v}}_{i}\vee
u)-K_{v}(u)+\underline{\smash{k}}_{i}(\underline{\smash{v}}_{i}\wedge
u)-\underline{\smash{k}}_{i}(\underline{\smash{v}}_{i})&\geq 0.\end{split}$
(23)

By (E4) and the identity
$\nabla(\underline{\smash{v}}_{i}-u)^{+}=\chi_{\\{\underline{\smash{v}}_{i}\geq
u\\}}\nabla(\underline{\smash{v}}_{i}-u)$ (where $\chi_{M}$ denotes the
indicator function of a set $M$) we deduce

$\langle
Eu-E\underline{\smash{v}}_{i},(\underline{\smash{v}}_{i}-u)^{+}\rangle\leq 0.$
(24)

Further, it follows from (18)

$\int_{\Omega}\left(\eta-\underline{\smash{\eta}}_{i}-h(\cdot,u)\right)(\underline{\smash{v}}_{i}-u)^{+}\leq
0,$ (25)

and $\underline{\smash{v}}_{i}\in\mathcal{D}(\underline{\smash{k}}_{i})$,
$u\in\mathcal{D}(K_{v})$ and $\underline{\smash{k}}_{i}\preccurlyeq K_{v}$
imply

$K_{v}(\underline{\smash{v}}_{i}\vee
u)-K_{v}(u)+\underline{\smash{k}}_{i}(\underline{\smash{v}}_{i}\wedge
u)-\underline{\smash{k}}_{i}(\underline{\smash{v}}_{i})\leq 0.$ (26)

Combining (23)—(26), we deduce

$\int_{\Omega}d(\cdot,u)(\underline{\smash{v}}_{i}-u)^{+}\geq 0.$ (27)

By definition of $d$, (27) implies
$\|(\underline{\smash{v}}_{i}-u)^{+}\|_{p}^{p}\leq 0$, which in turn implies
$\underline{\smash{v}}_{i}\leq u$, $i=1,2$. Since
$\underline{\smash{v}}=\underline{\smash{v}}_{1}\vee\underline{\smash{v}}_{2}$,
we have $\underline{\smash{v}}\leq u$.

In a similar way we conclude $u\leq\overline{v}$ by taking
$w=\overline{v}_{i}+(u-\overline{v}_{i})^{+}=\overline{v}_{i}\vee u$ as test
function in (15) and $w=u-(u-\overline{v}_{i})^{+}=u\wedge\overline{v}_{i}$ as
test function in (22).

In view of $\underline{\smash{v}}\leq u\leq\overline{v}$, (22) reduces to

$\langle Eu,w-u\rangle+\int_{\Omega}\eta(w-u)+K_{v}(w)-K_{v}(u)\geq
0\quad\text{for all }w\in W,$

and since $\eta\subset g(\cdot,u)=f(\cdot,u,v)$, we have $u\in S(v)$. ∎

Finally, existence of greatest solutions follows from Theorem 2.11, and since
all results remain true if we consider the dual order $\geq$, we obtain also a
smallest solution within the order interval
$[\underline{\smash{u}},\overline{u}]$:

######  4.17 Theorem.

Suppose (S), (E1)—(E4), (F1)—(F5) and (K1)—(K3). Then Problem (9) has both a
smallest and a greatest solution in $[\underline{u},\overline{u}]$.

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