Leray-Lions operators perturbed by state-dependent subdifferentials

SIEGFRIED CARL ^{∗}

Martin-Luther-Universität Halle-Wittenberg Fachbereich Mathematik und Informatik, Institut für Analysis 06099 Halle, Germany

Abstract: This paper deals with the Dirichlet problem for differential inclusions governed by an elliptic operator of Leray-Lions type and a multivalued term in form of a state-dependent subdifferential. The enclosure of weak solutions is proved by assuming appropriately defined upper and lower solutions. To this end the problem under consideration is shown to be equivalent to some quasi-variational inequality for which the existence of solutions lying between upper and lower solution is proved. Moreover, it can be shown that the solution set enclosed by upper and lower solution possesses extremal solutions with respect to the underlying partial ordering.

AMS (MOS) subject classification. 35B05, 35J65, 35R70, 49J40

1. Introduction

This note deals with existence results for quasilinear elliptic differential inclusions under Dirichlet boundary conditions of the form

Au + β(u, u) ∋ 0 in Ω ⊂ R^{N} ,  u = 0 on ∂Ω ,  (1.1)

in an open bounded domain Ω with Lipschitz boundary ∂Ω, where

Au = − ∑_{i=1}^{N} \frac{∂}{∂x_{i}} a_{i}(·, u, ∇u)

is an operator of Leray-Lions type from W_{0}^{1,p}(Ω) into W^{−1,q}(Ω) with 1 < p < ∞ and 1/p + 1/q = 1 , and β(u, ·) : R ↦ 2^{R} \ ∅ is a maximal monotone graph (m.m.g.) in R^{2} that, in addition, depends on the solution itself, and which is generated by some function f : R × R ↦ R.

∗The work on this paper was supported by a grant from the Deutsche Akademie der Naturforscher LEOPOLDINA

Various free boundary problems can be reduced to boundary value problems (BVP) with discontinuous nonlinearities which in turn can successfully be treated in a multi-valued setting such as given by the inclusion (1.1), cf.[1, 7, 10].

Our aim is to prove the existence of weak solutions of (1.1) lying between assumed appropriately defined upper and lower solutions. To this end the problem under consideration is shown to be equivalent to some quasi-variational inequality of the form

< Au, φ − u > +J(u, φ) − J(u, u) ≥ 0 .  (1.2)

By applying truncation and comparison techniques the quasi-variational inequality (1.2) can be solved using fixed point arguments.

Finally the solution set enclosed by upper and lower solution is proved to be directed which implies by means of Zorn’s lemma the existence of extremal solutions, i.e., the greatest and smallest solution in the sense of the underlying partial ordering.

2. Notations and hypotheses

Let V = W^{1,p}(Ω) and V_{0} = W_{0}^{1,p} denote the usual Sobolev spaces, and V^{∗} and V_{0}^{∗} = W^{−1,q}(Ω) , 1/p + 1/q = 1, their corresponding dual spaces, with the duality pairing between them denoted by < ·, · >. We introduce the natural partial ordering in L^{p}(Ω), that is u ≤ w if and only if w−u belongs to the set L_{+}^{p}(Ω) of all nonnegative elements of L^{p}(Ω), which induces also a partial ordering in the Sobolev space V. If u, w ∈ V and u ≤ w then

[u, w] = {v ∈ V | u ≤ v ≤ w}

denotes the order interval formed by u and w.

On the coefficients a_{i} (i = 1, ⋯, N) of the differential operator A we impose the following conditions of Leray-Lions type.

(A1) Each a_{i} : Ω × R × R^{N} ↦ R satisfies Carathéodory conditions, i.e., a_{i}(x, t, ξ) is measurable in x ∈ Ω for all (t, ξ) ∈ R × R^{N} and continuous in (t, ξ) for almost every (a.e.) x ∈ Ω. There exist a constant c_{0} > 0 and a function k_{0} ∈ L^{q}(Ω) , 1/p + 1/q = 1, such that

|a_{i}(x, t, ξ)| ≤ k_{0}(x) + c_{0}(|t|^{p−1} + |ξ|^{p−1}) ,

for a.e. x ∈ Ω and for all (t, ξ) ∈ R × R^{N}.

(A2)

∑_{i=1}^{N}(a_{i}(x, t, ξ) − a_{i}(x, t, ξ′)(ξ_{i} − ξ′_{i}) ≥ μ|ξ − ξ′|^{p}

for a.e. x ∈ Ω , for all t ∈ R, and for all ξ, ξ′ ∈ R^{N} with μ being some positive constant.

Let a denote the semilinear form associated with the differential operator A by

a(u, φ) = ∫_{Ω} ∑_{i=1}^{N} a_{i}(x, u, ∇u)\frac{∂φ}{∂x_{i}} dx ,

which due to (A1) is well defined on V × V.

Definition 2.1. A function u ∈ V_{0} is called a solution of the BVP (1.1) if there exists a function v ∈ L^{q}(Ω) such that

(i) v(x) ∈ β(u(x), u(x)) for a.e. x ∈ Ω,

(ii) a(u, φ) + ∫_{Ω} vφ dx = 0 for all φ ∈ V_{0}.

Definition 2.2. A function ū ∈ V is called an upper solution to the BVP (1.1) if there exists a function v̄ ∈ L^{q}(Ω) such that

(i) v̄(x) ∈ β(ū(x), ū(x)) for a.e. x ∈ Ω,

(ii) ū ≥ 0 on ∂Ω,

(iii) a(ū, φ) + ∫_{Ω} v̄φ dx ≥ 0 , for all φ ∈ V_{0} ∩ L_{+}^{p}(Ω).

A lower solution \underline{u} is defined similarly by reversing the inequality signs in (ii) and (iii) of definition 2.2. We shall make the following hypotheses.

(H1) Let there exist an upper and a lower solution ū and \underline{u} of the BVP (1.1), respectively, such that \underline{u} ≤ ū.

(H2) Let f : R × R ↦ R be a function satisfying

(i) the function r ↦ f(r, s) is equicontinuous with respect to s ∈ R, i.e., for any ε > 0 there is a δ(ε, r_{0}) independent on s such that

|f(r, s) − f(r_{0}, s)| < ε whenever |r − r_{0}| < δ(ε, r_{0}) ;

(ii) the function s ↦ f(r, s) is nondecreasing for each r ∈ R (possibly discontinuous), and is related with the m.m.g. β by

β(r, s) = [f(r, s − 0), f(r, s + 0)] ,  (2.1)

where the closed interval on the right-hand side of (2.1) is formed by the one-sided limits of f, i.e.,

f(r, s ± 0) = lim_{ε↓0} f(r, s ± ε) .

(H3) There is a function k_{1} ∈ L_{+}^{q}(Ω) and a constant α > 0 such that

|f(r, s)| ≤ k_{1}(x) ,

for all r ∈ [\underline{u}(x), ū(x)] and s ∈ [\underline{u}(x) − α, ū(x) + α] .

Remark 2.1. A function f that is, in addition, also dependent on the space variable x may also be taken into account which, however, has been omitted for the sake of simplicity. Remark 2.2. Hypothesis (H2) implies that the the function f is superpositionally measurable, i.e., whenever u, v : Ω ↦ R are measurable functions then the Nemytskij operator F defined by F (u, v)(x) := f(u(x), v(x)) generates a measurable function. Furthermore, by hypotheses (H3) the mapping u ↦ F (u, v) is equicontinuous and bounded from [\underline{u}, ū] ⊂ L^{p}(Ω) to L^{q}(Ω) with respect to v ∈ [\underline{u} − α, ū + α] . Remark 2.3. One possibility to determine upper and lower solutions to the inclusion (1.1) is to replace the problem by an BVP of the form

Au + f̃(u, u) = 0 in Ω , u = 0 on ∂Ω ,  (2.2)

where f̃: R × R ↦ R is any single-valued section of the m.m.g. β. Then obviously any upper (lower) solution of (2.2) is also an upper (lower) solution of (1.1).

For any constant k ≥ 0 we introduce the truncation mapping T_{k} by

T_{k}u(x) = \{ \begin{matrix}\overline{u}(x) + k if & u(x) > \overline{u}(x) + k \\ u(x) if & \underline{u}(x) − k ≤ u(x) ≤ \overline{u}(x) + k \\ \underline{u}(x) − k if & u(x) < \underline{u}(x) − k . \end{matrix}

It is well known that the truncation mapping T_{k} is bounded and continuous from V to itself (cf. [9]), which implies in particular that T_{k} : L^{p}(Ω) ↦ L^{p}(Ω) is bounded and continuous.

For constants l, k ≥ 0 we define composed operators F_{l,k} by

F_{l,k}(w, u)(x) := f_{l,k}(w(x), u(x)) , with f_{l,k}(w(x), u(x)) := f(T_{l}w(x), T_{k}u(x)) ,

and consider the following functional

J(w, u) = ∫_{Ω} ∫_{0}^{u(x)} f_{0,α}(w(x), s) ds dx ,  (2.3)

where α > 0 is given in hypothesis (H3). Then hypotheses (H2) and (H3) ensure that the functional

J : L^{p}(Ω) × L^{p}(Ω) ↦ R

is well defined.

Further, for any given w ∈ L^{p}(Ω) we get an estimate of the form

|J(w, u_{1}) − J(w, u_{2})| ≤ ∫_{Ω} k_{1}(x)|u_{1}(x) − u_{2}(x)|dx ≤ ∥k_{1}∥_{L^{q}(Ω)}∥u_{1} − u_{2}∥_{L^{p}(Ω)} . (2.4)

The function f_{0,α}(w(x), s) : Ω×R ↦ R is superpositionally measurable, nondecreasing in s and satisfies in view of (H3)

|f_{0,α}(w(x), s)| ≤ k_{1}(x) .

Hence the functional J(w, ·) : L^{p}(Ω) ↦ R is convex (cf.[8]) and by (2.4) Lipschitz continuous which implies the existence of the subdifferential of J with respect to its second argument denoted by ∂_{2}J(w, u) . Applying Lemma 2.1 and Theorem 2.3 due to Chang (cf.[8]) in a slightly modified form we obtain

∂_{2}J(w, u)(x) = [f_{0,α}(w(x), u(x) − 0), f_{0,α}(w(x), u(x) + 0)] ,  (2.5)

which holds in L^{p}(Ω) and V as well.

3. Existence result

We associate to the BVP (1.1) the following quasi-variational inequality (QVI): Find u ∈ V_{0} such that

a(u, φ − u) + J(u, φ) − J(u, u) ≥ 0  (3.1)

for all φ ∈ V_{0}, and with the functional J given by (2.3).

Lemma 3.1. Let hypotheses (A1), (A2) and (H1)-(H3) be satisfied. Then u ∈ [\underline{u}, \overline{u}] is a solution of the BVP (1.1) if and only if it is a solution of the QVI (3.1).

Proof: a) Let u ∈ [\underline{u}, \overline{u}] be a solution of the QVI (3.1) which means by definition of ∂_{2}J(u, u) that u satisfies the inclusion

− Au ∈ ∂_{2}J(u, u) in V_{0}^{∗} .  (3.2)

Since by (2.5) ∂_{2}J(u, u) ⊂ L^{q}(Ω) , there is a function v ∈ L^{q}(Ω) satisfying

v(x) ∈ [f_{0,α}(u(x), u(x) − 0), f_{0,α}(u(x), u(x) + 0)]  (3.3)

such that

−Au = v in V_{0}^{∗} ,

or equivalently

a(u, φ) + ∫_{Ω} vφ dx = 0  (3.4)

for all φ ∈ V_{0}. Further, since u ∈ [\underline{u}, \overline{u}] we get T_{0}u = u and T_{α}(u ± ε) = u ± ε for any ε ∈ (0, α) which implies

[f_{0,α}(u(x), u(x) − 0), f_{0,α}(u(x), u(x) + 0)] = [f(u(x), u(x) − 0), f(u(x), u(x) + 0)] .

(3.5)

which is equal to β(u(x), u(x)). Thus from (3.3), (3.4) and (3.5) it follows that u is a solution of the BVP (1.1).

b) Let u ∈ [\underline{u}, \overline{u}] be a solution of the BVP (1.1).Then there is a function v ∈ L^{q}(Ω) such that (3.3),(3.4) and (3.5) is satisfied. Since f_{0,α}(u(x), s) : Ω × R ↦ R is superpositionally measurable, nondecreasing in s and uniformly bounded due to (H3), the following inequality holds for a.e. x ∈ Ω and for any φ ∈ L^{p}(Ω):

v(x) (φ(x) − u(x)) ≤ ∫_{u(x)}^{φ(x)} f_{0,α}(u(x), s) ds ,  (3.6)

which yields by integration over Ω

∫_{Ω} v(φ − u)dx ≤ J(u, φ) − J(u, u)  (3.7)

for all φ ∈ L^{p}(Ω) , and in particular also for all φ ∈ V_{0}. Replacing φ ∈ V_{0} in (3.4) by φ − u we get in view of (3.7)

a(u, φ − u) + J(u, φ) − J(u, u) ≥ 0 ,

for all φ ∈ V_{0}, which completes the proof of the lemma.

The main result of this section is Theorem A. Let the hypotheses of lemma 3.1 be satisfied. Then the BVP (1.1) admits at least one solution u with \underline{u} ≤ u ≤ ū. Proof: We define an operator S that assigns to each w ∈ L^{p}(Ω) the solution u of the following auxiliary variational inequality:

Find u ∈ V_{0} such that

a_{w}(u, φ − u) + J(w, φ) − J(w, u) ≥ 0  (3.8)

for all φ ∈ V_{0}, where a_{w} is the semilinear form defined by

a_{w}(u, φ) = ∫_{Ω} ∑_{i=1}^{N} a_{i}(·, T_{0}w, ∇u)\frac{∂φ}{∂x_{i}} dx .

By hypotheses (A1) and (A2) the operator A_{w} : V_{0} ↦ V_{0}^{∗} associated with the semilinear form a_{w} is continuous, bounded, and uniformly monotone and thus in particular coercive. The functional J(w, ·) : V_{0} ↦ R is convex and even Lipschitz continuous. Hence the variational inequality (3.8) possesses a uniquely defined solution (cf.[14, Chapter 54]) which means that the operator S : L^{p}(Ω) ↦ V_{0} is well defined. Let c > 0 be some generic constant. From (3.8) with φ = 0 and using hypothesis (H3) we obtain

a_{w}(u, u) ≤ −J(w, u) ≤ ∥k_{1}∥_{L^{q}(Ω)}∥u∥_{L^{p}(Ω)} ,

which yields by means of (A2)

μ∥u∥_{V_{0}}^{p} ≤ c (∥k_{1}∥_{L^{q}(Ω)} + ∑_{i=1}^{N} ∥a_{i}(·, T_{0}w, 0)∥_{L^{q}(Ω)})∥u∥_{V_{0}} .  (3.9)

By (A1) we have

∥a_{i}(·, T_{0}w, 0)∥_{L^{q}(Ω)} ≤ c (∥k_{0}∥_{L^{q}(Ω)} +∥T_{0}w∥_{L^{p}(Ω)}^{p−1}) ≤ c (∥k_{0}∥_{L^{q}(Ω)} +∥ū∥_{L^{p}(Ω)}^{p−1} +∥\underline{u}∥_{L^{p}(Ω)}^{p−1})

which yields in view of (3.9) and 1 < p < ∞ a uniform bound for the range of S, i.e.,

∥u∥_{V_{0}} ≤ c  (3.10)

for all w ∈ L^{p}(Ω). This together with the compact imbedding V_{0} ⊂ L^{p}(Ω) shows that S provides a compact mapping of a ball B_{R} ⊂ L^{p}(Ω) into itself for sufficiently large radius R. In order to show the continuity of S let u = Sw and u_{0} = Sw_{0} . Taking in (3.8) as special test function φ = u_{0} for the solution u and φ = u for the solution u_{0} and adding the resulting inequalities we obtain

a_{w}(u, u − u_{0}) − a_{w}(u_{0}, u − u_{0}) ≤ a_{w_{0}}(u_{0}, u − u_{0}) − a_{w}(u_{0}, u − u_{0})

+ ∫_{Ω} ∫_{u(x)}^{u_{0}(x)}(f_{0,α}(w(x), s) − f_{0,α}(w_{0}(x), s)) ds dx , which results in

μ∥u − u_{0}∥_{V_{0}}^{p} ≤ c ∑_{i=1}^{N} ∥a_{i}(·, T_{0}w_{0}, ∇u_{0}) − a_{i}(·, T_{0}w, ∇u_{0})∥_{L^{q}(Ω)}∥u − u_{0}∥_{V_{0}}

+(∫_{0}^{1} ∥F_{0,α}(w, s(t)) − F_{0,α}(w_{0}, s(t))∥_{L^{q}(Ω)} dt)∥u − u_{0}∥_{L^{p}(Ω)} , (3.11)

where the following substitution has been used for the second term on the right-hand side of (3.11)

s(t)(x) = tu_{0}(x) + (1 − t)u(x) , t ∈ [0, 1] .

The equicontinuity of w ↦ F_{0,α}(w, v) (cf. Remark 2.2) and the continuity of w ↦ a_{i}(·, T_{0}w, ∇u_{0}) which follows by (A1) imply the continuity of the operator S in view of (3.11). Thus Schauder’s fixed point theorem can be applied which ensures the existence of a fixed point of the operator S, i.e., there exists a u ∈ V_{0} such that

a_{u}(u, φ − u) + J(u, φ) − J(u, u) ≥ 0 ,  (3.12)

for all φ ∈ V_{0}. To complete the proof we show that any solution of (3.12) belongs to the interval [\underline{u}, ū] since then T_{0}u = u and a_{u}(u, φ) = a(u, φ), and thus u is a solution of the BVP (1.1) in view of lemma 3.1. We prove that any solution of (3.12) satisfies u ≤ ū. The proof for u ≥ \underline{u} can be carried out in a similar way.

The upper solution ū of the BVP (1.1) satisfies by definition ū ≥ 0 on ∂Ω and

a(ū, φ) + ∫_{Ω} v̄φ dx ≥ 0  (3.13)

for all φ ∈ V_{0} ∩ L_{+}^{p}(Ω), where v̄ ∈ β(ū, ū). This latter inclusion implies (cf. (3.5), (3.6))

∫_{Ω} v̄(φ − ū) dx ≤ J(ū, φ) − J(ū, ū) ,  (3.14)

for all φ ∈ L^{p}(Ω). Let w^{+} = max(w, 0). Then by means of (3.14) we get from (3.13)

a(ū, (u − ū)^{+}) + J(ū, ū + (u − ū)^{+}) − J(ū, ū) ≥ 0 ,  (3.15)

where we have taken into account that (u − ū)^{+} ∈ V_{0} ∩ L_{+}^{p}(Ω). Taking in (3.12) the special test function φ = u − (u − ū)^{+} we obtain by means of (3.15)

a_{u}(u, (u − ū)^{+}) − a(ū, (u − ū)^{+}) ≤
J(ū, ū + (u − ū)^{+}) − J(ū, ū) + J(u, u − (u − ū)^{+}) − J(u, u) . (3.16)

By means of the definition of the functional J one can show by elementary means that the right-hand side of (3.16) is equal to zero. Thus we obtain

0 ≥ a_{u}(u, (u − ū)^{+}) − a(ū, (u − ū)^{+})
= ∫_{Ω} ∑_{i=1}^{N}(a_{i}(·, T_{0}u, ∇u) − a_{i}(·, ū, ∇ū))\frac{∂(u − ū)^{+}}{∂x_{i}}
≥ μ∥∇(u − ū)^{+}∥_{L^{p}(Ω)}^{p} ,

which implies (u − ū)^{+} = 0 ,i.e., u ≤ ū . This completes the proof of Theorem A.

4. Existence of extremal solutions

Let P denote the set of all solutions of the BVP (1.1) enclosed by the upper and lower solution ū and \underline{u}, respectively. Due to theorem A we have P ≠ ∅. In this section we are going to prove that P possesses extremal solutions. For this purpose an additional hypothesis is imposed on the coefficients a_{i}. We assume

(A3) |a_{i}(x, t, ξ)−a_{i}(x, t′, ξ)| ≤ [k_{2}(x)+|t|^{p−1}+|t′|^{p−1}+|ξ|^{p−1}] ω(|t−t′|) , i = 1, ⋯, N, for some function k_{2} ∈ L^{q}(Ω), for a.e. x ∈ Ω , for all t, t′ ∈ R and for all ξ ∈ R^{N} , where ω : [0, ∞) ↦ [0, ∞) is the modulus of continuity satisfying

∫_{0^{+}} \frac{dr}{ω^{q}(r)} = +∞ .  (4.1)

Remark 4.1. Hypothesis (A3) is satisfied for example in case that ω(|t − t′|) = c |t − t′|^{1/q} with some positive constant c, i.e., the coefficients a_{i}(x, t, ξ) satisfy a Hölder condition with respect to t.

Lemma 4.1. Let hypotheses (A1) - (A3) and (H1) - (H3) be satisfied. If u_{1}, u_{2} ∈ P then max(u_{1}, u_{2}) ∈ [\underline{u}, ū] is a lower solution of the BVP (1.1).

Proof: The proof adapts a special test function technique used in [6] combined with an approach given by Puel in [12].

Let u = max(u_{1}, u_{2}). We define a function v : Ω ↦ R by

v(x) = \{ \begin{matrix}v_{1}(x) if & x ∈ \{u_{1} ≥ u_{2}\} , \\ v_{2}(x) if & x ∈ \{u_{2} > u_{1}\} , \end{matrix}

where according to definition 2.1 the functions v_{j} ∈ L^{q}(Ω) satisfy v_{j} ∈ β(u_{j}, u_{j}) and

a(u_{j}, φ) + ∫_{Ω} v_{j}φ dx = 0 ,  (4.2)

for all φ ∈ V_{0} , j = 1, 2 . Obviously v ∈ L^{q}(Ω) and v(x) ∈ β(u(x), u(x)) for a.e. x ∈ Ω , such that the assertion of lemma 4.1 is proved provided we are able to show

a(u, φ) + ∫_{Ω} vφ dx ≤ 0 ,  (4.3)

for all φ ∈ V_{0} ∩ L_{+}^{p}(Ω) .

By (4.1) for any ε > 0 there exists a δ(ε) ∈ (0, ε) such that

∫_{δ(ε)}^{ε} \frac{dr}{ω^{q}(r)} = 1 .

We introduce the function γ_{ε} : R ↦ R defined by (cf.[6])

γ_{ε}(t) = \{ \begin{matrix}0  if & t < δ(ε) , \\ ∫_{δ(ε)}^{t} \frac{dr}{ω^{q}(r)} if & δ(ε) ≤ t ≤ ε , \\ 1  if & t > ε . \end{matrix}

For any ε > 0 the function γ_{ε} is Lipschitz continuous, nondecreasing and satisfies

γ_{ε}(t) ↦ χ_{{t>0}} as ε → 0 ,

where χ_{{t>0}} denotes the characteristic function of the set {t > 0}, as well as

0 ≤ γ′_{ε}(t) = \{ \begin{matrix}\frac{1}{ω^{q}(t)} , & for δ(ε) ≤ t ≤ ε , \\ 0 , & otherwise . \end{matrix}  (4.4)

Let ψ ∈ D_{+}(Ω) := {ψ ∈ C_{0}^{∞}(Ω) | ψ ≥ 0} . Taking as special test function in (4.2) φ = (1 − γ_{ε}(u_{2} − u_{1}))ψ ∈ V_{0} ∩ L_{+}^{p}(Ω) and φ = γ_{ε}(u_{2} − u_{1})ψ ∈ V_{0} ∩ L_{+}^{p}(Ω) for u_{1} and u_{2}, respectively, we get by adding the resulting equations

a(u_{1}, (1 − γ_{ε}(u_{2} − u_{1}))ψ) + a(u_{2}, γ_{ε}(u_{2} − u_{1})ψ) =
− ∫_{Ω}[v_{1}(1 − γ_{ε}(u_{2} − u_{1}))ψ + v_{2}γ_{ε}(u_{2} − u_{1})ψ] dx .  (4.5)

By means of (A2) the left-hand side of (4.5) denoted by I_{u}^{ε} can be estimated below by

I_{u}^{ε} ≥ a(u_{1}, ψ) + μ ∫_{Ω} |∇(u_{2} − u_{1})|^{p}γ′_{ε}(u_{2} − u_{1})ψ dx

+ ∫_{Ω} ∑_{i=1}^{N}(a_{i}(·, u_{2}, ∇u_{2}) − a_{i}(·, u_{1}, ∇u_{2}))\frac{∂(u_{2} − u_{1})}{∂x_{i}} ψγ′_{ε}(u_{2} − u_{1}) dx

+ ∫_{Ω} ∑_{i=1}^{N}(a_{i}(·, u_{2}, ∇u_{2}) − a_{i}(·, u_{1}, ∇u_{1}))\frac{∂ψ}{∂x_{i}} γ_{ε}(u_{2} − u_{1}) dx .  (4.6)

Using hypothesis (A3) then by means of Young’s inequality and (4.4) there exists a constant c_{μ} > 0 such that the third term on the right-hand side of (4.6) yields an estimate of the form

|∫_{Ω} ∑_{i=1}^{N}(a_{i}(·, u_{2}, ∇u_{2}) − a_{i}(·, u_{1}, ∇u_{2}))\frac{∂(u_{2} − u_{1})}{∂x_{i}} ψ γ′_{ε}(u_{2} − u_{1}) dx|

≤ \frac{μ}{2} ∫_{Ω} |∇(u_{2} − u_{1})|^{p}γ′_{ε}(u_{2} − u_{1})ψ dx

+  ∫_{{δ(ε)<u_{2}−u_{1}<ε}} c_{μ} [k_{2} + |u_{1}|^{p−1} + |u_{2}|^{p−1} + |∇u_{2}|^{p−1}]^{q} ψ dx .  (4.7)

Denoting the right-hand side of (4.5) by −I_{v}^{ε} from (4.5), (4.6) and (4.7) we obtain

−I_{v}^{ε} ≥ a(u_{1}, ψ) + ∫_{Ω} ∑_{i=1}^{N}(a_{i}(·, u_{2}, ∇u_{2}) − a_{i}(·, u_{1}, ∇u_{1}))\frac{∂ψ}{∂x_{i}} γ_{ε}(u_{2} − u_{1}) dx

−  ∫_{{δ(ε)<u_{2}−u_{1}<ε}} c_{μ} [k_{2} + |u_{1}|^{p−1} + |u_{2}|^{p−1} + |∇u_{2}|^{p−1}]^{q} ψ dx .  (4.8)

The last integral on the right-hand side of (4.8) tends to zero as ε → 0. Further, since

γ_{ε}(u_{2} − u_{1}) → χ_{{u_{2}>u_{1}}} and I_{v}^{ε} → ∫_{Ω} v ψ dx as ε → 0 ,

from (4.8) we finally get by passing to the limit as ε → 0

a(u, ψ) + ∫_{Ω} v ψ dx ≤ 0

for all ψ ∈ D_{+}, and thus by completion for all ψ ∈ V_{0} ∩ L_{+}^{p}(Ω) which proves lemma 4.1.

Lemma 4.2. Any well-ordered chain C ⊂ P possesses an upper bound in P.

Proof: Since P ⊂ [\underline{u}, ū], we have an uniform L^{p}(Ω)−boundedness of all u ∈ C . By lemma 3.1 any u ∈ C satisfies the QVI (3.1) from which due to hypothesis (A2) and the uniform L^{p}(Ω)−boundedness of the Nemytskij operator F_{0,α} it follows that the chain C is normbounded in V_{0}. Applying [11, Lemma 4.1.2] there is a nondecreasing sequence (u_{n}) of C which converges to w := sup C weakly in V_{0} and strongly in L^{p}(Ω). Each u_{n} ∈ [\underline{u}, ū] is a solution of the BVP (1.1) and thus satisfies the QVI

a(u_{n}, φ − u_{n}) + J(u_{n}, φ) − J(u_{n}, u_{n}) ≥ 0  (4.9)

for all φ ∈ V_{0} , n = 1, 2, · · · . By the equicontinuity of the Nemytskij operator F_{0,α} generating the functional J we obtain

J(u_{n}, φ) → J(w, φ) , as n → ∞ ,  (4.10)

and by (H3) we get

|J(u_{n}, u_{n}) − J(u_{n}, w)| ≤ | ∫_{Ω} ∫_{w(x)}^{u_{n}(x)}f_{0,α}(u_{n}(x), s) ds dx| ≤ ∥k_{1}∥_{L^{q}(Ω)}∥u_{n} − w∥_{L^{p}(Ω)} ,

which implies in view of (4.10)

J(u_{n}, u_{n}) → J(w, w) .  (4.11)

Taking in (4.9) as special test function φ = w we obtain by means of (4.10) and (4.11)

lim sup_{n→∞} a(u_{n}, u_{n} − w) ≤ 0 .  (4.12)

The strong convergence of the sequence (u_{n}) to w in V_{0} now follows from the (S^{+})- property of the operator A : V_{0} → V_{0}^{∗} generated by the semilinear form a(u, φ) (cf. [2, 13]) and relation (4.12). This allows the passage to the limit in inequality (4.9) as n → ∞ which proves the assertion of lemma 4.2.

Theorem B. Let hypotheses (A1)-(A3) and (H1)-(H3) be satisfied. Then the BVP (1.1) possesses the greatest solution u^{∗} and the least solution u_{∗} within the interval [\underline{u}, ū] in the sense of the underlying partial ordering, i.e., if u is any solution of (1.1) in [\underline{u}, ū] then u ∈ [u_{∗}, u^{∗}] . Proof: Due to lemma 4.2 Zorn’s lemma can be applied which yields the existence of a maximal element u_{max} of the partial ordered set P. From lemma 4.1 it readily follows that P is directed, i.e., if u_{1}, u_{2} ∈ P then there is an element u_{3} ∈ P such that u_{1} ≤ u_{3} , u_{2} ≤ u_{3} . This is because max(u_{1}, u_{2}) is a lower solution and thus from Theorem A there must be a solution u_{3} of the BVP (1.1) satisfying u_{3} ∈ [max(u_{1}, u_{2}), ū]. The directedness of P, however, implies that the maximal element u_{max} is uniquely defined and must be the greatest one, i.e., u_{max} = u^{∗}. The Proof for the existence of the least element u_{∗} ∈ P is carried out in a similar way. Remark 4.2. The results of this paper combined with results recently obtained in [5] allow to treat BVP of the form

Au + β(u, u) ∋ g(u, u) in Ω , u = 0 on ∂Ω ,  (4.13)

where A and β are as given above and g : R × R ↦ R may be a function that is continuous in its first and nondecreasing (possibly discontinuous) in its second argument. Inclusions of the form (4.13) cover a wide range of problems in the applied science having nonlinear source terms that may be discontinuous, e.g. cf.[1, 3, 4, 5].

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