Journal of Convex Analysis Volume 2 (1995), No.1/2, 241–261

Existence of Solutions for Unilateral Problems With Multivalued Operators

Pirro Oppezzi, Anna Maria Rossi

Dipartimento di Matematica, Università di Genova, Via L.B.Alberti 4, 16132 Genova, Italy. e-mail: oppezzi@cartesio.dima.unige.it, rossia@cartesio.dima.unige.it

Received 13 July 1994 Revised manuscript received 2 May 1995

Dedicated to R. T. Rockafellar on his 60th Birthday

We prove some new results about the existence of solutions of variational inequalities with quasilinear operators having the generalized pseudo-monotone property. We also consider the case where data are Radon measures or L1 elements.

Introduction

Let Ω be a bounded open subset of IR^{n}, 1 < p < n, ψ ∈ L^{∞}(Ω). The problem we are going to consider, when f lies in the dual space of H_{0}^{1,p}(Ω), has the form

\{ \begin{matrix}find u ∈ H^{1,p}(Ω), u ≥ ψ \\ find ζ ∈ A(u) satisfying \\ ⟨ζ, v − u⟩ ≥ ⟨f, v − u⟩ for any v ∈ H_{0}^{1,p}(Ω), v ≥ ψ, \end{matrix}  (0)

where A is a multivalued operator defined on H_{0}^{1,p}(Ω) with values in its dual H^{−1,p′}(Ω) (p′ = p/(p − 1)). More precisely we take a multivalued map a defined on Ω × IR × IR^{n} with values in IR^{n}, maximal monotone with respect to the last variable. Then ζ ∈ A(u) if and only if ζ = − div g and g is a measurable selection of the map

x ∈ Ω ↦ a(x, u(x), Du(x)) ⊂ IR^{n}.

When a is single-valued and strictly monotone with respect to the last variable, some particular case of this problem was developed for example by Pascali and Sburlan in [8], ch. VI. When f ∈ L^{1}(Ω), after a suitable formulation of the inequality in (0), various existence results, which consider particular cases of single valued and strictly monotone operators, were obtained for example by L. Boccardo and T. Gallouet in [2] and L. Boccardo and

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

242 P. Oppezzi, A. M. Rossi / Existence of solutions for unilateral problems

G.R. Cirmi in [3]. Another kind of existence result for single valued operators was provided by J.M. Rakotoson in [9] for the case of f Radon measure. In this work we first give, in section 2, an existence theorem with f in H^{−1,p′}(Ω) and a multivalued operator A of the above kind. The form of the problem (0) is adopted for example by R.T. Rockafellar in [10] and by G. Dal Maso - A. Defranceschi in [5]. In section 3 we deal with the cases where f is a bounded Radon measure or an L^{1} function. In the first case we give an existence theorem for the problem stated in analogy with the one in [9]. Then when we take f ∈ L^{1}(Ω), we prove another existence theorem after stating the problem in analogy with the one in [2].

1. Formulation of the problem

Notation and Hypotheses

Let Ω be a bounded open set of IR^{n}, n ≥ 2, we denote by H_{0}^{1,p}(Ω), for 1 < p < ∞, the usual Sobolev space, by H^{−1,p′}(Ω) its dual; by ⟨., .⟩ the scalar product on IR^{n} or the duality between H_{0}^{1,p}(Ω) and H^{−1,p′}(Ω). The symbol ∥∥_{L^{p}} will denote the norm in L^{p}(Ω)^{n} or in L^{p}(Ω) and x_{h} ⇀ x will mean that the sequence (x_{h})_{h∈N} of a certain dual of a Banach space, converges to x in the weak topology. Let a: Ω×IR×IR^{n} → 2^{IR^{n}} be L(Ω)×B(IR)×B(IR^{n})-measurable, namely for every open set U ⊂ IR^{n}, a^{−1}(U): = {(x, s, ξ) ∈ Ω × IR × IR^{n}: U ∩ a(x, s, ξ) ≠ ∅} ∈ L(Ω) × B(IR) × B(IR^{n}). For a single or multivalued map F we also denote by Γ(F) its graph. We assume p ∈ (1, n) and α, β ∈ IR_{+} such that

(1.1.1) β < p − 1, (1.1.2) \frac{α}{p − 1 − β} < \frac{p^{∗}}{p} , (1.1.3) \frac{α}{p − 1 − β} < p′  (1.1)

with p^{∗} = np/(n − p), p′ = p/(p − 1). Besides we suppose a to be closed and convex valued and satisfying the following conditions:

i) for a.e. x ∈ Ω and every s ∈ IR, a(x, s, ·): IR^{n} → 2^{IR^{n}} is maximal monotone; ii) there exist μ ∈ L^{p′}(Ω), ν ∈ L^{n/(n−1)}(Ω), c, c_{1}, c_{2} ∈ IR_{+} such that

|η| ≤ μ(x) + c_{1}|ξ|^{p−1} + c_{2}|s|^{α}|ξ|^{β}  (ii1)
⟨η, ξ⟩ ≥ ν(x) + c|ξ|^{p}  (ii2)

for a.e. x ∈ Ω, for every s ∈ IR, ξ ∈ IR^{n}, η ∈ a(x, s, ξ). iii) if u, v, (u_{h})_{h∈N}, η are given, being u ∈ H_{0}^{1,p}(Ω), v ∈ H^{1,p}(Ω), (u_{h})_{h∈N} in H_{0}^{1,p}(Ω)

such that u_{h} → u a.e. in Ω and η an L(Ω)-measurable selection of the map

x ∈ Ω ↦ a(x, u(x), Dv(x)) ⊂ IR^{n},

then there exists (η_{h})_{h∈N} converging a.e. to η in Ω, such that, for every h ∈ N, η_{h} is an L(Ω)-measurable selection of the map x ∈ Ω ↦ a(x, u_{h}(x), Dv(x)) ⊂ IR^{n}.

From now on “measurability” will have the meaning adopted above for the multivalued function a.

Remark 1.1. The assumption iii) is for instance satisfied by all multivalued functions of the form: a(x, s, ξ) = a_{0}(x, ξ)+a_{1}(x, s, ξ) with a_{0}: Ω×IR^{n} → 2^{IR^{n}}, a_{1}: Ω×IR×IR^{n} → IR^{n} measurable and a_{1}(x, ·, ξ) continuous for a.e. x ∈ Ω and every ξ ∈ IR^{n}. Indeed if η is an L(Ω)-measurable selection of the map x ∈ Ω ↦ a(x, u(x), Dv(x)) ∈ IR^{n}, the sequence (η_{h})_{h∈N} defined by

η_{h}(x) = η(x) − a_{1}(x, u(x), Dv(x)) + a_{1}(x, u_{h}(x), Dv(x))

satisfies iii). Another case in which iii) is true is given by a: Ω × IR × IR^{n} → 2^{IR^{n}} defined by a(x, s, ξ) = {λa_{0}(x, s, ξ) + (1 − λ)a_{1}(x, s, ξ), λ ∈ [0, 1]}, where a_{i}: Ω × IR × IR^{n} → IR^{n}, i = 1, 2, are measurable functions, continuous in the second variable for a.e. x ∈ Ω and every ξ ∈ IR^{n}.

Remark 1.2. If u: Ω → IR, w: Ω → IR^{n} are L(Ω)-measurable,

x ∈ Ω ↦ a(·, u, w)(x) = a(x, u(x), w(x)) ∈ 2^{IR^{n}}

turns out to be measurable as well. Indeed, if U ⊂ IR^{n} is an open set,

a^{−1}(U)
= {x ∈ Ω: ∃η ∈ U ∩ a(x, u(x), w(x))}
= {x ∈ Ω: ∃η ∈ U, s ∈ IR, ξ ∈ IR^{n} | (x, s, ξ, η) ∈ Γ(a) ∩ (Γ((u, w)) × IR^{n})}
= pr_{Ω}(Γ(a) ∩ (Γ((u, w)) × IR^{n}) ∩ (Ω × IR × IR^{n} × U)),

so that the assertion follows by Theorem III.23 in [4] (here pr_{Ω} denotes the projection from Ω × IR × IR^{n} × IR^{n} into Ω).

Remark 1.3. If g: Ω → 2^{IR^{n}} is a measurable selection of a(·, u, Dv), u, v ∈ H_{0}^{1,p}(Ω), then g ∈ L^{p′}(Ω)^{n}. Indeed, by hypothesis (ii1) and Young’s inequality we have:

|g(x)| ≤ μ(x)+c_{1}|Dv(x)|^{p−1}+c_{2} \frac{p − 1 − β}{p − 1} |u(x)|^{α(p−1)/(p−1−β)}+c_{2} \frac{β}{p − 1}|Dv(x)|^{p−1} (1.2)

where by (1.1.2) we have αp′(p − 1)/(p − 1 − β) < p^{∗}, so that our assertion follows by Sobolev’s inequality.

Problem. Let A: H_{0}^{1,p}(Ω) → 2^{L^{p′}(Ω)^{n}} be defined by

A(u) = {g ∈ (L^{p′}(Ω))^{n}: g(x) ∈ a(x, u(x), Du(x)) for a.e. x ∈ Ω},

and A: H_{0}^{1,p}(Ω) → 2^{H^{−1,p′}(Ω)} by A(u) = {− div g: g ∈ A(u)}. If a non-empty closed convex set K ⊂ H_{0}^{1,p}(Ω) and f ∈ H^{−1,p′}(Ω) are given, our first problem is stated as follows:

\{ \begin{matrix}u ∈ K \\ ∃ζ ∈ A(u) such that ⟨ζ, v − u⟩ ≥ ⟨f, v − u⟩ for every v ∈ K. \end{matrix}  (I)

Remark 1.4. It is easy to show that problem (I) is equivalent to the relation

f ∈ (A + ∂I_{K})u,

where ∂I_{K} is the subdifferential of the indicator function I_{K} of K.

2. Existence of solutions for the problem (I)

Lemma 2.1. If ((u_{h}, f_{h}))_{h∈N} is a sequence in H_{0}^{1,p}(Ω)×H^{−1,p′}(Ω) such that f_{h} ∈ A(u_{h}) for h ∈ N, u_{h} → u ∈ H_{0}^{1,p}(Ω) strongly and f_{h} ⇀ f ∈ H^{−1,p′}(Ω) weakly, then f ∈ A(u).

Proof. By the definition of A, for every h ∈ N there exists g_{h} ∈ L^{p′}(Ω)^{n} such that g_{h}(x) ∈ a(x, u_{h}(x), Du_{h}(x)) for a.e. x ∈ Ω and f_{h} = − div g_{h}. Taking (1.2) and the boundedness of (u_{h})_{h∈N} in H_{0}^{1,p}(Ω) into account, we obtain that the sequence (g_{h})_{h∈N} is bounded in L^{p′}(Ω)^{n}. Therefore there exists g ∈ L^{p′}(Ω)^{n} such that, by passing to a subsequence if necessary, g_{h} ⇀ g in L^{p′}(Ω)^{n}. Now we prove that f = − div g and g(x) ∈ a(x, u(x), Du(x)) for a.e. x ∈ Ω. Since ⟨f, v⟩ = lim_{h→∞}⟨− div g_{h}, v⟩ = lim_{h→∞} ∫_{Ω}⟨g_{h}, Dv⟩dx = ∫_{Ω}⟨g, Dv⟩dx =

⟨− div g, v⟩ for every v ∈ H_{0}^{1,p}(Ω), we get f = − div g. To conclude we show that for every ξ ∈ IR^{n} and ζ ∈ a(x, u(x), ξ) is ⟨g(x)−ζ, Du(x)−ξ⟩ ≥ 0 for a.e. x ∈ Ω. Then from maximal monotonicity of the map ξ ∈ IR^{n} ↦ a(x, u(x), ξ) ∈ 2^{IR^{n}} for x ∈ Ω\Ω_{0} where Ω_{0} has Lebesgue measure zero, it follows that g(x) ∈ a(x, u(x), Du(x)). Let η be a measurable selection of a(·, u, ξ). Since, by passing to a subsequence if necessary, (u_{h})_{h∈N} converges a.e. to u, by hypothesis iii), there exists a sequence (η_{h})_{h∈N} converging a.e. to η on Ω, such that for every h ∈ N η_{h} is an L(Ω)-measurable selection of the map x ∈ Ω ↦ a(x, u_{h}(x), ξ) ∈ 2^{IR^{n}}. We note first that from (ii1) it follows that η_{h} → η in L^{p′}(Ω)^{n}, indeed for every h ∈ N and a.e. x ∈ Ω we have:

|η_{h}(x)| ≤ μ(x) + c_{1}|ξ|^{p−1} + c_{2}|u_{h}(x)|^{α}|ξ|^{β}

where the right hand side converges in L^{p′}(Ω)^{n} as αp′ < p^{∗}. If φ ∈ C(Ω), φ ≥ 0, since the map ξ ∈ IR^{n} ↦ a(x, u_{h}(x), ξ) ∈ 2^{IR^{n}} is monotone for a.e. x ∈ Ω and for every h ∈ N, we have 0 ≤ ∫_{Ω}⟨g_{h}−η_{h}, Du_{h}−ξ⟩φdx. Taking the convergence of (g_{h})_{h∈N}, (η_{h})_{h∈N}, (Du_{h})_{h∈N} into account, we have lim_{h→∞} ∫_{Ω}⟨g_{h} −η_{h}, Du_{h} −ξ⟩φdx = ∫_{Ω}⟨g −η, Du−ξ⟩φdx and since φ is arbitrary, we obtain ⟨g(x) − η(x), Du(x) − ξ⟩ ≥ 0 for a.e. x ∈ Ω. But the selection η is also arbitrary, so that from Theorem III.9 in [4], which ensures the existence of a sequence (σ_{h})_{h∈N} of measurable selections of a(·, u, ξ) such that {σ_{h}(x): h ∈ N} is dense in a(x, u(x), ξ), it follows that ⟨g(x) − ζ, Du(x) − ξ⟩ ≥ 0 for a.e. x ∈ Ω and every ζ ∈ a(x, u(x), ξ).

Lemma 2.2. If u ∈ H_{0}^{1,p}(Ω) and A is the operator defined in 1.1, then A(u) is closed,

convex, nonempty and bounded. Moreover A: H_{0}^{1,p}(Ω) → 2^{H^{−1,p′}(Ω)} is a bounded operator.

Proof. From hypothesis (ii1) about a we immediately get boundedness of A(u): indeed if − div g ∈ A(u), g ∈ A(u), taking (1.2) and αp/(p − 1 − β) < p^{∗} into account, from Sobolev’s inequality it follows that for a suitable constant K

∥g∥_{L^{p′}} ≤ K(∥μ∥_{L^{p′}} + (∥Du∥_{L^{p}})^{p/p′} + (∥Du∥_{L^{p}})^{α(p−1)/(p−1−β)}).  (2.1)

Since ∥ − div g∥_{H^{−1,p′}} ≤ ∥g∥_{L^{p′}}, by (2.1) we obtain also boundedness of A. Moreover a is convex valued, so that the same is true for A. Now let us prove that A(u) is nonempty. a(x, u(x), ·), and consequently a(x, u(x), ·)^{−1}, is maximal monotone for a.e. x ∈ Ω. Moreover, by (ii1), (a(x, u(x), ·)^{−1})^{−1} is locally bounded according to definition in [8] 2.2 ch.III, hence the theorem on page 147 in [8] ensures that a(x, u(x), ·)^{−1} is surjective. Then for a.e. x ∈ Ω and every ξ ∈ IR^{n} is a(x, u(x), ξ) ≠ ∅. Hence for a.e. x ∈ Ω is a(x, u(x), Du(x)) ≠ ∅ and by Theorem 9 in [4] ch.III there exists a measurable selection g: Ω → IR^{n} of a(·, u, Du). By Remark 1.2 − div g ∈ A(u). Lastly, A(u) is closed in H^{−1,p′}(Ω): indeed if (f_{h})_{h∈N} is a sequence in A(u) such that f_{h} ⇀ γ ∈ H^{−1,p′}(Ω) applying Lemma 2.1 we get γ ∈ A(u).

Remark 2.3. Given u ∈ H_{0}^{1,p}(Ω), as in Remark 1.3 we can see that if g: Ω → 2^{IR^{n}} is a measurable selection of a(·, u, Dv), v ∈ H_{0}^{1,p}(Ω), we have g ∈ L^{p′}(Ω)^{n}.

Therefore we may define the operator B_{u}: H_{0}^{1,p}(Ω) → 2^{H^{−1,p′}(Ω)}, by

H_{0}^{1,p}(Ω) ∋ v ↦ B_{u}(v): = {− div η : η(x) ∈ a(x, u(x), Dv(x)) for a.e. x ∈ Ω}.

Lemma 2.1 and 2.2 hold, with analogous and more simple proof (hypothesis iii) doesn’t need in this case), if we replace A by B_{u}.

Definition 2.4. ([8]) If X is a Banach space and X^{∗} its dual, then T: X → 2^{X^{∗}} is coercive if for every selection τ: X → X^{∗}, τ(x) ∈ T (x) for every x ∈ X, we have

lim_{∥x∥→∞} \frac{⟨τ(x), x⟩}{∥x∥} = ∞.

If T is single valued, “coercivity” is given by the same condition with τ(x) = T (x).

Definition 2.5. ([8]) Let X be a reflexive Banach space. T: X → 2^{X^{∗}} is said to have the generalized pseudo monotone property if for any sequence ((x_{h}, f_{h}))_{h∈N} in Γ(T) such that x_{h} ⇀ x in X, f_{h} ⇀ f in X^{∗} and for which lim sup⟨f_{h}, x_{h}− x⟩ ≤ 0, we have (x, f) ∈ Γ(T) and ⟨f_{h}, x_{h}⟩ → ⟨f, x⟩.

Definition 2.6. ([8]) Let X be a strictly convex reflexive Banach space and X^{∗} its dual.

a) P: X → X^{∗} is said to be smooth if it is bounded, coercive, maximal monotone.

b) T: X → 2^{X^{∗}} with the generalized pseudo monotone property is called regular if T +P is surjective for any smooth operator P: X → X^{∗}.

Definition 2.7. ([8]) Let X be a reflexive Banach space and T: X → 2^{X^{∗}}. a) The map T is upper semicontinuous in x ∈ D(T): = {x ∈ X: T x ≠ ∅}, if for any sequence (x_{h})_{h∈N} in D(T), with x_{h} → x in X and for any neighbourhood V of T (x), there exists h_{0} ∈ N such that T (x_{h}) ⊂ V for h > h_{0}.

b) The map T is said to be of type (M) provided that: M_{1}) T (x) is bounded, convex, closed and nonempty for each x ∈ X; M_{2}) for each sequence ((x_{h}, f_{h}))_{h∈N} in Γ(T) such that x_{h} ⇀ x in X, f_{h} ⇀ f in X^{∗} and

for which lim sup⟨f_{h}, x_{h} − x⟩ ≤ 0, we have (x, f) ∈ Γ(T); M_{3}) T is upper semicontinuous from the finite-dimensional subspaces of X to X^{∗} endowed

with the weak topology.

Remark 2.8. If X is a reflexive Banach space, T: X → 2^{X^{∗}} is bounded and satisfies condition M_{2}), then it satisfies condition M_{3}). Indeed let x_{h}, x ∈ Y, h ∈ N, Y be a finite dimensional subspace of X, x_{h} → x, V be a weakly open neighbourhood of T (x) in X^{∗}. If for infinitely many values of h ∈ N we have T (x_{h})\V ≠ ∅, then there exists an increasing sequence (h_{k})_{k∈N} in N such that for every k ∈ N there exists λ_{k} ∈ T (x_{h_{k}})\V. By boundedness of T, by passing to a further subsequence if necessary, the sequence (λ_{k})_{k∈N} converges weakly in X^{∗}, so that lim_{k→∞}⟨λ_{k}, x_{h_{k}} − x⟩ = 0 and by M_{2}) lim_{k→∞} λ_{k} ∈ T (x) ⊂ V. This contradicts the fact that, as λ_{k} ∉ V for every k ∈ N, lim_{k→∞} λ_{k} ∉ V.

Lemma 2.9. If u_{0} ∈ H_{0}^{1,p}(Ω), define A_{0}: H_{0}^{1,p}(Ω) → 2^{H^{−1,p′}(Ω)} by

A_{0}(v) = A(v + u_{0}).

A being the operator in (I). If P: H_{0}^{1,p}(Ω) → H^{−1,p′}(Ω) is any smooth operator, then P + A_{0} is of type (M).

Proof. From Lemma 2.2 it follows that M_{1}) is satisfied by P + A_{0}. In order to obtain M_{2}) we first establish the following propositions:

I) If u_{0} ∈ H_{0}^{1,p}(Ω) is given, the operator B_{u_{0}} defined in 2.3 is maximal monotone. II) A_{0} satisfies M_{2}). Proof of I). On account of 2.3, if v ∈ H_{0}^{1,p}(Ω) then B_{u_{0}}(v) is closed, convex, non empty and bounded. Moreover it is upper semicontinuous from the line segments in H_{0}^{1,p}(Ω) to H^{−1,p′}(Ω) endowed with the weak topology: if V is a weakly open neighborhood of B_{u_{0}}(v) and (v_{h})_{h∈N} strongly converges to v in H_{0}^{1,p}(Ω), we can find k ∈ N such that B_{u_{0}}(v_{h}) ⊂ V when h > k. On the contrary let us suppose that for an increasing sequence (h_{k})_{k∈N} in N, be f_{k} ∈ B_{u_{0}}(v_{h_{k}})\V. Being B_{u_{0}}, as remarked in 2.3, a bounded operator, we can extract a subsequence from (f_{k})_{k∈N}, whose limit, Lemma 2.1 being true with B_{u_{0}} in place of A, belongs to B_{u_{0}}(v). Then it belongs to V, contrary to the fact that f_{k} ∉ V for each k ∈ N. Monotonicity of B_{u_{0}} follows from the fact that a(x, u_{0}(x), ·) is monotone for a.e. x ∈ Ω. Then B_{u_{0}} verifies all the hypothesis of [8] theorem in 2.3 ch.III, so that it is maximal monotone.

Proof of II). Let ((u_{h}, f_{h}))_{h∈N} be a given sequence in H_{0}^{1,p}(Ω) × H^{−1,p′}(Ω) such that f_{h} ∈ A(u_{h}) for h ∈ N, u_{h} ⇀ u in H_{0}^{1,p}(Ω), f_{h} ⇀ f in H^{−1,p′}(Ω) and lim sup⟨f_{h}, u_{h} − u⟩ ≤ 0.

If g_{h} ∈ L^{p′}(Ω)^{n}, g_{h}(x) ∈ a(x, u_{h}(x), Du_{h}(x)) for a.e. x ∈ Ω and f_{h} = − div g_{h}, inequality (2.1) and boundedness of (u_{h})_{h∈N} in H_{0}^{1,p}(Ω) guarantee that (g_{h})_{h∈N} is bounded in L^{p′}(Ω)^{n}. Therefore there exists g ∈ L^{p′}(Ω)^{n} such that, by extracting a subsequence if necessary, g_{h} ⇀ g in L^{p′}(Ω)^{n}. In analogy to the proof in 2.1, we obtain − div g = f.

Now we conclude by proving that if v ∈ H_{0}^{1,p}(Ω) and − div η ∈ B_{u}(v), then 0 ≤ ⟨− div g − (− div η), u − v⟩; this fact implies − div g ∈ B_{u}(u) = A(u) by maximal monotonicity of B_{u}. Let η ∈ L^{p′}(Ω) be such that η(x) ∈ a(x, u(x), Dv(x)) for a.e. x ∈ Ω. By Rellich’s theorem there exists a subsequence of (u_{h})_{h∈N} converging to u strongly in L^{p}(Ω). Hence, by passing to a further subsequence if necessary, we can suppose (u_{h})_{h∈N} converging a.e. on Ω to u. Let (η_{h})_{h∈N} be given by hypothesis iii) in connection with u, v, η, (u_{h})_{h∈N}, such that η_{h} → η a.e. on Ω and η_{h} is, for each h ∈ N, an L(Ω)-measurable selection of x ∈ Ω ↦ a(x, u_{h}(x), Dv(x)) ∈ IR^{n}. By (1.2) we have for h ∈ N and a.e. x ∈ Ω:

|η_{h}(x)| ≤ μ(x) + c_{1}|Dv(x)|^{p−1} + c_{2} \frac{p − 1 − β}{p − 1} |u_{h}(x)|^{α(p−1)/(p−1−β)} + c_{2} \frac{β}{p − 1}|Dv(x)|^{p−1}.

The sequence on the right hand side converges in L^{p′}(Ω) strongly as (1.1.1) and (1.1.2) give αp′(p − 1)/(p − 1 − β) < p^{∗}. So η_{h} → η in L^{p′}(Ω)^{n} strongly.

Having η_{h}(x) ∈ a(x, u_{h}(x), Dv(x)), from monotonicity we get:

0 ≤ lim sup ∫_{Ω}⟨g_{h}(x) − η_{h}(x), D(u_{h} − v)(x)⟩dx
= lim sup (∫_{Ω}⟨g_{h}, D(u_{h} − u)⟩dx + ∫_{Ω}⟨g_{h}, D(u − v)⟩dx − ∫_{Ω}⟨η_{h}, D(u_{h} − v)⟩dx)
= lim sup⟨f_{h}, u_{h} − u⟩ + ∫_{Ω}(⟨g, D(u − v)⟩ − ⟨η, D(u − v)⟩)dx
≤ ⟨− div g − (− div η), u − v⟩.

This concludes that A, and consequently A_{0}, satisfies condition M_{2}).

We return to the proof of the lemma. Now we prove property M_{2}) for P + A_{0}, where P is assumed to be smooth. Let ((v_{h}, f_{h}))_{h∈N} in H_{0}^{1,p}(Ω) × H^{−1,p′}(Ω) be such that f_{h} ∈ (P + A_{0})(v_{h}) for each h ∈ N, v_{h} ⇀ v in H_{0}^{1,p}(Ω), f_{h} ⇀ f in H^{−1,p′}(Ω) and lim sup⟨f_{h}, v_{h} − v⟩ ≤ 0. We will show that f ∈ (P +A_{0})(v). Taking b_{h} ∈ A_{0}(v_{h}) so that f_{h} = P (v_{h})+b_{h}, from boundedness of P and A_{0}, due to Lemma 2.2, we get b, d ∈ H^{−1,p′}(Ω) such that, by extracting a subsequence if necessary, b_{h} ⇀ b and P (v_{h}) ⇀ d. Hence from monotonicity of P we get:

lim sup⟨b_{h}, v_{h} − v⟩ =
= lim sup(⟨P (v_{h}) + b_{h}, v_{h} − v⟩ − ⟨P (v_{h}) − P (v), v_{h} − v⟩ − ⟨P (v), v_{h} − v⟩)
≤ lim sup(⟨P (v_{h}) + b_{h}, v_{h} − v⟩ − ⟨P (v), v_{h} − v⟩
= lim sup⟨f_{h}, v_{h} − v⟩
≤ 0.

Proposition II) gives b ∈ A_{0}(v), hence we may take γ: Ω → IR^{n} measurable such that γ(x) ∈ a(x, (v + u_{0})(x), D(v + u_{0})(x)) for a.e. x ∈ Ω and b = − div γ. Extracting a subsequence if necessary, we can suppose (v_{h} + u_{0})_{h∈N} a.e. convergent to v + u_{0}. Then, like in the similar case shown in the proof of II) above, by iii) there exists, in connection with v + u_{0}, v + u_{0}, γ, (v_{h} + u_{0})_{h∈N}, a sequence (γ_{h})_{h∈N} converging strongly to γ in L^{p′}(Ω)^{n} and such that γ_{h} is, for each h ∈ N, a measurable selection of the map

x ∈ Ω ↦ a(x, (v_{h} + u_{0})(x), D(v + u_{0})(x)) ∈ IR^{n}.

Thus, if g_{h} ∈ L^{p′}(Ω)^{n}, g_{h}(x) ∈ a(x, (v_{h} + u_{0})(x), D(v_{h} + u_{0})(x)) for a.e. x ∈ Ω and b_{h} = − div g_{h} for h ∈ N, from monotonicity of a(x, (v_{h} + u_{0})(x), ·), we get:

⟨P (v_{h}), v_{h} − v⟩ = ⟨P (v_{h}) + b_{h}, v_{h} − v⟩ − ⟨b_{h} − (− div γ_{h}), v_{h} − v⟩ + ⟨div γ_{h}, v_{h} − v⟩
= ⟨P (v_{h}) + b_{h}, v_{h} − v⟩ − ∫_{Ω}⟨g_{h} − γ_{h}, D(v_{h} + u_{0} − (v + u_{0}))⟩dx
− ∫_{Ω}⟨γ_{h}, D(v_{h} − v)⟩dx
≤ ⟨f_{h}, v_{h} − v⟩ − ∫_{Ω}⟨γ_{h}, D(v_{h} − v)⟩dx

and therefore lim sup⟨P (v_{h}), v_{h} − v⟩ ≤ 0. On the other hand by Proposition in 5.2 ch.III of [8], the operator P, being maximal monotone, satisfies M_{2}), consequently the previous inequality ensures that P (v) = d. Thus we conclude that f = d + b ∈ (P + A_{0})(v) and P + A_{0} satisfies M_{2}). Property M_{3}) follows from boundedness of P + A_{0} and Remark 2.8.

Lemma 2.10. If u_{0} ∈ H_{0}^{1,p}(Ω), the operator A_{0}, defined in 2.9, is coercive.

Proof. Let G: H_{0}^{1,p}(Ω) → H^{−1,p′}(Ω) be such that G(v) ∈ A_{0}(v) for each v ∈ H_{0}^{1,p}(Ω), and g_{v} ∈ L^{p′}(Ω)^{n}, g_{v}(x) ∈ a(x, (v + u_{0})(x), D(v + u_{0})(x)) for a.e. x ∈ Ω, be such that G(v) = − div g_{v}. By (2.1) we have

∥g_{v}∥_{L^{p′}} ≤ K(∥μ∥_{L^{p′}} + (∥v + u_{0}∥_{H_{0}^{1,p}})^{p/p′} + (∥v + u_{0}∥_{H_{0}^{1,p}})^{α(p−1)/(p−1−β)}).

Then by using coercivity ii2), Sobolev’s and Hölder’s inequalities:

⟨G(v), v⟩ = ∫_{Ω}⟨g_{v}, D(v + u_{0})⟩dx − ∫_{Ω}⟨g_{v}, Du_{0}⟩dx
≥ ∫_{Ω}(ν + c|D(v + u_{0})|^{p})dx − ∥g_{v}∥_{L^{p′}}∥Du_{0}∥_{L^{p}}
≥ − ∥ν∥_{L^{1}} + \overline{c}(∥v + u_{0}∥_{H_{0}^{1,p}})^{p} − K∥Du_{0}∥_{L^{p}}(∥μ∥_{L^{p′}} + (∥v + u_{0}∥_{H_{0}^{1,p}})^{p/p′}+
+ (∥v + u_{0}∥_{H_{0}^{1,p}})^{α(p−1)/(p−1−β)}),

where \overline{c} > 0 is a suitable constant.

Since from condition (1.1.3) it follows that α(p − 1)/(p − 1 − β) < p, the above inequality gives the desired result.

Definition 2.11. ([8]) Let X be a normed space and X^{∗} its dual. A map T: X → 2^{X^{∗}} is quasi bounded if to each M > 0 there corresponds a C > 0 such that for each x ∈ X, ∥x∥ ≤ M, if f ∈ T x satisfies ⟨f, x⟩ ≤ M∥x∥ then ∥f∥ ≤ C.

Theorem 2.12. ([8] theorem 3.5 ch.III) Let X be a reflexive strictly convex Banach space and X^{∗} its dual. If T: X → 2^{X^{∗}} is maximal monotone, 0 ∈ D(T) and H: X → 2^{X^{∗}} is quasi bounded, regular and coercive, then T + H is surjective.

Theorem 2.13. Assuming the hypotheses described in section 1, then there exists a solution of the problem (I).

Proof. By Remark 1.4 it suffices to prove that if f ∈ H^{−1,p′}(Ω) is given, there exists u ∈ H_{0}^{1,p}(Ω) such that f ∈ (A + ∂I_{K})u. For this purpose we show that if u_{0} ∈ K is given, then Theorem 2.12 can be applied to the case X = H_{0}^{1,p}(Ω), T (v) = ∂I_{K}(v + u_{0}), H(v) = A(v + u_{0}) for every v ∈ H_{0}^{1,p}(Ω). First we show that A has the generalized pseudo-monotone property (Def. 2.5). Let ((v_{h}, f_{h}))_{h∈N} be a sequence in H_{0}^{1,p}(Ω) × H^{−1,p′}(Ω) such that f_{h} ∈ A(v_{h}) for h ∈ N, v_{h} ⇀ v in H_{0}^{1,p}(Ω), f_{h} ⇀ f in H^{−1,p′}(Ω) and lim sup⟨f_{h}, v_{h} − v⟩ ≤ 0. We proved in Lemma 2.9 that A satisfies M_{2}), hence f ∈ A(v). For every h ∈ N we may write f_{h} = − div g_{h}, f = − div g, g_{h}(x) ∈ a(x, v_{h}(x), Dv_{h}(x)), g(x) ∈ a(x, v(x), Dv(x)) for a.e. x ∈ Ω. Like in the similar case shown in the proof of II) in Lemma 2.9, by hypothesis iii) there exists a sequence (γ_{h})_{h∈N} in connection with v, v, g, (v_{h})_{h∈N}, such that γ_{h} → g in L^{p′}(Ω)^{n} and γ_{h} is a measureable selection of the map x ∈ Ω ↦ a(x, v_{h}(x), Dv(x)) ∈ IR^{n}; then, by using monotonicity of a(x, v_{h}(x), ·) we get:

⟨− div g_{h}, v_{h}⟩ = ∫_{Ω}⟨g_{h}, Dv_{h}⟩dx =
= ∫_{Ω}⟨g_{h} − γ_{h}, Dv_{h} − Dv⟩dx + ∫_{Ω}⟨γ_{h}, Dv_{h} − Dv⟩dx + ∫_{Ω}⟨g_{h}, Dv⟩dx
≥ ∫_{Ω}⟨γ_{h}, Dv_{h} − Dv⟩dx + ⟨f_{h}, v⟩.

It follows that lim inf⟨f_{h}, v_{h}⟩ ≥ ⟨f, v⟩ and thus lim_{h→∞}⟨f_{h}, v_{h}⟩ = ⟨f, v⟩.

It can be easily seen that the operator H, defined by H(v) = A(v+u_{0}), has the generalized pseudo-monotone property, too.

We finally see that H is regular. If P: H_{0}^{1,p}(Ω) → H^{−1,p′}(Ω) is smooth, by Lemma 2.9, P + A_{0} is of type (M); moreover it is coercive and bounded because P and A_{0} are. Theorem in [8] 5.4, ch.III and subsequent remarks ([8] page 156) applied to P +A_{0} ensure its surjectivity, so that H is regular. On the other hand H is bounded and coercive by Lemmas 2.2 and 2.10 so that it satisfies the hypotheses in 2.12. The domain of ∂I_{K} is K, then by applying Proposition 2.13 of [8] ch.III, we obtain that ∂I_{K} is maximal monotone.

Thus the same is true for T, defined by T (u) = ∂I_{K}(u + u_{0}), and Theorem 2.12 can be applied to conclude the proof.

3. Existence theorems for problems with measure or L^{1} data

Notation 3.1. If any ψ ∈ L^{∞}(Ω) is given, let:

K(ψ) = {v ∈ H_{0}^{1,1}(Ω): v ≥ ψ a.e. on Ω}

and V_{0}^{∞}(Ω, ψ) = {φ ∈ D(Ω): ∀(v_{h})_{h∈N} in H_{0}^{1,p}(Ω) ∩ K(ψ), ∃(φ_{h})_{h∈N} in D(Ω) such that

φ_{h} → φ in D(Ω) and v_{h} + φ_{h} ∈ K(ψ)∀h ∈ N}.

For k > 0 denote τ_{k}: IR → IR: τ_{k}(s) = (s ∧ k) ∨ (−k) and v^{k}: = τ_{k} ∘ v for any v ∈ H_{loc}^{1,1}(Ω).

Remark 3.2. Let g: Ω → IR^{n} be a measurable selection of a(·, u, Du). If, together with the assumptions in section 1 on the multivalued map a, we suppose p ∈ (2 − \frac{1}{n}, n)

and u ∈ H_{0}^{1,r}(Ω) for every r ∈ [1, \frac{n(p−1)}{n−1}), then by (1.2) and (1.1.2), it turns out that

g ∈ L^{1}(Ω).

Weakly formulated problems

Let ψ ∈ L^{∞}(Ω) and K(ψ) like in Notation 3.1; we assume W_{0}^{1,∞}(Ω) ∩ K(ψ) to be non empty. We denote p_{0} = n(p−1)/(n−1) and assume p ∈ (2 − \frac{1}{n}, n). We suppose moreover

that for a: Ω × IR × IR^{n} → 2^{IR^{n}} all conditions stated in section 1 are true but replacing (1.1.2) by:

\frac{α}{p − 1 − β} < \frac{n − 1}{n − p}.  (3.1)

Problem with measure data. Let f: B(Ω) → IR be a bounded Radon measure and V_{0}^{∞}(Ω, ψ) like in Notation 3.1. We consider the following problem:

\{ \begin{matrix}find u ∈ K(ψ), u ∈ H_{0}^{1,r}(Ω) for each r ∈ [1, p_{0}) and \\ g selection of a(·, u, Du), g ∈ L^{r/(p−1)}(Ω)^{n} for each r ∈ [(p − 1) ∨ 1, p_{0}) \\ such that ∫_{Ω}⟨g, Dφ⟩ ≥ ∫_{Ω} φdf for each φ ∈ V_{0}^{∞}(Ω, ψ). \end{matrix} (II)

Problem with L^{1} data. Let f ∈ L^{1}(Ω). We consider the following problem:

\{ \begin{matrix}find u ∈ K(ψ), u ∈ H_{0}^{1,r}(Ω) for each r ∈ [1, p_{0}) and g selection of a(·, u, Du) \\ g ∈ L^{r/(p−1)}(Ω)^{n} for each r ∈ [(p − 1) ∨ 1, p_{0}) such that when k ≥ ∥ψ∥_{L^{∞}} : \\ u^{k} ∈ H_{0}^{1,p}(Ω), ⟨g, Du^{k}⟩ ∈ L^{1}(Ω), ∫_{Ω}⟨g, D(u^{k} − v)⟩dx ≤ ∫_{Ω} f(u^{k} − v)dx \\ for v ∈ W_{0}^{1,∞}(Ω) ∩ K(ψ). \end{matrix} (III)

P. Oppezzi, A. M. Rossi / Existence of solutions for unilateral problems 251

Theorem 3.3. With the assumptions in subsection 3.1, let (f_{h})_{h∈N} be a sequence in H^{−1,p′}(Ω) ∩ L^{1}(Ω) such that sup{∥f_{h}∥_{L^{1}}: h ∈ N} < ∞. For each h ∈ N let u_{h} ∈ H_{0}^{1,p}(Ω) ∩ K(ψ) be a solution of the problem (I) relative to K = H_{0}^{1,p}(Ω) ∩ K(ψ) and f = f_{h}. Then (u_{h})_{h∈N} is bounded in H_{0}^{1,r}(Ω) for each r ∈ [1, p_{0}); besides if g_{h} ∈ A(u_{h}), (A as defined in our first problem (I)), satisfies ⟨− div g_{h}, v − u_{h}⟩ ≥ ⟨f_{h}, v − u_{h}⟩ for every v ∈ H_{0}^{1,p}(Ω) ∩ K(ψ), then (|g_{h}|^{r/(p−1)})_{h∈N} is bounded in L^{1}(Ω). If moreover k ≥ ∥ψ∥_{L^{∞}} then (u_{h}^{k})_{h∈N} is bounded in H_{0}^{1,p}(Ω).

Proof. Let Ω(h, k) = {x ∈ Ω: k ≤ |u_{h}(x)| < k + 1}, we prove that if k ≥ ∥ψ∥_{L^{∞}} it follows that

∫_{Ω(h,k)} |Du_{h}|^{p}dx ≤ F^{p} where F^{p} = \frac{1}{c} (∫_{Ω} |ν|dx + sup \{∫_{Ω} |f_{h}|dx: h ∈ N\}); (3.2)

and:

∫_{Ω_{h,k}} |Du_{h}|^{r}dx ≤ H(r, k) (∫_{Ω} |u_{h}|^{r^{∗}}dx)^{(p−r)/p} , r ∈ [1, p_{0})  (3.3)

where Ω_{h,k} = \{x ∈ Ω: |u_{h}(x)| ≥ k\} and H(r, k) = F^{r} (∑_{j=k}^{∞} (\frac{1}{j^{r^{∗}}} )^{(p−r)/r})^{r/p} (H(r, k) is a positive real number because r^{∗}(p − r)/r > 1 if r ∈ [1, p_{0})). For k ≥ 0 let φ_{k}: IR → IR be the odd function defined by

φ_{k}(t) = \{ \begin{matrix}0 & if t ∈ [0, k] \\ t − k & if t ∈ (k, k + 1] \\ 1 & if t ∈ (k + 1, ∞) \end{matrix} .

Observing that u_{h} − φ_{k}(u_{h}) ∈ H_{0}^{1,p}(Ω) ∩ K(ψ) if k ≥ ∥ψ∥_{L^{∞}}, from the inequality ∫_{Ω}⟨g_{h}, D(u_{h} − v)⟩dx ≤ ∫_{Ω} f_{h}(u_{h} − v)dx which holds for v ∈ H_{0}^{1,p}(Ω) ∩ K(ψ), h ∈ N, we get ∫_{Ω}⟨g_{h}, Dφ_{k}(u_{h})⟩dx ≤ ∫_{Ω} f_{h}φ_{k}(u_{h})dx ≤ ∫_{Ω} |f_{h}|dx. Moreover by (ii2) we have

\begin{matrix}∫_{Ω}⟨g_{h}, Dφ_{k}(u_{h})⟩dx = & ∫_{Ω}⟨g_{h}, φ′_{k}(u_{h})Du_{h}⟩dx \\ = & ∫_{Ω(h,k)}⟨g_{h}, Du_{h}⟩dx \\ ≥ & ∫_{Ω(h,k)}(ν + c|Du_{h}|^{p})dx, \end{matrix}

hence ∫_{Ω(h,k)} |Du_{h}|^{p}dx ≤ \frac{1}{c} (∫_{Ω} |ν|dx + sup \{∫_{Ω} |f_{h}|dx: h ∈ N\}), namely (3.2).

252 P. Oppezzi, A. M. Rossi / Existence of solutions for unilateral problems

Now from (3.2):

\begin{matrix}∫_{Ω(h,k)} |Du_{h}|^{r}dx ≤ & (∫_{Ω(h,k)} |Du_{h}|^{p}dx)^{r/p} |Ω(h, k)|^{(p−r)/p} \\ ≤ & F^{r} (∫_{Ω(h,k)} |u_{h}|^{r^{∗}}dx)^{(p−r)/p} (\frac{1}{k^{r^{∗}}} )^{(p−r)/p} . \end{matrix}

It follows that:

\begin{matrix}∫_{Ω_{h,k}} |Du_{h}|^{r}dx = & ∑_{j=k}^{∞} ∫_{Ω(h,j)} |Du_{h}|^{r}dx \\ ≤ & F^{r} ∑_{j=k}^{∞} (∫_{Ω(h,j)} |u_{h}|^{r^{∗}}dx)^{(p−r)/p} (\frac{1}{j^{r^{∗}}} )^{(p−r)/p} \\ ≤ & F^{r} (∑_{j=k}^{∞} ∫_{Ω(h,j)} |u_{h}|^{r^{∗}}dx)^{(p−r)/p} (∑_{j=k}^{∞} (\frac{1}{j^{r^{∗}}} )^{(p−r)/r})^{r/p} \\ ≤ & H(r, k) (∫_{Ω} |u_{h}|^{r^{∗}}dx)^{(p−r)/p} , \end{matrix}

and (3.3) is proved. Now observing that u_{h} − u_{h}^{k} + w ∈ H_{0}^{1,p}(Ω) ∩ K(ψ) when k ≥ ∥ψ∥_{L^{∞}}, we have ∫_{Ω}⟨g_{h}, D(u_{h}^{k} − w)⟩dx ≤ ∫_{Ω} f_{h}(u_{h}^{k} − w)dx, thus ∫_{Ω}⟨g_{h}, Du_{h}^{k}⟩dx ≤ ∫_{Ω}⟨g_{h}, Dw⟩dx + (k + ∥w∥_{L^{∞}}) sup{∥f_{h}∥_{L^{1}}: h ∈ N}. As p − 1 is less than p_{0}, to prove our theorem we may suppose r > p − 1, so that by Holder’s inequality:

∫_{Ω}⟨g_{h}, Du_{h}^{k}⟩dx ≤ (∫_{Ω} |g_{h}|^{\frac{r}{p−1}} dx)^{\frac{p−1}{r}} (∫_{Ω} |Dw|^{\frac{r}{r−p+1}} dx)^{\frac{r−p+1}{r}} +
+ (k + ∥w∥_{L^{∞}}) sup{∥f_{h}∥_{L^{1}}: h ∈ N} for any k ≥ ∥ψ∥_{L^{∞}}.

(3.4)

By (1.2) there exists K_{1}, K_{2} ∈ IR_{+} and m ∈ L^{1}(Ω) such that

|g_{h}(x)|^{p′} ≤ m(x) + K_{1}|Du_{h}(x)|^{p} + K_{2}|u_{h}(x)|^{αp/(p−1−β)} for a.e. x ∈ Ω which by means of (ii2), letting ω = m − ν \frac{1}{c}K_{1} and K = \frac{1}{c}K_{1}, gives

|g_{h}(x)|^{p′} ≤ ω(x) + K⟨g_{h}(x), Du_{h}(x)⟩ + K_{2}|u_{h}(x)|^{αp/(p−1−β)} for a.e. x ∈ Ω. Let k ≥ ∥ψ∥_{L^{∞}} and r ∈ ((p − 1) ∨ 1, p_{0}) be fixed and Ω^{h,k} = {x ∈ Ω: |u_{h}(x)| ≤ k}, hence:

∫_{Ω^{h,k}} |g_{h}|^{p′}dx ≤ ∫_{Ω^{h,k}}(ω + K⟨g_{h}, Du_{h}^{k}⟩ + K_{2}|u_{h}|^{αp/(p−1−β)})dx

P. Oppezzi, A. M. Rossi / Existence of solutions for unilateral problems 253

and by (3.4):

\begin{matrix}∫_{Ω^{h,k}} |g_{h}|^{p′}dx ≤ \\ ≤ ∫_{Ω}(|ω| + K_{2}k^{αp/(p−1−β)})dx + K(k + ∥w∥_{L^{∞}}) sup\{∥f_{h}∥_{L^{1}}: h ∈ N\}+ \\ + K (∫_{Ω} |g_{h}|^{\frac{r}{p−1}} dx)^{\frac{p−1}{r}} (∫_{Ω} |Dw|^{\frac{r}{r−p+1}} dx)^{\frac{r−p+1}{r}} . \end{matrix}  (3.5)

On the other hand by r < p_{0} and p < n we have r/(p − 1) < p′, thus letting c(k, r) = |Ω|^{(p−r)/p}(K(k + ∥w∥_{L^{∞}}) sup{∥f_{h}∥_{L^{1}}: h ∈ N} + ∫_{Ω}(|ω| + K_{2}k^{αp/(p−1−β)})dx)^{r/p}, c′(r) = |Ω|^{(p−r)/p}K^{r/p} (∫_{Ω} |Dw|^{\frac{r}{r−p+1}} dx)^{\frac{r−p+1}{p}}, it turns out that

∫_{Ω^{h,k}} |g_{h}|^{\frac{r}{p−1}} dx ≤ (∫_{Ω^{h,k}} |g_{h}|^{p′}dx)^{r/p} |Ω|^{(p−r)/p} ≤ c(k, r) + c′(r) (∫_{Ω} |g_{h}|^{\frac{r}{p−1}} dx)^{\frac{1}{p′}} . Again by (1.2) there exist M_{1}, M_{2} ∈ IR_{+} such that

|g_{h}(x)|^{\frac{r}{p−1}} ≤ |μ(x)|^{\frac{r}{p−1}} + M_{1}|Du_{h}(x)|^{r} + M_{2}|u_{h}(x)|^{αr/(p−1−β)} for a.e. x ∈ Ω. Since (3.1) gives n (1 − \frac{p−1−β}{α} ) < p_{0}, by choosing r > n (1 − \frac{p−1−β}{α} ), we have \frac{αr}{p−1−β} < r^{∗}. Hence by Holder’s and Sobolev’s inequalities:

\begin{matrix}(∫_{Ω} |g_{h}|^{\frac{r}{p−1}} dx)^{\frac{1}{p′}} ≤ \\ ≤ (∫_{Ω}(|μ|^{\frac{r}{p−1}} + M_{1}|Du_{h}|^{r})dx + M_{3} (∫_{Ω} |u_{h}|^{r^{∗}}dx)^{\frac{αr}{(p−1−β)r^{∗}}} )^{\frac{1}{p′}} \\ ≤ M_{0} + M ((∫_{Ω} |Du_{h}|^{r}dx)^{\frac{1}{p′}} + (∫_{Ω} |Du_{h}|^{r}dx)^{\frac{α}{(p−1−β)p′}} ) , \end{matrix}  (3.6)

where M_{3}, M, M_{0} ∈ IR_{+} are suitable constants. In the coercivity condition (ii2) we may suppose \frac{|ν|}{c} ≥ 1, so that

254 P. Oppezzi, A. M. Rossi / Existence of solutions for unilateral problems

|Du_{h}(x)|^{r} ≤ (\frac{2}{c})^{r/(p−1)} (|g_{h}(x)|^{r/(p−1)} + |ν(x)|^{r/(p−1)}) for a.e. x ∈ Ω. Then by (3.6) ∫_{Ω^{h,k}} |Du_{h}|^{r}dx ≤

≤ (\frac{2}{c})^{r/(p−1)} [c(k, r) + c′(r) (∫_{Ω} |g_{h}|^{\frac{r}{p−1}} dx)^{\frac{1}{p′}} + ∫_{Ω} |ν|^{\frac{r}{p−1}} dx]

≤ (\frac{2}{c})^{\frac{r}{p−1}} [c(k, r) + c′(r) (M_{0} + M ((∫_{Ω} |Du_{h}|^{r}dx)^{\frac{1}{p′}} + (∫_{Ω} |Du_{h}|^{r}dx)^{\frac{α}{(p−1−β)p′}} ))

+ ∫_{Ω} |ν|^{\frac{r}{p−1}} dx]. By (3.3) and Sobolev’s inequality: ∫_{Ω_{h,k}} |Du_{h}|^{r}dx ≤ H(r, k)S (∫_{Ω} |Du_{h}|^{r}dx)^{r^{∗}(p−r)/rp}, so that by adding the above inequality: ∫_{Ω} |Du_{h}|^{r}dx ≤

≤ (\frac{2}{c})^{\frac{r}{p−1}} [c(k, r) + c′(r) (M_{0} + M ((∫_{Ω} |Du_{h}|^{r}dx)^{\frac{1}{p′}} + (∫_{Ω} |Du_{h}|^{r}dx)^{\frac{α}{(p−1−β)p′}} )) +

+ ∫_{Ω} |ν|^{\frac{r}{p−1}} dx] + H(r, k)S (∫_{Ω} |Du_{h}|^{r}dx)^{r^{∗}(p−r)/rp} . Since \frac{α}{(p−1−β)p′} < 1 by (1.1.3) and r^{∗}(p − r)/rp < 1 as p < n, the last inequality gives boundedness of (∫_{Ω} |Du_{h}|^{r}dx)_{h∈N}. Hence by Sobolev’s inequality, (u_{h})_{h∈N} is bounded in H_{0}^{1,r}(Ω) if r ∈ [1, p_{0}). By using (1.2) again, if M_{4} > 0 is a suitable constant:

∫_{Ω} |g_{h}|^{r/(p−1)}dx ≤ M_{4} ∫_{Ω}(|μ|^{r/(p−1)} + |Du_{h}|^{r} + |u_{h}|^{αr/(p−1−β)})dx.

Now (3.1) involves that αr/(p − 1 − β) < p_{0}^{∗}, so that (|g_{h}|^{r/(p−1)})_{h∈N} is bounded in L^{1}(Ω) if r ∈ [1, p_{0}) as (u_{h})_{h∈N} is bounded in L^{q}(Ω) for every q ∈ [1, p_{0}^{∗}). Finally from (3.5), being k ≥ ∥ψ∥_{L^{∞}}, we obtain that (g_{h}1_{Ω^{h,k}})_{h∈N} is bounded in (L^{p′}(Ω))^{n}. By coercivity condition ii2), the same is true for (u_{h}^{k})_{h∈N} in H_{0}^{1,p}(Ω). Definition 3.4. ([1]) Let Ω be a bounded open set in IR^{n}, the norm-capacity of H_{0}^{1,p}(Ω) is the map c_{p}: 2^{Ω} → [0, ∞] defined as follows:

c_{p}(K) = inf{∥u∥_{H_{0}^{1,p}}: u ∈ C_{0}^{1}(Ω), u ≥ 0 on Ω, u ≥ 1 on K} if K ⊂ Ω is compact, c_{p}(U) = sup{c_{p}(K): K compact ⊂ U} if U ⊂ Ω is open,

c_{p}(E) = inf{c_{p}(U): U open ⊃ E} for arbitrary E ⊂ Ω.

Definition 3.5. ([1]) We say that u: Ω → IR is c_{p}-quasicontinuous, with c_{p} defined in 3.4, if for every ϵ > 0 there exists an open set U_{ϵ} ⊂ Ω, with c_{p}(U_{ϵ}) < ϵ, such that u_{|Ω\U_{ϵ}} is continuous.

Proposition 3.6. Let Ω be a bounded open set in IR^{n} and c_{p} the norm-capacity as in definition 3.4.

i) If E ∈ L(Ω) and c_{p}(E) = 0, then |E| = 0. ii) If u ∈ H_{0}^{1,p}(Ω) there exists ũ: Ω → IR c_{p}-quasi continuous, such that u = ũ a.e. on

Ω.

Proof. i) If E ∈ L(Ω) and |E| > 0, there exists a compact K ⊂ E with |K| > 0, hence by definition of c_{p} and its monotonicity c_{p}(E) ≥ c_{p}(K) ≥ |K|^{1/p}. ii) For this proof we refer to [1], Proposition 2.8, or, for a more immediate statement, to [6] Proposition 7.7. Indeed it is possible to verify that, in the case of H_{0}^{1,p}(Ω), the two definitions of capacity given in [1] and in [6] coincide with c_{p} introduced above and moreover the two quasi-continuous representatives coincide except on a set of zero capacity at most. See also [11] and [7] for a general overview on the notion of capacity and relative properties.

Notation 3.7. If u ∈ H_{0}^{1,p}(Ω) we denote henceforth by ũ a c_{p}-quasi continuous representative of u.

Definition 3.8. ([1]) A Radon measure μ: L(Ω) → IR is said to be of finite energy relative to H_{0}^{1,p}(Ω) if it is continuous on (C_{0}^{1}(Ω), ∥∥_{H^{1,p}}).

Proposition 3.9.

i) If μ: L(Ω) → IR is a positive and finite energy Radon measure relative to H_{0}^{1,p}(Ω) then μ(E) = 0 if E ∈ L(Ω) and c_{p}(E) = 0. Moreover ũ ∈ L^{1}(Ω, μ) for any u ∈ H_{0}^{1,p}(Ω).

ii) Let φ ∈ H^{−1,p′}(Ω) be a positive functional. Then there exists a positive finite energy Radon measure μ_{φ} such that ⟨φ, u⟩ = ∫_{Ω} ũdμ_{φ} for every u ∈ H_{0}^{1,p}(Ω).

Proof. It follows from [1], Propositions 2.20, 2.21, 2.22.

Definition 3.10. ([1]) A convex set K ⊂ H_{0}^{1,p}(Ω) is unilateral if it is closed, nonempty and:

- u ∧ v ∈ K for every u, v ∈ K;

- u + ṽ ∈ K for every u ∈ K, v ∈ H_{0}^{1,p}(Ω), with ṽ ≥ 0 c_{p} − a.e.

Theorem 3.11. Let K ⊂ H_{0}^{1,p}(Ω) be an unilateral convex set.

a) There exists χ: Ω → [−∞, ∞] c_{p}-quasi upper semicontinuous (i.e. for every ϵ > 0 there is an open set A ⊂ Ω with c_{p}(A) < ϵ such that χ|_{Ω\A} is upper semicontinuous) such that

K = {u ∈ H_{0}^{1,p}(Ω): ũ ≥ χ c_{p} − a.e. on Ω}.

b) Let f ∈ H^{−1,p′}(Ω) be given and u ∈ H_{0}^{1,p}(Ω) be a solution of (I) where K is supposed to be unilateral. Then if g ∈ A(u) is such that ⟨− div g, v − u⟩ ≥ ⟨f, v − u⟩ for every v ∈ K, and χ is related to K as in a), there exists a positive and finite energy Radon measure μ such that:

∫_{Ω} ṽdμ = ⟨− div g − f, v⟩ for every v ∈ H_{0}^{1,p}(Ω) and ∫_{Ω}(ũ − χ)dμ = 0.

Proof. See [1] Théorème 3.2 and Généralization.

Lemma 3.12. Let f ∈ H^{−1,p′}(Ω) be given and u ∈ H_{0}^{1,p}(Ω) be a solution of the problem (I) where K ⊂ H_{0}^{1,p}(Ω) is assumed to be an unilateral convex set. If g ∈ A(u) satisfies ⟨− div g, v − u⟩ ≥ ⟨f, v − u⟩ for every v ∈ K, then:

i) ∫_{Ω}⟨g, D(φ(u − v))⟩dx ≤ ⟨f, φ(u − v)⟩ for every v ∈ K and φ ∈ C^{1}(Ω) ∩ W^{1,∞}(Ω), φ ≥ 0 on Ω.

ii) If v ∈ K, u − v ∈ L^{∞}(Ω), φ ∈ H_{0}^{1,p}(Ω) ∩ L^{∞}(Ω), φ ≥ 0 a.e. on Ω, then φ(u − v) ∈ H_{0}^{1,p}(Ω) and

∫_{Ω}⟨g, D(φ(u − v))⟩dx ≤ ⟨f, φ(u − v)⟩.

Proof. i) Let χ and μ be as in theorem 3.11, φ ∈ C^{1}(Ω) ∩ W^{1,∞}(Ω), φ ≥ 0 and v ∈ K. It can be easily seen that φ(u − v) ∈ H_{0}^{1,p}(Ω), thus if χ and μ are given by Theorem 3.11:

∫_{Ω}⟨g, D(φ(u − v))⟩dx = ⟨− div g, φ(u − v)⟩
= ⟨− div g − f, φ(u − v)⟩ + ⟨f, φ(u − v)⟩ = ∫_{Ω} φ(ũ − ṽ)dμ + ⟨f, φ(u − v)⟩
= ∫_{Ω} φ(ũ − χ)dμ + ∫_{Ω} φ(χ − ṽ)dμ + ⟨f, φ(u − v)⟩ ≤ ⟨f, φ(u − v)⟩.

The last inequality depends also on Proposition 3.9i) which, by ṽ ≥ χ c_{p}-a.e. on Ω, gives ṽ ≥ χ μ-a.e. on Ω. ii) If φ ∈ H_{0}^{1,p}(Ω) ∩ L^{∞}(Ω), φ ≥ 0 a.e. on Ω, there exists a sequence (φ_{i})_{i∈N} in C_{0}^{1}(IR^{n}), φ_{i} → φ in H^{1,p}(Ω), with ∥φ_{i}∥_{L^{∞}} ≤ ∥φ∥_{L^{∞}}, φ_{i} ≥ 0 for every i ∈ N. To see this we may take φ_{i} = J_{ϵ_{i}} ∗ φ where J_{ϵ_{i}} are the usual mollifiers and ϵ_{i} ↘ 0. Therefore, by passing to a subsequence if necessary, φ_{i}(u − v) → φ(u − v) in H_{0}^{1,p}(Ω). Hence ⟨f, φ_{i}(u − v)⟩ → ⟨f, φ(u − v)⟩, ∫_{Ω}⟨g, D(φ_{i}(u − v))⟩dx → ∫_{Ω}⟨g, D(φ(u − v))⟩dx and from i) follows ii).

Remark 3.13. With the assumptions in subsection 3.1, for a given (f_{h})_{h∈N} such that sup{∥f_{h}∥_{L^{1}}: h ∈ N} < ∞ let us take (u_{h})_{h∈N}, (g_{h})_{h∈N} as in Theorem 3.3 and fix r ∈ ((p−1)∨1, p_{0}) and k ≥ ∥ψ∥_{L^{∞}}. Then let u ∈ H_{0}^{1,r}(Ω) be the weak limit of a subsequence of (u_{h})_{h∈N}, Ω^{k} = {x ∈ Ω: |u(x)| < k}, Ω^{h,k} = {x ∈ Ω: |u_{h}(x)| < k}, g_{h,k} = g_{h}1_{Ω^{h,k}}.

P. Oppezzi, A. M. Rossi / Existence of solutions for unilateral problems 257

From boundedness of (u_{n}^{k})_{h∈N} in H_{0}^{1,p}(Ω), due to Theorem 3.3, by using (1.2) it follows that (g_{h,k})_{h∈N} is bounded in L^{p′}(Ω)^{n}. Moreover if (h_{j})_{j∈N} is an increasing sequence in N such that g_{h_{j}} ⇀ g in L^{r/(p−1)}(Ω)^{n}, u_{h_{j}} ⇀ u in H_{0}^{1,r}(Ω) and g_{h_{j},k} ⇀ g̃_{k} in L^{p′}(Ω)^{n}, then

g̃_{k}(x) = g(x) for a.e. x ∈ Ω^{k}.

By Rellich’s theorem we may assume also u_{h_{j}} → u a.e. in Ω. Thus, if ϵ, δ ∈ IR_{+} there exists Ω(ϵ) ⊂ Ω^{k−δ} such that u_{h_{j}} → u uniformly on Ω(ϵ) and |Ω^{k−δ}\Ω(ϵ)| < ϵ. Let j_{ϵ,δ} ∈ N be such that |u_{h_{j}}(x)| < k when j > j_{ϵ,δ} and x ∈ Ω(ϵ), hence Ω(ϵ) ⊂ Ω^{h_{j},k} if j > j_{ϵ,δ}. Since if x ∈ Ω(ϵ) is g_{h_{j},k}(x) = g_{h_{j}}(x), it follows that ∫_{E} gdx = lim_{j→∞} ∫_{E} g_{h_{j},k}dx = ∫_{E} g̃_{k}dx for any E ⊂ Ω(ϵ) measurable. Therefore g = g̃_{k} a.e. on Ω(ϵ) and from the arbitrariness of ϵ, δ it easily follows that g̃_{k} = g a.e. on Ω^{k}.

Theorem 3.14. With the assumptions in subsection 3.1, let (f_{h})_{h∈N}, (u_{h})_{h∈N}, (g_{h})_{h∈N} as in Theorem 3.3. If u ∈ H_{0}^{1,r}(Ω), g ∈ L^{r/(p−1)}(Ω)^{n} for some r ∈ ((p − 1) ∨ 1, p_{0}), and u_{h} ⇀ u in H_{0}^{1,r}(Ω), g_{h} ⇀ g in L^{r/(p−1)}(Ω)^{n}, then

u ∈ K(ψ) and g(x) ∈ a(x, u(x), Du(x)) for a.e. x ∈ Ω.

Proof. By passing to a subsequence if necessary, by Rellich’s theorem we may suppose that u_{h} → u a.e. on Ω, thus, being (u_{h})_{h∈N} in K(ψ), we get u ∈ K(ψ). Let w ∈ W_{0}^{1,∞}(Ω) ∩ K(ψ), ϑ ∈ C_{0}^{1}(Ω), ϑ ≥ 0, k ≥ ∥ψ∥_{L^{∞}} and Ω^{k}, Ω^{h,k}, g_{h,k} be like in 3.13. Moreover let us consider, for ϵ ∈ IR_{+}, τ_{ϵ}: IR → IR defined as in Notation 3.1 and, for k ∈ (1, ∞), the even function σ_{k}: IR → IR defined by

σ_{k}(t) = \{ \begin{matrix}1 & if 0 ≤ t < k − 1 \\ 0 & if t ≥ k \\ −t + k & if k − 1 ≤ t < k \end{matrix}

We observe that u_{h}^{k} = τ_{k} ∘ u_{h} ⇀ u^{k} = τ_{k} ∘ u in H_{0}^{1,p}(Ω). Indeed as (u_{h}^{k})_{h∈N} is bounded in H_{0}^{1,p}(Ω), due to Theorem 3.3, every subsequence has a subsequence converging in H_{0}^{1,p}(Ω), whose limit is u^{k} because u_{h} → u a.e. on Ω. Now let ξ ∈ IR^{n}, η be a measurable selection of a(·, u^{k}, ξ) and (η_{h})_{h∈N} be given by hypothesis iii) in connection with u^{k}, (u_{h}^{k})_{h∈N}, v(x) = ⟨ξ, x⟩ and η, such that η_{h} → η a.e. in Ω and η_{h} is a measurable selection of a(·, u_{h}^{k}, ξ) for every h ∈ N. Letting φ_{h,k} = (σ_{k} ∘ u_{h})(σ_{k} ∘ u)ϑ, from monotonicity of a(x, u_{h}(x), ·) for a.e. x ∈ Ω and from definition of φ_{h,k} it follows that:

\begin{aligned}0 & ≤ ∫_{Ω}⟨g_{h} − η_{h}, Du_{h} − ξ⟩τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k}dx = \\ & = ∫_{Ω}⟨g_{h}, D(u_{h} − u)⟩τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k}dx + ∫_{Ω}⟨g_{h}, Du⟩τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k}dx+ \\ & − ∫_{Ω}⟨g_{h}, ξ⟩τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k}dx − ∫_{Ω}⟨η_{h}, Du_{h} − ξ⟩τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k}dx. \end{aligned} (3.7)

Now, by Proposition 3.6i), K(ψ) ∩ H_{0}^{1,p}(Ω) is an unilateral convex set. Since u_{h}−τ_{ϵ}∘(u_{h}−u) ∈ K(ψ), Lemma 3.12ii) may be applied with φ_{h,k}, u_{h}, u_{h}−τ_{ϵ}∘(u_{h}−u) instead of φ, u and v respectively, getting:

∫_{Ω}⟨g_{h}, D(u_{h} − u)⟩τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k}dx = ∫_{Ω}⟨g_{h}, Dτ_{ϵ} ∘ (u_{h} − u)⟩φ_{h,k}dx

= ∫_{Ω}⟨g_{h}, D(φ_{h,k}τ_{ϵ} ∘ (u_{h} − u))⟩dx − ∫_{Ω}⟨g_{h,k}, Dφ_{h,k}⟩τ_{ϵ} ∘ (u_{h} − u)dx

≤ ∫_{Ω}⟨f_{h}, φ_{h,k}τ_{ϵ} ∘ (u_{h} − u)⟩dx − ∫_{Ω}⟨g_{h,k}, (σ′_{k} ∘ u_{h})Du_{h}⟩(σ_{k} ∘ u)ϑτ_{ϵ} ∘ (u_{h} − u)dx

− ∫_{Ω}⟨g_{h,k}, (σ′_{k} ∘ u)Du⟩(σ_{k} ∘ u_{h})ϑτ_{ϵ} ∘ (u_{h} − u)dx+
− ∫_{Ω}⟨g_{h,k}, Dϑ⟩(σ_{k} ∘ u_{h})(σ_{k} ∘ u)τ_{ϵ} ∘ (u_{h} − u)dx

≤ ϵ sup_{h∈N} ∥f_{h}∥_{L^{1}}∥ϑ∥_{L^{∞}} + ϵ sup_{h∈N} ∥g_{h,k}∥_{L^{p′}} sup_{h∈N} ∥Du_{h}^{k}∥_{L^{p}}∥ϑ∥_{L^{∞}}+

+ ϵ sup_{h∈N} ∥g_{h,k}∥_{L^{p′}}∥Du^{k}∥_{L^{p}}∥ϑ∥_{L^{∞}} + ∥τ_{ϵ} ∘ (u_{h} − u)Dϑ∥_{L^{p}} sup_{h∈N} ∥g_{h,k}∥_{L^{p′}},

where, by Remark 3.13, sup_{h∈N} ∥g_{h,k}∥_{L^{p′}} < ∞. We observe that Du^{k}τ′_{ϵ} ∘ (u_{h} − u)(σ_{k} ∘ u_{h})(σ_{k} ∘ u)ϑ → Du^{k}(σ_{k} ∘ u)^{2}ϑ a.e. on Ω and hence strongly in L^{p}(Ω)^{n}. Thus if g̃_{k} is the weak limit in L^{p′}(Ω)^{n} of a subsequence of (g_{h,k})_{h∈N} which we shall denote by (g_{h,k})_{h∈N} as well, by Remark 3.13, we have:

lim_{h→∞} ∫_{Ω}⟨g_{h}, Du⟩τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k}dx

= lim_{h→∞} ∫_{Ω}⟨g_{h,k}, Du^{k}⟩τ′_{ϵ} ∘ (u_{h} − u)(σ_{k} ∘ u_{h})(σ_{k} ∘ u)ϑdx

= ∫_{Ω}⟨g̃_{k}, Du^{k}⟩(σ_{k} ∘ u)^{2}ϑdx = ∫_{Ω}⟨g, Du^{k}⟩(σ_{k} ∘ u)^{2}ϑdx.

Now since g_{h} ⇀ g in L^{r/(p−1)}(Ω)^{n}, ξτ′_{ϵ} ∘ (u_{h} − u)φ_{h,k} → ξ(σ_{k} ∘ u)^{2}ϑ a.e. on Ω and consequently in the norm of L^{q}(Ω)^{n} for every q ∈ (1, ∞), it follows that:

lim_{h→∞} ∫_{Ω}⟨g_{h}, −ξ⟩τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k}dx = ∫_{Ω}⟨g, −ξ⟩(σ_{k} ∘ u)^{2}ϑdx.

P. Oppezzi, A. M. Rossi / Existence of solutions for unilateral problems 259

The growth condition ii1) involves that

|η_{h}τ′_{ϵ} ∘ (u_{h_{j}} − u)φ_{h,k}| ≤ (μ + c_{1}|ξ|^{p−1} + c_{2}|u_{h}^{k}|^{α}|ξ|^{β})ϑ1_{Ω^{h,k}} a.e. on Ω.

Thus since η_{h}τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k} → η(σ_{k} ∘ u)^{2}ϑ a.e. on Ω such a convergence is also strong in L^{p′}(Ω)^{n}. On the other hand (Du_{h}^{k} − ξ) ⇀ (Du^{k} − ξ) in L^{p}(Ω)^{n}, so that:

lim_{h→∞} ∫_{Ω}⟨−η_{h}, Du_{h} − ξ⟩τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k}dx

= lim_{h→∞} ∫_{Ω}⟨−η_{h}, Du_{h}^{k} − ξ⟩τ′_{ϵ} ∘ (u_{h} − u)φ_{h,k}dx

= ∫_{Ω}⟨−η, Du^{k} − ξ⟩(σ_{k} ∘ u)^{2}ϑdx = ∫_{Ω}⟨−η, Du − ξ⟩(σ_{k} ∘ u)^{2}ϑdx.

From (3.7) and all above inequalities, being lim_{h→∞} ∥τ_{ϵ} ∘ (u_{h} − u)Dϑ∥_{L^{p}} = 0, we get:

0 ≤ ∫_{Ω}⟨g − η, Du − ξ⟩(σ_{k} ∘ u)^{2}ϑdx + ϵ(sup_{h∈N} ∥f_{h}∥_{L^{1}} + sup_{h∈N} ∥g_{h,k}∥_{L^{p′}} sup_{h∈N} ∥Du_{h}^{k}∥_{L^{p}}+

+ sup_{h∈N} ∥g_{h,k}∥_{L^{p′}}∥Du^{k}∥_{L^{p}})∥ϑ∥_{L^{∞}}.

Since ϵ and ϑ are arbitrary it follows that: 0 ≤ ⟨g − η, Du^{k} − ξ⟩(σ_{k} ∘ u)^{2} a.e. on Ω, hence 0 ≤ ⟨g − η, Du^{k} − ξ⟩ a.e. on Ω^{k−1}. Also ξ ∈ IR^{n} and the measurable selection η of a(·, u^{k}, ξ) are arbitrarily choosen, so that, like in the last part of the proof of Lemma 2.1, we get ⟨g(x) − ζ, Du(x) − ξ⟩ ≥ 0 for a.e. x ∈ Ω^{k−1}, every ξ ∈ IR^{n} and every ζ ∈ a(x, u^{k}(x), ξ). Finally the maximal monotonicity of a(x, u(x), ·) for a.e. x ∈ Ω, ensures that g(x) ∈ a(x, u(x), Du(x)) for a.e. x ∈ Ω^{k−1}, which concludes the proof as k is also arbitrary.

Theorem 3.15. With the assumptions in subsection 3.1, there exists a solution of the problem (II) where f: B(Ω) → IR is a bounded Radon measure.

Proof. Let us define f_{h}: Ω → IR by f_{h}(x) = ∫_{Ω} J_{ϵ_{h}}(x − y)df(y), where J_{ϵ_{h}} are the usual mollifiers and ϵ_{h} ↘ 0. Then (f_{h})_{h∈N} is a sequence of H^{−1,p′}(Ω) ∩ L^{1}(Ω) such that sup{∥f_{h}∥_{L^{1}}: h ∈ N} < ∞ and f_{h} → f in the distributional sense. Corresponding to each f_{h}, by Theorem 2.13 there exist u_{h} ∈ H_{0}^{1,p}(Ω) ∩ K(ψ) and g_{h} ∈ A(u_{h}), solving the problem (I), i.e. ⟨− div g_{h}, v − u_{h}⟩ ≥ ⟨f_{h}, v − u_{h}⟩ for every v ∈ H_{0}^{1,p}(Ω) ∩ K(ψ). By Theorem 3.3, fixing r ∈ ((p−1)∨1, p_{0}), there exist u ∈ H_{0}^{1,r}(Ω)∩K(ψ), g ∈ L^{r/(p−1)}(Ω)^{n} and an increasing sequence (h_{j})_{j∈N} in N such that u_{h_{j}} ⇀ u in H_{0}^{1,r}(Ω) and g_{h_{j}} ⇀ g in L^{r/(p−1)}(Ω)^{n}. Then for any s ∈ ((p − 1) ∨ 1, p_{0}) we have u_{h_{j}} ⇀ u in H_{0}^{1,s}(Ω) and g_{h_{j}} ⇀ g in L^{s/(p−1)}(Ω)^{n}. Indeed from every subsequence of (u_{h_{j}})_{j∈N} we can extract a further subsequence which weakly converges in H_{0}^{1,s}(Ω), whose limit still is u, as H^{−1,r′}(Ω) ⊂ H^{−1,s′}(Ω). We can apply the same argument to (g_{h_{j}})_{j∈N}.

260 P. Oppezzi, A. M. Rossi / Existence of solutions for unilateral problems

Moreover by Theorem 3.14, g(x) ∈ a(x, u(x), Du(x)) for a.e. x ∈ Ω. Now if some φ ∈ V_{0}^{∞}(Ω, ψ) is given, by definition of V_{0}^{∞}(Ω, ψ) there exists (φ_{h})_{h∈N} in D(Ω), φ_{h} → φ in the topology of D(Ω), such that for every h ∈ N u_{h} + φ_{h} ∈ K(ψ). Thus from ∫_{Ω}⟨g_{h}, Dφ_{h}⟩dx ≥ ∫_{Ω} f_{h}φ_{h}dx letting h → ∞, we get ∫_{Ω}⟨g, Dφ⟩dx ≥ ∫_{Ω} φdf.

Theorem 3.16. With the assumptions made in subsection 3.1, there exists a solution of problem (III) where f is supposed to be an element of L^{1}(Ω) and ψ ∈ L^{∞}(Ω).

Proof. Let (f_{h})_{h∈N} be a sequence in H^{−1,p′}(Ω) ∩ L^{1}(Ω) such that f_{h} → f in L^{1}(Ω). For every h ∈ N we consider a solution u_{h} ∈ H_{0}^{1,p}(Ω) ∩ K(ψ) of problem (I) corresponding to f_{h}, which exists by Theorem 2.13. Then, for every h ∈ N, let g_{h} ∈ A(u_{h}) be such that ⟨− div g_{h}, v − u_{h}⟩ ≥ ⟨f_{h}, v − u_{h}⟩ for any v ∈ H_{0}^{1,p}(Ω) ∩ K(ψ). For a fixed k ≥ ∥ψ∥_{L^{∞}}, like in the proof of the previous theorem, in virtue of Theorems 3.3 and 3.14 and of Remark 3.13, there exist u ∈ H_{0}^{1,r}(Ω) and a selection g ∈ L^{r/(p−1)}(Ω)^{n} of a(·, u, Du), such that, by passing to a subsequence if necessary, u_{h} ⇀ u in H_{0}^{1,r}(Ω) and a.e. in Ω, g_{h} ⇀ g in L^{r/(p−1)}(Ω)^{n} for every r ∈ ((p − 1) ∨ 1, p_{0}). Moreover, if u_{h}^{k} = τ_{k} ∘ u_{h} and u^{h} = τ_{k} ∘ u, with τ_{k} defined as in 3.1, then u_{h}^{k} ⇀ u^{k} in H_{0}^{1,p}(Ω) and a.e. on Ω. When k ≥ ∥ψ∥_{L^{∞}} then u_{h} − u_{h}^{k} + v ∈ H_{0}^{1,p}(Ω) ∩ K(ψ), so that for any v ∈ W_{0}^{1,∞}(Ω) ∩ K(ψ), h ∈ N:

∫_{Ω}⟨g_{h}, D(u_{h}^{k} − v)⟩dx ≤ ∫_{Ω} f_{h}(u_{h}^{k} − v)dx.  (3.8)

Like in Remark 3.13 let Ω^{h,k} = {x ∈ Ω: |u_{h}(x)| < k}, g_{h,k} = g_{h}1_{Ω^{h,k}}, so that

∫_{Ω}⟨g_{h}, Du_{h}^{k}⟩dx = ∫_{Ω}⟨g_{h} − g_{h,k}, Du_{h}^{k}⟩dx + ∫_{Ω}⟨g_{h,k}, Du_{h}^{k}⟩dx = ∫_{Ω}⟨g_{h,k}, Du_{h}^{k}⟩dx.

Let now γ be a measurable selection of a(·, u^{k}, Du^{k}) such that γ|_{Ω^{k}} = g|_{Ω^{k}}, where Ω^{k} = {x ∈ Ω: |u(x)| < k}. By hypothesis (iii) for every h ∈ N we may take a measurable selection γ_{h} of a(·, u_{h}^{k}, Du^{k}) such that γ_{h} → γ a.e. in Ω.

Moreover taking monotonicity of a(x, u_{h}(x), ·) into account:

∫_{Ω}⟨g_{h,k}Du_{h}^{k}⟩dx
= ∫_{Ω^{h,k}}⟨g_{h,k} − γ_{h}, Du_{h}^{k} − Du^{k}⟩dx + ∫_{Ω}⟨g_{h,k}, Du^{k}⟩dx + ∫_{Ω^{h,k}}⟨γ_{h}, Du_{h}^{k} − Du^{k}⟩dx
≥ ∫_{Ω^{h,k}}⟨γ_{h}, Du_{h}^{k} − Du^{k}⟩dx + ∫_{Ω}⟨g_{h,k}, Du^{k}⟩dx.

By Remark 3.13 there exists g̃_{k} ∈ L^{p′}(Ω)^{n} such that, by passing to a further subsequence if necessary, g_{h,k} ⇀ g̃_{k} in L^{p′}(Ω)^{n} and g̃_{k}(x) = g(x) for a.e. x ∈ Ω^{k}. Hence ⟨g, Du^{k}⟩ ∈ L^{1}(Ω) and ∫_{Ω}⟨g_{h,k}, Du^{k}⟩dx → ∫_{Ω}⟨g, Du^{k}⟩dx.

P. Oppezzi, A. M. Rossi / Existence of solutions for unilateral problems 261

Now we see that (γ_{h})_{h∈N} is strongly convergent to γ in L^{p′}(Ω)^{n}: indeed from growth condition (ii1) it follows that |γ_{h}| ≤ μ + c_{1}|Du^{k}|^{p−1} + c_{2}|u_{h}^{k}|^{α}|Du^{k}|^{β} a.e. on Ω and |Du^{k}|^{β} ∈ L^{p′}(Ω) as, by (1.1.3), βp′ < p. Moreover (Du_{h}^{k} −Du^{k})1_{Ω^{h,k}} ⇀ 0 in L^{p}(Ω)^{n} as Du^{k}(1_{Ω^{h,k}} −1_{Ω^{k}}) → 0 a.e. on Ω. Therefore

lim_{h→∞} ∫_{Ω^{h,k}}⟨γ_{h}, Du_{h}^{k} − Du^{k}⟩dx = 0.

Then from (3.8) it follows that for a suitable increasing sequence (h_{j})_{j∈N} in N we get:

∫_{Ω}⟨g, D(u^{k} − v)⟩dx

= lim_{h→∞} ∫_{Ω^{h,k}}⟨γ_{h}, Du_{h}^{k} − Du^{k}⟩dx + ∫_{Ω}⟨g_{h,k}, Du^{k}⟩dx − ∫_{Ω}⟨g_{h}, Dv⟩dx ≤

≤ lim_{j→∞} ∫_{Ω}⟨g_{h_{j}}, D(u_{h_{j}}^{k} − v)⟩dx ≤ lim_{j→∞} ∫_{Ω} f_{h_{j}}(u_{h_{j}}^{k} − v)dx = ∫_{Ω} f(u^{k} − v)dx.

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