On pseudomonotone elliptic operators with functional dependence on unbounded domains

On pseudomonotone elliptic operators with functional dependence on unbounded domains

Mihály Csirik

Department of Applied Analysis, Eötvös Loránd University, Hungary, H-1117 Budapest, Pázmány Péter sétány 1/C.

Received 17 February 2016, appeared 1 May 2016 Communicated by László Simon

Abstract. We generalize F. E. Browder’s results concerning pseudomonotone elliptic partial differential operators defined on unbounded domains. Browder treated equa- tions for quasilinear operators of divergence form

|α|≤k

∑

Dαaα(x,u(x), . . . ,D^βu(x)) = f(x),

on an arbitrary unbounded domain Ω, where |β| ≤ k for some k ≥ 1. We show that under suitable assumptions, Browder’s result holds true if the functions a_α are functionals ofu.

Keywords: elliptic operators, nonlocal, nonlinear, pseudomonotone operators.

2010 Mathematics Subject Classification: 35J88, 35J87.

1 Introduction

The theory of pseudomonotone operators proved to be highly useful for establishing existence and uniqueness theorems for divergence-form elliptic problems with standard growth condi- tions (see [10] too, for general growth conditions). The concept of pseudomonotonicity was introduced by H. Brezis [2] in 1968.

Definition 1.1. Let X be a Banach space. A bounded operator A: X → X^∗ is said to be pseudomonotoneif for any sequence{u_j} ⊂X, such that

u_j *u (_in X) _{and lim sup}

j→_∞

hA(u_j)_,u_j−ui ≤_0, then

(PM1) hA(u_j),u_j−ui →0 as j→_∞and (PM2) A(u_j)* A(u)in X^∗ as j→_∞.

BEmail: csirik@gmail.com

As usual, the symbol*denotes weak convergence.

The following abstract surjectivity result [9, Theorem 2.12] is widely used in the literature for proving the existence of a weak solution to a nonlinear elliptic partial differential equation.

Theorem 1.2. Let X be a reflexive separable Banach space and A: X → X^∗ a bounded, coercive and pseudomonotone operator. Then for arbitrary F∈X^∗, there exists u∈X, such that A(u) =F in X^∗.

In this context, coercivity is defined as follows:

Definition 1.3. An operatorA: X→X^∗is calledcoerciveif hA(u),ui

kuk →+_∞ (askuk →_∞).

Guaranteeing boundedness and coercivity is usually a trivial matter. The proof of pseu- domonotonicity usually involves the Rellich–Kondrachov compactness theorem as a crucial step. On unbounded domains however, a compact embedding result seems to require more complicated conditions on the domain, see e.g. [1, Theorem 6.52]. F. E. Browder managed to avoid the use of such compactness results in [3]. To establish pseudomonotonicity, it turns out that the main task is to prove the a.e. convergence of the sequences{D^αu_j}^∞_j=1. Browder’s idea is a natural one: let the unbounded domainΩbe exhausted by an increasing sequence{_Ωi} of bounded domains with smooth boundary – such that on eachΩi the Rellich–Kondrachov theorem holds. Combining this with a diagonal argument, we extract a subsequence of the lower-order derivatives{D^αu_j}converging a.e. toD^αu(|α| ≤k−1). Proving a.e. convergence of the highest-order derivativesD^αu_j →D^αu(|α|=k) is more involved.

The results of F. E. Browder on nonlinear elliptic equations on unbounded domains have been extended in [10], [4] and [6] to strongly nonlinear elliptic equations, i.e. equations con- taining a term which is arbitrarily quickly increasing with respect to the values of unknown functionu. Further, there are some results in [7] and [8] on elliptic problems where the lower order terms or the boundary condition contains nonlocal (e.g. integral type) dependence onu.

The aim of this paper is to extend Browder’s theorem to elliptic operators with nonlocal dependence in the main (highest order) terms, too: we shall modify the assumptions and the proof of the original theorem for 2k-order divergence-type nonlinear functional elliptic equations. After formulating sufficient conditions for such a nonlocal operator to be bounded, coercive and pseudomonotone, we prove our main result. Finally, we give concrete examples that satisfy our assumptions.

2 Problem formulation

Let Ω ⊂ _Rⁿ be a possibly unbounded domain with sufficiently smooth boundary, and let W₀^k,p(_Ω) ⊂ V ⊂ W^k,p(_Ω) be a closed linear subspace with 1 < p < _∞ and k ≥ 1. Let

A:V →V^∗ be defined by hA(u),vi=

∑

|α|≤k

Z

Ωaα(x,u(x), . . . ,D^βu(x), . . . ;u)D^αv(x)dx (2.1) for allu,v∈ V, where |β| ≤k is a multiindex. The functionaα may depend on the pointwise values of any of the partial derivatives ofu. Furthermore, “;u” notation signifies thata_α may be afunctionalofu. In other words,a_α may depend on thewholesolutionu.

The arguments of the functions a_α are denoted as a_α(x,η;u), and we sometimes split η as η = (ζ,ξ) where ζ ∈ _R^N1 and ξ ∈ _R^N2, so that η ∈ _R^N with N = N₁+N2 and write aα(x,ζ,ξ;u), where the numbers N₁andN₂ denote number of multiindexesβsuch that|β| ≤ k−1 and|β|=k, respectively. Furthermore, the notation

η<sup>(`)</sup>={η<sub>β</sub> : |β|=`}

is used, where`=0, 1, . . . ,k. Note that

ζ =ⁿη<sup>(`)</sup> : `=0, 1, . . . ,k−1o

and ξ =ⁿη<sup>(`)</sup> : `= ko . We impose the following assumptions on the structure of AandΩ.

(A0)Suppose that there exist a sequence{_Ωi} ⊂_Rⁿof bounded domains such thatΩi ⊂_Ωi+1 (i = 1, 2, . . .) andΩ = ^S∞_i=1Ωi. Furthermore, assume that each ∂Ωi is sufficiently smooth so that the Rellich–Kondrachov theorem holds: W^k,p(_Ωi)⊂⊂W^k−1,p(_Ωi)(i=1, 2, . . .).

(A1) Let a_α be Carathéodory functions for fixed u ∈ V and all multiindex |α| ≤ k, i.e. let a_α(·,η;u)be measurable for every fixedη∈ _R^N, and leta_α(x, ·;u)be continuous for almost every fixedx ∈_Ω.

(A2)Suppose that there exist a bounded functionalg₁: V→_R+and a compact map k^α₁: V →L^r0`(_Ω)

with k^α₁(u)≥0, where p⁰ = p/(p−1),r⁰_{`</sub>= r<sub>`}/(r<sub>`</sub>−1)and p≤r` < p^∗_{`</sub>, p<sup>∗</sup><sub>`} =







np

n−(k−`)p, if n> (k−`)p

>0, otherwise.

such that

|aα(x,η;u)| ≤g₁(u)^h|η^{(`)</sup>|<sup>p−1</sup>+|η<sup>(`)}|^r`−1i+ [k^α₁(u)](x)

for each multiindex ` = |α| ≤ k, almost all x ∈ _Ω, allη ∈ _R^N and all u ∈ V. Note that for

|α|= `=k, we must haver<sub>k</sub> = p. Here, we introduce the notation [K<sub>1</sub><sup>(`)</sup>(u)](x) =max

|α|=`[k<sup>α</sup><sub>1</sub>(u)](x) for all `=1, . . . ,k.

(A3)Suppose that

|α

∑

a_α(x,ζ,ξ;u)−a_α(x,ζ,ξ⁰;u)(ξ_α−ξ⁰_α)>0 for almost allx ∈_{Ω, all}ζ ∈_R^N1,ξ 6= ξ⁰ ∈_R^N2 and allu∈V.

(A4) Suppose that there exist a bounded and lower semicontinuous functional g₂: V → _R+ and a compact mapk₂:V →L¹(_Ω)such that

|α

∑

aα(x,η;u)ηα ≥ g₂(u)|ξ|^p−[k₂(u)](x) for almost allx ∈_{Ω, every}u∈V, and allη= (ζ,ξ)∈_R^N1×_R^N2.

Note that the preceding coercivity-like assumption requires the inequality to hold forall u ∈ V andη – contrary to usual asymptotic version, which is prescribed only forlarge kuk_V

and |η|. The reason for this is that the proof of pseudomonotonicity employs a certain in- equality which is needed for alluandηand is derived from this coercivity estimate. We now state a significant strengthening of(A4)that ensures coercivity in the sense of Definition1.3.

(A4’)Suppose that there exist a bounded functionalg₂: V→_R+and a compact mapk₂: V → L¹(_Ω)_{such that}

|α

∑

a_α(x,η;u)η_α ≥







g₂(u)|ξ|^p−[k₂(u)](x), for everyu∈V

g₂(u)^h|ξ|^p+_∑^k<sub>`=</sub><sup>−1</sup><sub>0</sub>(|η<sup>(`)</sup>|^p+|η^{(`)</sup>|<sup>r`})ⁱ−[k₂(u)](x), for largekuk_V for almost allx∈_{Ωand all}η= (_ζ,ξ)∈ _R^N. Here, the functionalg₂ satisfies the estimate

g₂(u)≥c^∗kuk⁻_V^σ∗

for all u ∈ V with sufficiently large kuk_V, with some c^∗ > 0 and 0 ≤ σ^∗ < p−1. Also, the mapk₂satisfies

kk₂(u)k_L1(_Ω)≤c^∗kuk^σ_V

for allu∈V with sufficiently largekuk_V and some 0≤σ < p−σ^∗.

(A5)Whenever u_j * u in V and{η_j} ⊂ _R^N with η_j → η, then aα(x,η_j;u_j) → aα(x,η;u)for a.e.x∈ _Ωup to a subsequence.

3 Pseudomonotonicity

Theorem 3.1. Assume (A0), (A1), (A2), (A3)and(A4). Then the operator A: V → V^∗ defined in (2.1)is pseudomonotone.

Proof. Let{u_j} ⊂Vbe a sequence that satisfiesu_j *uinVand lim sup

j→_∞

hA(u_j),u_j−ui ≤0. (3.1)

Assumption (A0) implies that there exists a sequence {_Ωi} ⊂ _Rⁿ of bounded domains such that Ωi ⊂ _Ωi+1, Ω = ^S∞_i=1Ωi and the Rellich–Kondrachov theorem holds on each Ωi: W^k,p(_Ωi)⊂⊂W^k−1,p(_Ωi). For everyi∈_Nthere is a subsequence{u⁽_jⁱ⁾}^∞_j=1⊂ {u_j}^∞_j=1(indexed by the samejfor simplicity) such that {u⁽_jⁱ⁾}^∞_j=1 ⊃ {u⁽_jⁱ⁺¹⁾}^∞_j=1 andu⁽_jⁱ⁾ → u inW^k−1,p(_Ωi)as j → ∞. The diagonal sequence {u_j}^∞_j=1 = {u⁽_j^j)}^∞_j=1 satisfies u_j → u in W^k−1,p(_Ωi) for any i∈_{N. Then}

D^γu_j → D^γu a.e. inΩfor all|γ| ≤k−1 (3.2) up to a subsequence. Further, by(A2)and(A4)we may assume that the sequences{K₁^(`)(u_j)}⊂

L<sup>r0`</sup>(<sub>Ω</sub>) (for every` = 1, . . . ,k) and{k₂(u_j)} ⊂ L¹(_Ω)are convergent. Note however, that we do nothaveu_j→ uinW^k−1,p(_Ω)_.

The following notations are used throughout the proof:

ζ(x) ={D^βu(x) : |β| ≤k−1}, ζ_j(x) ={D^βu_j(x) : |β| ≤k−1},

ξ(x) ={D^βu(x) : |β|=k}, ξ_j(x) ={D^βu_j(x) : |β|=k} η<sup>(`)</sup>(x) ={D<sup>β</sup>u(x) : |β|=`}, η<sup>(`)</sup><sub>j</sub> (x) ={D<sup>β</sup>u<sub>j</sub>(x) <sub>:</sub> |β|=`},

η(x) ={η<sup>(`)</sup>(x) : `=1, . . . ,k}, η_j(x) ={η<sup>(`)</sup><sub>j</sub> (x) : `=1, . . . ,k}.







































(3.3)

Using these, we may write

hA(u_j)−A(u),u_j−ui=

Z

Ωp_j, where

p_j(x) =

∑

|α|≤k

aα(x,ζ_j(x),ξ_j(x);u_j)−aα(x,ζ(x),ξ(x);u)(D^αu_j−D^αu), Also, (3.2) may be written as ζ_j →ζ a.e. orη^{(`)</sup><sub>j</sub> →η<sup>(`)} a.e. for all`=0, 1, . . . ,k−1.

First we derive conclusion (PM1) of pseudomonotonicity. The following trivial lemma is well-known.

Lemma 3.2. Relation(3.1)implies lim sup

j→_∞

hA(u_j)−A(u),u_j−ui ≤0.

Proof. We have lim sup

j→_∞

hA(u_j)−A(u)_,u_j−ui ≤_{lim sup}

j→_∞

hA(u_j)_,u_j−ui −_{lim inf}

j→_∞ hA(u)_,u−u_ji_.

By (3.1), the first term is nonpositive. For the second term, note that the functional v 7→

hA(u),u−viis weakly lower semicontinuous, so lim infhA(u),u−u_ji ≥0.

The conclusion of Lemma3.2can be written briefly as lim sup

j→_∞ Z

Ωp_j ≤0. (3.4)

Using the positive-negative decomposition p_j(x) = p⁺_j (x)−p⁻_j (x), we have 0 ≤ p⁺_j (x) = p_j(x) +p⁻_j (x)hence (3.4) immediately implies

Z

Ωp⁺_j →0 (3.5)

as j → ∞. Hence, the convergence R

Ωp⁻_j → 0 (j → ∞) needs to be established, so that R

Ωp_j → 0 (j → ∞) holds, which implies (PM1). This will be done via Vitali’s convergence theorem (see TheoremA.6) applied to the sequence{p⁻_j }.

Lemma 3.3. The sequence {p⁻_j }is equiintegrable and tight over Ω. Furthermore, there exist C1 > 0 and an a.e. bounded functionβ: Ω→_R+such that for a.a. x ∈_Ω,

p_j(x)≥C₁|ξ_j(x)|^p−β(x) (3.6) Proof. Expand p_j(x)as

p_j(x) =

∑

|α|=k

aα(x,ζ_j,ξ_j;u_j)D^αu_j+

∑

|α|≤k−1

aα(x,ζ_j,ξ_j;u_j)D^αu_j−w_j(x), where

w_j(x) =

∑

|α|≤k

h

aα(x,ζ,ξ;u) D^αu_j−D^αu) +aα(x,ζ_j,ξ_j;u_j)D^αui

=_:

∑

`=0

w^(`)_j (x)_.

We prove that{w_j}is equiintegrable and tight. Assumption(A2)implies that

|w^(`)_j (x)| ≤C₂

g₁(u)^h|η^{(`)</sup>|<sup>p−1</sup>+|η<sup>(`)}|^{r`−1i</sup>+ [K<sub>1</sub><sup>(`)}(u)](x) |η^{(`)</sup><sub>j</sub> |+|η<sup>(`)}| +C₂

g₁(u_j)^h|η_j^{(`)</sup>|<sup>p−1</sup>+|η<sup>(`)}_j |^{r`−1i</sup>+ [K<sup>(`)}₁ (u_j)](x)|η^(`)|

(3.7)

≤C₃

|η^{(`)</sup>|<sup>p−1</sup>|η<sup>(`)}_j |+|η^{(`)</sup>|<sup>p</sup>+|η<sup>(`)}|^{r`−1</sup>|η<sup>(`)}_j |+|η^{(`)</sup>|<sup>r`} +|η^{(`)</sup><sub>j</sub> |<sup>p−1</sup>|η<sup>(`)}|+|η^{(`)</sup><sub>j</sub> |<sup>r`−1}|η^(`)|

+ [K^{`</sup><sub>1</sub>(u)](x) |η<sub>j</sub><sup>(`)}|+|η^{(`)</sup>|+ [K<sup>(`)}₁ (u_j)](x)|η^(`)|

(3.8)

where C₂,C₃ > 0 are constants. We shall apply Proposition A.3 to prove that the function dominatingw^{(`)</sup><sub>j</sub> (x)is equiintegrable and tight. The weak convergenceu<sub>j</sub> *uinV⊂W<sup>k,p</sup>(<sub>Ω</sub>) implies that the sequence {η<sup>(`)}_j } ⊂W^{k−`,p</sup>(<sub>Ω</sub>)is bounded, hence by the Sobolev embedding W<sup>k−`,p}(_Ω) ⊂ L^q(_Ω) (where p ≤ q ≤ p^∗<sub>`</sub>) we have that {η<sub>j</sub><sup>(`)</sup>} ⊂ L^q(_Ω) is bounded. In particular,{|η^{(`)</sup><sub>j</sub> |<sup>r`}},{|η^(`)_j |^p} ⊂L¹(_Ω)are bounded.

The second and fourth terms in (3.8) are equiintegrable and tight by part (1) of Propo- sition A.3. Further, the first term is equiintegrable and tight by part (3) of Proposition A.3 applied to the constant sequence|η^{(`)</sup>|<sup>p−1</sup> ∈ L<sup>p0</sup>(<sub>Ω</sub>)(with|η<sup>(`)}|^p ∈ L¹(_Ω)being equiintegrable and tight by part (1) of the said Proposition) and to the bounded sequence{|η^{(`)</sup><sub>j</sub> |} ⊂ L<sup>p</sup>(<sub>Ω</sub>). The third term is similar. The fifth term is also equiintegrable and tight by part (3) of Propo- sition A.3 applied to the bounded {|η<sup>(`)}_j |^p−1} ⊂ L^p0(_Ω) and the constant |η^{(`)</sup>| ∈ L<sup>p</sup>(<sub>Ω</sub>) sequences. The sixth term is handled in a similar way. Finally, {K<sup>(`)}₁ (u_j)^r0`} ⊂ L¹(_Ω)is con- vergent by construction. Therefore the last two terms are equiintegrable and tight, too.

Moreover, assumption(A4)implies that

p_j(x)≥g₂(u_j)|ξ_j|^p−k₂(u_j)(x)− |w_j(x)| ≥ −k₂(u_j)(x)− |w_j(x)|_{. (3.9)} It follows that

0≤ p⁻_j (x)≤[k2(u_j)](x) +|w_j(x)|,

hence{p⁻_j }is equiintegrable and tight, where we have used the fact that{k₂(u_j)}is equiinte- grable and tight, since it is convergent inL¹(_Ω)_.

Finally, we turn to the proof of inequality (3.6). Young’s inequality applied to the products on the right side of (3.7) implies that

|w^{(`)</sup><sub>j</sub> (x)| ≤K<sub>3</sub>(ε)|η<sup>(`)}|^{(p−1)r0`</sup>+|η<sup>(`)}|^{r`</sup>+ [K<sub>1</sub><sup>(`)}(u)](x)^r0`+C₃ε

|η^{(`)</sup><sub>j</sub> |<sup>r`}+|η^{(`)</sup>|<sup>r`} +C₄ε

|η_j^{(`)</sup>|<sup>(p−1)r0`}+|η^{(`)</sup><sub>j</sub> |<sup>r`}+ [K^{(`)</sup><sub>1</sub> (u<sub>j</sub>)](x)<sup>r0`}+K₄(ε)|η^{(`)</sup>|<sup>r`}. By summing over j=0, 1, . . . ,k, and noting thatr_k = p, we get

|w_j(x)| ≤C₅ε|ξ_j|^p+K₅(ε)2|ξ|^p+ [K₁^(k)(u)](x)^p0+ [K⁽₁^k)(u_j)](x)^p0+

k−1

`=

∑

|w^(`)_j (x)|

≤C₅ε|ξ_j|^p+K(ε)

2|ξ|^p+ [K₁^(k)(u)](x)^p0+ [K⁽₁^k)(u_j)](x)^p0 +

k−1

`=

∑

h|η^{(`)</sup>|<sup>r`}+|η^{(`)</sup>|<sup>(p−1)r0`}+|η^{(`)</sup><sub>j</sub> |<sup>r`}+|η^{(`)</sup><sub>j</sub> |<sup>(p−1)r0`} + [K₁^{(`)</sup>(u)](x)<sup>r0`}+ [K^{(`)</sup><sub>1</sub> (u<sub>j</sub>)](x)<sup>r0`i}

=:C₅ε|ξ_j|^p +K(ε)

2|ξ|^p+

k−1

`=

∑

h|η^{(`)</sup>|<sup>r`}+|η^{(`)</sup><sub>j</sub> |<sup>r`}+|η^{(`)</sup>|<sup>(p−1)r0`}+|η^{(`)</sup><sub>j</sub> |<sup>(p−1)r`0i}+ [K₃(u,u_j)](x)

,

where {K₃(u,u_j)} ⊂ L¹(_Ω) is convergent, hence it is convergent a.e. up to a subsequence, thus it is a.e. bounded. Therefore, using the a.e. convergence η<sup>(`)</sup> → η (` = 0, . . . ,k−1) we have that the function

β₁(x) =₂|ξ|^p+

k−1

`=

∑

h|η^{(`)</sup>|<sup>r`}+|η^{(`)</sup><sub>j</sub> |<sup>r`}+|η^{(`)</sup>|<sup>(p−1)r0`}+|η_j^{(`)</sup>|<sup>(p−1)r0`i}+ [K₃(u,u_j)](x)

is bounded a.e.

The first inequality of (3.9) combined with the preceding estimate and assumption (A4) leads to

p_j(x)≥ g₂(u_j)|ξ_j|^p−[k₂(u_j)](x)− |w_j(x)|

≥ g₂(u_j)|ξ_j|^p−C₅ε|ξ_j|^p−[k₂(u_j)](x)−K(ε)β₁(x)

≥ |ξ_j|^p(A−C₅ε)−β(x)

where g2(u_j) ≥ A > 0 (due to the weak lower semicontinuity of g2: V → _R+ and the weak convergence u_j * u) and β(x) = K(ε)β₁(x) + [k₂(u_j)](x)is still bounded a.e., because {k₂(u_j)} ⊂ L¹(_Ω)is bounded and therefore convergent a.e. up to a subsequence. The desired inequality follows by choosingε = A/(2C5).

Claim 3.4. The convergence p⁻_j →0a.e. holds.

Proof. Split p_j(x)as

p_j(x) =

∑

|α|=k

aα(x,ζ_j,ξ_j;u_j)−aα(x,ζ_j,ξ;u_j)(D^αu_j−D^αu)

+

∑

|α|=k

aα(x,ζ_j,ξ;u_j)−aα(x,ζ,ξ;u)(D^αu_j−D^αu)

+

∑

|α|≤k−1

aα(x,ζ_j,ξ_j;u_j)−aα(x,ζ,ξ;u)(D^αu_j−D^αu)

=_:q_j(x) +r_j(x) +s_j(x)

(3.10)

Letχ_j be the characteristic function of the level set{x ∈_Ω : p⁻_j (x)>0}and write

−p⁻_j =χ_jq_j+χ_jr_j+χ_js_j.

First, note thatχ_jq_j ≥0 a.e. due to the monotonicity assumption(A3), so it is enough to prove χ_jr_j → 0 a.e. and χ_js_j → 0 a.e. Lemma 3.3 ensures that there exists β: Ω → _R a.e. bounded such that

|ξ_j(x)|^p ≤β(x),

for allx∈ _Ωsuch that p_j(x)<0. Therefore{χ_j(x)ξ_j(x)}is bounded for a.e. x ∈_{Ω. By}(A2), (A5)andζ_j →ζa.e. (from (3.2)), we find thatχ_jr_j →0 a.e. andχ_js_j →0 a.e. for a subsequence, from whichp⁻_j →0 a.e. follows.

In summary, we have that{p⁻_j }is equiintegrable and tight, andp⁻_j →0 a.e. A corollary of the Vitali convergence theorem (TheoremA.6below) yields that these conditions are actually necessary and sufficientto ensure the convergence

Z

Ωp⁻_j →0, asj→∞. Recalling (3.5), we have in summary

Z

Ωp_j →0 (3.11)

asj→∞. Then conclusion(PM1)of pseudomonotonicity is established:

hA(u_j),u_j−ui=hA(u_j)−A(u),u_j−ui+hA(u),u_j−ui

=

Z

Ωp_j+hA(u),u_j−ui →0.

Turning to the proof of (PM2), first note that (3.11) implies that p_j → 0 a.e. up to a subsequence.

Claim 3.5. The convergenceξ_j →ξa.e. holds.

Proof. It follows from estimate (3.6) that{ξ_j}is bounded a.e. Fix anx₀ ∈_Ωsuch that {ξ_j(x₀)}

is bounded andp_j(x₀)→0. Assume for contradiction thatξ_j(x₀)→ξ⁰ for a subsequence and some ξ⁰ such thatξ⁰ 6= ξ(x0). Since we have ζ_j → ζ a.e., by using decomposition (3.10) and (A1), it follows thatr_j →0 ands_j →0 a.e. But then the continuity assumption (A5)implies

p_j(x₀)→0=

∑

|α|=k

a_α(x₀,ζ,ξ⁰;u)−a_α(x₀,ζ,ξ;u)(ξ⁰_α−D^αu(x₀)) Thus(A3)yields ξ⁰_α =D^αu(x₀), which is a contradiction.

Finally, we proveA(u_j)* A(u)inV^∗. By the Vitali convergence theorem hA(u_j),vi=

∑

|α|≤k

Z

Ωaα(x,η_j;u_j)D^αv(x)dx

→

∑

|α|≤k

Z

Ωa_α(x,η;u)D^αv(x)dx,

because the integrand is equiintegrable and tight by PropositionA.3 (3) and the a.e. conver- gencea_α(x,η_j;u_j)→a_α(x,η;u)follows from(A5).

Proposition 3.6. If(A4’)holds then A:V →V^∗is coercive.

Proof. We have foru∈V with sufficiently largekuk_V, hA(_u)_,ui ≥ g2(_u)

Z

Ω|ξ|^p+

k−1

`=

∑

(|η^{(`)</sup>|<sup>r`}+|η^(`)|^p)_dx−

Z

Ω[_k2(_u)](_x)_dx

≥Ckuk⁻_V^σ∗kuk^p_V−c^∗kuk_V^σ

≥C⁰kuk_V^p−σ∗

for someC,C⁰ >0. ThereforehA(u),ui/kuk_V→+_∞ifkuk_V →_{∞, because} p−σ^∗ >1.

4 Examples

Here we formulate examples satisfying(A1)–(A5)and(A4’). For all|α|= `, with`=0, 1, . . . ,k consider

a_α(x,η;u) =_{Ψ`</sub>(H<sub>`}(u))^ha_{`</sub>(x)χ<sub>`}(G<sub>`</sub>(u)) |η<sup>(`)</sup>|^{r`−2</sup>+|η<sup>(`)}|^p−2η_α+b_α(x)M_α(u)ⁱ_,

where p≤ r_{`</sub> ≤ p<sup>∗</sup><sub>`} andm≤ a_`(x)≤ M for some constantsm,M >0. (We remind the reader that η^(k)=ξ andp^∗_k = p, so that the highest order aα reads

aα(x,η;u) =_Ψk(H_k(u))a_k(x)χ_k(G_k(u))|ξ|^p−2ξα+bα(x)Mα(u),

where |α| = k, which is reminiscent of the p-Laplacian.) We propose the following two possibilities for the choice ofΨ` andH`.

1. Let H`: W<sup>k−1,p</sup>(<sub>Ω</sub><sup>0</sup>) → L<sup>∞</sup>(<sub>Ω</sub>)be a bounded linear map (with Ω<sup>0</sup> ⊂ <sub>Ω</sub> a bounded do- main) and let Ψ`: R → _R+ be continuous with Ψ`(ν) ≥ C_Ψ/(1+|ν|)^−σ∗ for some C_Ψ >0 and large|ν|.

2. LetH`:V →<sub>R</sub>be a bounded linear functional and letΨ`: R→_R+be continuous with Ψ`(ν)≥C_Ψ/(1+|ν|^σ∗)for someC_Ψ>0.

Again, we may chooseχ` andG` as follows.

1. LetG`: W<sup>k−1,p</sup>(<sub>Ω</sub><sup>0</sup>)→ L<sup>p0</sup>(<sub>Ω</sub>)be a bounded linear map and letχ`: R→ _R+ be contin- uous withm≤ χ_`(ν)≤ M for some constantsm,M>0.

2. LetG`: V →<sub>R</sub>be a bounded linear functional and let χ`: R→_R+ be continuous with m≤χ_`(ν)≤ Mfor some constantsm,M>_0.

Finally, for fixed any|α|= `, let 2≤ p₁≤ p,m=1, . . . ,k and let Mα: V→W^m,p1(_Ω) (or R) be a bounded map such that

kM_α(u)k_Wm,p1(_Ω) ≤constkuk^γ_V^α, (4.1) kM_α(u)−M_α(v)k_Wm,p1(_Ω) ≤constku−vk^γ_V^α, (4.2) where 0<γα < _r^p0

`; also, let λα =qα/r<sup>0</sup><sub>`</sub> andbα ∈ L^r0`λ0α(_Ω)where (p₁ <qα < _n−^np_mp¹

1 if m< _pⁿ

1

qα >0 otherwise (or, ifMα: V→_{R, then}

|M_α(u)| ≤constkuk^γ_V^α, (4.3)

|M_α(u)−M_α(v)| ≤constku−vk^γ_V^α, (4.4) withγ_α =σ/r⁰<sub>`</sub>,b<sub>α</sub>∈ L<sup>r0`</sup>(_Ω)_).

Under these hypotheses,(A1)and(A3)are satisfied. Note that the continuous embeddings W^m,p1(_Ω)⊂ L^qα(_Ω)hold, so

kMα(u)k_Lqα(_Ω) ≤constkMα(u)k_W^m,p1(_Ω) ≤constkuk^γ_V^α. Therefore, by Hölder’s inequality and (4.1)

Z

Ω|bα(x)|^{r0`</sup>|Mα(u)|<sup>r0`}dx≤ kbαk^r0`

L^r0`λ0α(_Ω)

h^Z

Ω|Mα(u)|^{r0`λαi1/λα}

=kbαk^r0`

L^r0`λ0α(_Ω)kMα(u)k^q_L^αq^/λα(_Ω^α)

≤constkb_αk^r0`

L^r0`λ0α(_Ω)kuk^q_V^αγα/λα

≤c^∗kuk^σ_V,

(4.5)

where σ = qαγα/λα = r⁰_`γα < p for the case Mα: V → W^m,p1(_Ω). The case Mα: V → _R is treated similarly.

Claim 4.1. Assumption(A2)holds.

Proof. The growth condition reads

|a_α(x,η,ξ;u)| ≤_{Ψ`</sub>(H<sub>`}(u))|a_{`</sub>(x)|χ<sub>`}(G<sub>`</sub>(u)) |η<sup>(`)</sup>|^{r`−1</sup>+|η<sup>(`)}|^p−1 +_{Ψ`</sub>(H<sub>`}(u))|b_α(x)M_α(u)|.

Theng₁(u) =<sub>Ψ`</sub>(H`(u))M²is a bounded functional by assumption. Letting [k^α₁(u)](x) =<sub>Ψ`</sub>(H`(u))|b_α(x)M_α(u)|,

we find by (4.5) thatk^α₁: V→ L^r0`(_Ω)is bounded.

Proving the compactness ofk^α₁requires more effort (except whenM_α: V→R). To this end, suppose that{u_j} ⊂ V is a bounded sequence. Let{_Ωi}be the sequence guaranteed to exist by assumption (A0). Then kbαk

L^r0`λ0α(_Ω\_Ωi) →0. Using the compact embeddingW^m,p1(_Ωi)⊂⊂

L^qα(_Ωi) we can choose subsequences of {u_j}as follows. Let {u_1j} ⊂ {u_j}be a subsequence such that

kMα(u_1j)−Mα(u_1m)k_Lqα(_Ω1) <1 for j,m=1, 2, 3, . . . Let{u_2j} ⊂ {u_1j}be a subsequence such that

kM_α(u_2j)−M_α(u_2m)k_Lqα(_Ω2)< ¹

2 forj,m=2, 3, . . . Continuing this way, for fixedilet{u_ij} ⊂ {u_i−1,j}be a subsequence such that

kM_α(u_ij)−M_α(u_im)k_Lqα(_Ωi)< ¹

i for j,m=i,i+1, . . . It follows that the diagonal sequence {u_jj}satisfies

kMα(u_jj)−Mα(umm)k_Lqα(_Ωi)< ¹

i forj,m= i,i+1, . . . Using Hölder’s inequality, we find for j,m≥i

Z

Ω|bα(x)|^{r`0</sup>|Mα(u<sub>jj</sub>)−Mα(umm)|<sup>r`0}dx

= _Z

Ω\_Ωi

+

Z

Ωi

|b_α(x)|^{r0`</sup>|M<sub>α</sub>(u<sub>jj</sub>)−M<sub>α</sub>(u<sub>mm</sub>)|<sup>r0`}dx

≤constkb_αk

L^r0`λ0α(_Ω\_Ωi)

_Z

Ω\_Ωi

|M_α(u_jj)−M_α(u_mm)|^qαdx 1/λα

+constkb_αk

L^r0`λ0α(_Ωi)

_Z

Ωi

|M_α(u_jj)−M_α(u_mm)|^qαdx 1/λα

. Here, kb_αk

L^r0`λ0α(_Ω\_Ωi) → 0 and kb_αk

L^r0`λ0α(_Ωi) is bounded. By assumption (4.2), the first integral is bounded and for the second integral we have

Z

Ωi

|Mα(u_jj)−Mα(umm)|^qαdx≤ ¹ i^qα →0 if j,m≥iandi→_∞.

We now show that(A4’)holds. It is enough to estimate the terms of

|α

∑

aα(x,η;u)ηα

=<sub>Ψ`</sub>(H`(u))a`(x)χ`(G`(u))(|η<sup>(`)</sup>|^{r`</sup>+|η<sup>(`)}|^p) +

∑

|α|=`

Ψ`(H`(u))bα(x)Mα(u)ηα

for all `=0, 1, . . . ,k. The first term may be estimated from below by CΨ`(H<sub>`</sub>(u))|η<sup>(`)</sup>|^{r`</sup>+|η<sup>(`)}|^p

for some constantC>0. Here, the quantity Ψ`(H`(u))satisfies Ψ`(H<sub>`</sub>(u))≥ ^CΨ

|H`(u)|<sup>σ∗</sup>+1 ≥ <sup>CΨ</sup> kH`(u)k^σ∗

L^∞(_Ω)+1 ≥ ^C

Ψ0

kuk^σ∗

W^k−1,p(_Ω⁰)+1 ≥ ^C

Ψ0

kuk^σ_V^∗+1. The terms of the sum may be bounded from above by Young’s inequality,

Ψ`(H`(u))|bα(x)Mα(u)ηα| ≤<sub>εΨ`</sub>(H`(u))|ηα|^{r`</sup>+C<sup>∗</sup>(ε)|bα(x)|<sup>r0`}|Mα(u)|^r0`

≤ εΨ`(H<sub>`</sub>(u))|η^{(`)</sup>|<sup>r`}+C^∗(ε)|b_α(x)|^{r0`</sup>|M<sub>α</sub>(u)|<sup>r0`}.

Choosing a sufficiently smallε >0, it turns out that it is enough to estimate the L¹(_Ω)-norm of the expression

[k^α₂(u)](x) =|bα(x)|^{r0`</sup>|Mα(u)|<sup>r0`}, which, using (4.5), satisfies

kk^α₂k_L1(_Ω) ≤c^∗kuk^σ_V.

The proof of compactness of k^α₂ is analogous to that of k^α₁. The required k₂ in Assumption (A4’)is given by the pointwise maximum ofk^α₂ over all|α| ≤k.

To finish the argument, note that assumption (A5)is satisfied since the functions Φ`, χ`

and Ψα are continuous and the operators H`, G` and M_α are continuous in the respective Sobolev and Lebesgue spaces. Thus ifu_j * u in V, then for a subsequence H`(u<sub>j</sub>), G`(u_j), Mα(u_j)are convergent a.e. in Ω.

Example 4.2. For a more concrete example toMα, consider the following. In the caseMα: V → W^m,p1(_Ω), let Mα(u) = Heα(u)where Heα: V → W^m,p1(_Ω)is a continuous linear operator. For a more concrete example, consider

[Heα(u)](x) =

∑

|α|≤k

Z

ΩGα(x,y)D^αu(y)dy, where the functionsGα: Ω×_Ω→_Rsatisfy

x7→

_Z

Ω|D^βG_α(x,y)|^p0dy 1/p⁰

∈ L^p1(_Ω) for |β| ≤m.

In the case M_α: V → _R, let M_α(u) = _Φα(He_α(u)), where He_α: V → _R+ is a bounded linear functional andΦα: R+ → _R+ is continuous with |_Φα(ν₁)−_Φα(ν2)| ≤ C_Φ|ν₁−ν2|^{σ/r0`</sup>. Note thatΦα(ν)≤C<sub>Φ</sub>|ν|<sup>σ/r`0} follows automatically.

The operators H`: W<sup>k−1,p</sup>(<sub>Ω</sub><sup>0</sup>)→ L<sup>∞</sup>(<sub>Ω</sub>)andG`: W^k−1,p(_Ω⁰)→ L^p0(_Ω)can be defined by the formula

(Bu)(x) =

∑

|α|≤k−1

Z

Ω⁰Gα(x,y)D^αu(u)dy, where the measurable functionsGα: Ω×_Ω⁰ →_Rsatisfy

x7→

_Z

Ω⁰|G_α(x,y)|^p0dy 1/p⁰

∈ L^∞(_Ω) and L^p0(_Ω), respectively.