On Pseudomonotone Elliptic Operators with Functional Dependence on Unbounded Domains

We generalize F. E. Browder's results concerning pseudomonotone elliptic partial differential operators defined on unbounded domains. Browder treated equations for quasilinear operators of divergence form ∑ |α|≤k D α a α 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 of u.


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 conditions (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 pseudomonotone if for any sequence {u j } ⊂ X, such that u j u (in X) and lim sup j→∞ A(u j ), u j − u ≤ 0, then (PM1) A(u j ), u j − u → 0 as j → ∞ and (PM2) A(u j ) A(u) in X * as j → ∞.
Email: 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: Guaranteeing boundedness and coercivity is usually a trivial matter.The proof of pseudomonotonicity 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 α u j } ∞ j=1 .Browder's idea is a natural one: let the unbounded domain Ω be exhausted by an increasing sequence {Ω i } of bounded domains with smooth boundary -such that on each Ω i the Rellich-Kondrachov theorem holds.Combining this with a diagonal argument, we extract a subsequence of the lower-order derivatives {D α u j } converging a.e. to D α u (|α| ≤ k − 1).Proving a.e.convergence of the highest-order derivatives 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 containing a term which is arbitrarily quickly increasing with respect to the values of unknown function u.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 on u.
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.

Problem formulation
Let Ω ⊂ R n be a possibly unbounded domain with sufficiently smooth boundary, and let for all u, v ∈ V, where |β| ≤ k is a multiindex.The function a α may depend on the pointwise values of any of the partial derivatives of u.Furthermore, "; u" notation signifies that a α may be a functional of u.In other words, a α may depend on the whole solution u.
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 N 1 and ξ ∈ R N 2 , so that η ∈ R N with N = N 1 + N 2 and write a α (x, ζ, ξ; u), where the numbers N 1 and N 2 denote number of multiindexes β such that |β| ≤ k − 1 and |β| = k, respectively.Furthermore, the notation We impose the following assumptions on the structure of A and Ω. (A0) Suppose that there exist a sequence {Ω i } ⊂ R n of bounded domains such that Ω i ⊂ Ω i+1 (i = 1, 2, . ..) and Ω = ∞ i=1 Ω i .Furthermore, assume that each ∂Ω i is sufficiently smooth so that the Rellich-Kondrachov theorem holds: (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 fixed η ∈ R N , and let a α (x, • ; u) be continuous for almost every fixed x ∈ Ω. (A2) Suppose that there exist a bounded functional g 1 : V → R + and a compact map with k α 1 (u) ≥ 0, where p = p/(p − 1), r = r /(r − 1) and for each multiindex = |α| ≤ k, almost all x ∈ Ω, all η ∈ R N and all u ∈ V.Note that for |α| = = k, we must have r k = p.Here, we introduce the notation [K ( ) (A4) Suppose that there exist a bounded and lower semicontinuous functional g 2 : V → R + and a compact map for almost all x ∈ Ω, every u ∈ V, and all η = (ζ, Note that the preceding coercivity-like assumption requires the inequality to hold for all u ∈ V and η -contrary to usual asymptotic version, which is prescribed only for large u V and |η|.The reason for this is that the proof of pseudomonotonicity employs a certain inequality which is needed for all u and η and is derived from this coercivity estimate.We now state a significant strengthening of (A4) that ensures coercivity in the sense of Definition 1.3.(A4') Suppose that there exist a bounded functional g 2 : V → R + and a compact map Here, the functional g 2 satisfies the estimate for all u ∈ V with sufficiently large u V , with some c * > 0 and 0 ≤ σ * < p − 1.Also, the map k 2 satisfies

Pseudomonotonicity Theorem 3.1. Assume (A0), (A1), (A2), (A3) and (A4). Then the operator A
Assumption (A0) implies that there exists a sequence i=1 Ω i and the Rellich-Kondrachov theorem holds on each (indexed by the same j for simplicity) such that {u up to a subsequence.Further, by (A2) and (A4) we may assume that the sequences {K ( ) The following notations are used throughout the proof: Using these, we may write where Also, (3.2) may be written as e. for all = 0, 1, . . ., k − 1.First we derive conclusion (PM1) of pseudomonotonicity.The following trivial lemma is well-known.
By (3.1), the first term is nonpositive.For the second term, note that the functional v → A(u), u − v is weakly lower semicontinuous, so lim inf A(u), u − u j ≥ 0.
The conclusion of Lemma 3.2 can be written briefly as Using the positive-negative decomposition as j → ∞.Hence, the convergence Ω p − j → 0 (j → ∞) needs to be established, so that Ω p j → 0 (j → ∞) holds, which implies (PM1).This will be done via Vitali's convergence theorem (see Theorem A.6) applied to the sequence {p − j }.
Lemma 3.3.The sequence {p − j } is equiintegrable and tight over Ω.Furthermore, there exist C 1 > 0 and an a.e.bounded function β : Ω → R + such that for a.a.x ∈ Ω, Proof.Expand p j (x) as where We prove that {w j } is equiintegrable and tight.Assumption (A2) implies that |w ( ) where C 2 , C 3 > 0 are constants.We shall apply Proposition A.3 to prove that the function dominating w ( ) j (x) is equiintegrable and tight.The weak convergence u j u in V ⊂ W k,p (Ω) implies that the sequence {η Ω) is bounded, hence by the Sobolev embedding W k− ,p (Ω) ⊂ L q (Ω) (where p ≤ q ≤ p * ) we have that {η ( ) The second and fourth terms in (3.8) are equiintegrable and tight by part (1) of Proposition A.3.Further, the first term is equiintegrable and tight by part (3) of Proposition A.3 applied to the constant sequence |η ( ) | p−1 ∈ L p (Ω) (with |η ( ) | p ∈ L 1 (Ω) being equiintegrable and tight by part (1) of the said Proposition) and to the bounded sequence {|η ( ) The third term is similar.The fifth term is also equiintegrable and tight by part (3) of Proposition A.3 applied to the bounded {|η ( ) The sixth term is handled in a similar way.Finally, {K ( ) Moreover, assumption (A4) implies that j } is equiintegrable and tight, where we have used the fact that {k 2 (u j )} is equiintegrable and tight, since it is convergent in L 1 (Ω).
Proof.Split p j (x) as Let χ j be the characteristic function of the level set {x ∈ Ω : p − j (x) > 0} and write −p − j = χ j q j + χ j r j + χ j s j .First, note that χ j q j ≥ 0 a.e.due to the monotonicity assumption (A3), so it is enough to prove χ j r j → 0 a.e. and χ j s j → 0 a.e.Lemma 3.3 ensures that there exists β : Ω → R a.e.bounded such that for all x ∈ Ω such that p j (x) < 0. Therefore {χ j (x)ξ j (x)} is bounded for a.e.x ∈ Ω.By (A2), (A5) and ζ j → ζ a.e.(from (3.2)), we find that χ j r j → 0 a.e. and χ j s j → 0 a.e. for a subsequence, from which p − j → 0 a.e.follows.
In summary, we have that {p − j } is equiintegrable and tight, and p − j → 0 a.e.A corollary of the Vitali convergence theorem (Theorem A.6 below) yields that these conditions are actually necessary and sufficient to ensure the convergence as j → ∞.Recalling (3.5), we have in summary as j → ∞.Then conclusion (PM1) of pseudomonotonicity is established: Turning to the proof of (PM2), first note that (3.11) implies that p j → 0 a.e. up to a subsequence.

Finally, we prove A(u j )
A(u) in V * .By the Vitali convergence theorem because the integrand is equiintegrable and tight by Proposition A.3 (3) and the a.e.convergence a α (x, η j ; u j ) → a α (x, η; u) follows from (A5).
Proof.We have for u ∈ V with sufficiently large u V ,

Examples
Here we formulate examples satisfying (A1)-(A5) and (A4').For all |α| = , with = 0, 1, . . ., k consider where p ≤ r ≤ p * and m ≤ a (x) ≤ M for some constants m, M > 0. (We remind the reader that η (k) = ξ and p * k = p, so that the highest order a α reads where |α| = k, which is reminiscent of the p-Laplacian.)We propose the following two possibilities for the choice of Ψ and H .
2. Let H : V → R be a bounded linear functional and let Ψ : R → R + be continuous with Again, we may choose χ and G as follows.
2. Let G : V → R be a bounded linear functional and let χ : R → R + be continuous with m ≤ χ (ν) ≤ M for some constants m, M > 0.

M. Csirik
Finally, for fixed any |α| = , let 2 ≤ p 1 ≤ p, m = 1, . . ., k and let be a bounded map such that where 0 < γ α < p r ; also, let λ α = q α /r and b α ∈ L r λ α (Ω) where |M α (u) with Under these hypotheses, (A1) and (A3) are satisfied.Note that the continuous embeddings Therefore, by Hölder's inequality and (4.1) where σ = q α γ α /λ α = r γ α < p for the case Proof.The growth condition reads Proving the compactness of k α 1 requires more effort (except when M α : V → R).To this end, suppose that {u j } ⊂ V is a bounded sequence.Let {Ω i } be the sequence guaranteed to exist by assumption (A0).Then b α L r λ α (Ω\Ω i ) → 0. Using the compact embedding W m,p 1 (Ω i ) ⊂⊂ L q α (Ω i ) we can choose subsequences of {u j } as follows.Let {u 1j } ⊂ {u j } be a subsequence such that M α (u 1j ) − M α (u 1m ) L qα (Ω 1 ) < 1 for j, m = 1, 2, 3, . . .Let {u 2j } ⊂ {u 1j } be a subsequence such that for j, m = 2, 3, . . .Continuing this way, for fixed i let {u ij } ⊂ {u i−1,j } be a subsequence such that It follows that the diagonal sequence {u jj } satisfies Using Hölder's inequality, we find for j, m ≥ i Here, b α L r λ α (Ω\Ω i ) → 0 and b α L r λ α (Ω i ) is bounded.By assumption (4.2), the first integral is bounded and for the second integral we have We now show that (A4') holds.It is enough to estimate the terms of for all = 0, 1, . . ., k.The first term may be estimated from below by for some constant C > 0. Here, the quantity Ψ (H (u)) satisfies The terms of the sum may be bounded from above by Young's inequality, Choosing a sufficiently small ε > 0, it turns out that it is enough to estimate the The proof of compactness of k α 2 is analogous to that of k α 1 .The required k 2 in Assumption (A4') is given by the pointwise maximum of k α 2 over all |α| ≤ k.To finish the argument, note that assumption (A5) is satisfied since the functions Φ , χ and Ψ α are continuous and the operators H , G and M α are continuous in the respective Sobolev and Lebesgue spaces.Thus if u j u in V, then for a subsequence H (u j ), G (u j ), M α (u j ) are convergent a.e. in Ω.
Example 4.2.For a more concrete example to M α , consider the following.In the case M α : V → W m,p 1 (Ω), let M α (u) = H α (u) where H α : V → W m,p 1 (Ω) is a continuous linear operator.For a more concrete example, consider where the functions The operators H : where the measurable functions G a α jk (x)D j vD k w + c α (x)vw dx, where v, w ∈ V 1 and a α jk ∈ L ∞ (Ω) form a uniformly elliptic coefficient matrix and c α (x) ≥ c 0 > 0. The strong form of this operator is "−divA α Dv + c α v", where A α = (a α jk ).Then we may take (Ω) is a continuous linear operator.(A function v ∈ H 1 0 (Ω ) belongs to W 1,2 (Ω) if it is extended by 0 in Ω \ Ω .)Example 4.4.More generally, let V 1 ⊂ W m,p 1 (Ω) be a closed subspace (which may depend on α) and let N α : V 1 → V * 1 be a bounded, strictly monotone and coercive operator that satisfies and Then for every w ∈ V * 1 there exists a unique element v ∈ V 1 such that N α (v) = w and the mapping N −1 α : V * 1 → V 1 is Hölder continuous:

A Equiintegrability and tightness
This appendix collects some results used in the paper; see e.g.[5] for proofs.
Definition A.1.A sequence { f j } of measurable functions f j : Ω → R is said to be equiintegrable over Ω if for all ε > 0 there exists δ > 0 such that E | f j | < ε for all j ∈ N and all E ⊂ Ω measurable with |E| < δ.
is convergent by construction.Therefore the last two terms are equiintegrable and tight, too.