## 1 Introduction

In their seminal paper [6] Brézis and Lieb prove a
result about the decoupling of certain integral expressions,
which has been used extensively in the calculus of variations.
Using concentration compactness arguments in the spirit of
Lions [16, 14, 15], we prove a variant of
this lemma under weaker assumptions on the nonlinearity than
known before. To describe a special case of the Brézis–Lieb
lemma, suppose that Ω is an unbounded domain in *u*. If one denotes by
*f*, i.e.,

The same conclusion is obtained in that paper for more general
functions, imposing conditions that are satisfied for continuous
convex *f* with

A different approach to the decoupling of superposition operators
along sequences of functions rests on certain regularity
assumptions on *f*. For example, assume that

Then the proof of [19, Lemma 8.1] can easily be
extended to obtain (1.1). See also the slightly more
general [7, Lemma 1.3], where *f* is allowed to
depend on *x* explicitly.

Our aim is to give a decoupling result under a different set of
hypotheses that applies to a much larger class of functions *f*
than considered above, within a certain range of exponents *p*.
In particular, we do neither impose any convexity type assumptions on
*f* as was done in [6], nor any regularity
assumptions as in [19, 7] apart from
continuity. The price we pay for relaxing the hypotheses on *f*
is that we need to restrict the range of allowed growth exponents
*p* in comparison with [6], that we need to assume
some type of translation invariance for Ω, and that the
decoupling result only applies to a smaller set of admissible
sequences, namely sequences that converge weakly in

To keep the presentation simple and highlight the main idea, we
only treat the case *f* on the
space variable.

To explain our results we formalize the notion of decoupling.

Suppose that *X* and *Y* are Banach spaces. Consider a map
*BL-splits along * (BL being an abbreviation for Brézis–Lieb) if

We say that *almost BL-splits along * if, starting with any subsequence of

If *u* is a limit of *with respect to u*.

By [6], the map

then the induced superposition operator

We illustrate the distinction between BL-splitting and almost BL-splitting by the following examples.

If *u* such that the induced
continuous superposition operator *u*. On the other hand, *f* in these examples.

The sequences mentioned in the example are provided in Section 4 below.

Our main interest is to avoid condition (1.2), or any
other conditions on *f* that ensure uniform continuity on bounded
subsets of

In this context we now formulate our main theorem, in a slightly
more general setting than what we considered above.
A function *Caratheodory function* if *f* is measurable and if
*A* is a
real invertible *f* is said to be
*A*-periodic in its first argument if

Denote by

*Consider *

*and which is A-periodic in its first argument, for some
invertible matrix *

Our proof of Theorem 1.3 has similarities with the
proof of [18, Theorem 3.1] but involves an
intermediate cut-off step in the proof of
Theorem 2.1. Essentially, we first
prove almost BL-splitting of *U*. Then we collect the possible mass loss
at infinity along subsequences with the help of Lions’ Vanishing
Lemma, employing the assumption

Theorem 1.3 applies in particular to the functions
considered in Example 1.2 when

Of course, by Sobolev’s embedding theorem, a map

The result also holds true in a slightly restricted sense for
functions *f* that are sums of functions as in
Theorem 1.3, i.e., functions that satisfy merely

where

The uniform continuity of operators *U* is bounded, by the
compact Sobolev embedding

We now discuss additional aspects and applications of the results
presented above. To this end we return to a simple setting on

To prove the existence of a minimizer in typical variational
problems involving Φ, Lions [14, 15]
introduces the concentration-compactness principle. It is a tool
to exclude the possibility of vanishing and of dichotomy along a
minimizing sequence *local*
functionals like Φ it then follows easily that *f*. On the other
hand, the arguments are more involved than when using
BL-splitting because one has to insert cut-off functions to
obtain sequences

To explain the advantage of an abstract presentation using BL-splitting, we note that to treat nonlocal functionals of convolution type, e.g.,

the property of disjoint supports is not as effective anymore.
In the convolution, the supports get “smeared out” and one has
to control the interaction with more involved estimates, see
[14, p. 123]. This is aggravated when one also
has to consider the decoupling of *derivatives* of Ψ.
We have shown in [2] that using BL-splitting is
effective in situations involving nonlocal functionals.
Moreover, BL-splitting even survives certain nonlocal operations,
like the saddle point reduction;
see [2, Theorem 5.1].

For particular cases there are other approaches to avoid
conditions on *f* besides continuity and growth bounds. We
reformulate and simplify the following cited results slightly to
adapt them to our setting and notation. In [3] we
proved, for

BL-splits along a weakly convergent sequence if the weak limit is
a function tending to 0 as

is uniformly continuous on bounded subsets of *f*). In both cases our result here
is stronger, since we show uniform continuity and BL-splitting
into the spaces

A different application of Theorem 1.3, that is independent of variational methods, is the general study of maps that are uniformly continuous on a subset of an infinite dimensional Hilbert space. These play a role in infinite dimensional potential theory [12, 5] or, more generally, in the theory of stochastic equations in infinite dimensions [17, 9, 8].

The paper is structured as follows: In
Section 2 we treat almost BL-splitting of

## 2 Almost BL-splitting

In this section we prove a result on the almost BL-splitting of
superposition operators in *x* is not needed.

If

*Consider *

*f*. Then we have the following results:

- (a)
*If*$\left({u}_{n}\right)\subseteq {L}^{p}$ *is bounded and converges in*${L}_{\mathrm{loc}}^{p}$ *to a function**u**, then*$u\in {L}^{p}$ *and*$\mathcal{F}:{L}^{p}\to {L}^{\nu}$ *almost BL-splits along*$\left({u}_{n}\right)$ *with respect to**u*. - (b)
*If*$p\in [2,{2}^{*})$ *and*${u}_{n}\rightharpoonup u$ *in*${H}^{1}$ *, then*$\mathcal{F}:{H}^{1}\to {L}^{\nu}$ *almost BL-splits along*$\left({u}_{n}\right)$ *with respect to**u*. - (c)
*In*(b)*, if in addition*$\left({\overline{u}}_{n}\right)\subseteq {H}^{1}$ *converges weakly and*${\left|{u}_{n}-{\overline{u}}_{n}\right|}_{p}\to 0$ *as*$n\to \infty $ *, then*${\overline{u}}_{n}\rightharpoonup u$ *in*${H}^{1}$ *and*$\mathcal{F}$ *almost BL-splits along*$\left({u}_{n}\right)$ *and*$\left({\overline{u}}_{n}\right)$ *with respect to**u**, preserving subsequences and the auxiliary sequence*$\left({v}_{n}\right)$ *in the following sense: for any subsequence*${n}_{k}$ *there are a subsequence*${n}_{{k}_{\ell}}$ *and*$\left({v}_{\ell}\right)$ *such that*${v}_{\ell}\to u$ *in*${H}^{1}$ *and, writing*${u}_{\ell}:={u}_{{n}_{{k}_{\ell}}}$ *and*${\overline{u}}_{\ell}:={\overline{u}}_{{n}_{{k}_{\ell}}}$ *, we have*$\mathcal{F}\left({u}_{\ell}\right)-\mathcal{F}\left({u}_{\ell}-{v}_{\ell}\right)\to \mathcal{F}\left(u\right)$ *and*$\mathcal{F}\left({\overline{u}}_{\ell}\right)-\mathcal{F}\left({\overline{u}}_{\ell}-{v}_{\ell}\right)\to \mathcal{F}\left(u\right).$

For the proof, let *R*.

We prove the assertions one by one.
(a): From (1.4) and from the theory of
superposition operators [4] it follows that
*U* of

The functions *Q* (see [14]). It is easy to build a sequence

Hence, for all

Consider a smooth cut off function

From the continuity of

Since *R* chosen accordingly, as in
(2.1),

where *C* is independent of ε. Letting
ε tend to 0 and using (2.2), we obtain

(b): The continuous embedding
*U* implies that

and *u* by (a).

(c): Since *R* large enough, (2.1) also holds
true if we replace

## 3 Uniform continuity

Here we prove uniform continuity on bounded subsets of *f* in *x*. As a consequence,
we also obtain BL-splitting along weakly convergent sequences in

For simplicity we will only prove the case

Let

and let

We first recall a functional consequence of Lions’ Vanishing Lemma [15, Lemma I.1.].

*Suppose for a sequence *

Note first that

If the claim were not true there would exist

Pick *n*. With

for all *n*. We reach a contradiction since

The claim of the theorem now follows from
[15, Lemma I.1.] with

We start by proving the uniform continuity. Let

and that there is

Successively we will define infinitely many sequences

and

Here,

We need to say something about the extraction of subsequences.
In order to obtain

For

If

Pick

There are

By (3.5) and by
Theorem 2.1
(b) and (c)
there exists a sequence

and

Set

By the equivariance of

and, since by (3.11) the map

Equations (3.13) and (3.4) (for *k*) imply that

hence (3.4) for *k*) imply that

hence (3.5) for *k*) that

hence (3.6) for *k*), (3.9), and (3.11) it
follows that

Since (3.8) is true for *k*, together with
(3.16) we obtain (3.8) for *k*) and (3.10) yield

By the definition of

This proves (3.7) for

We now skip a finite number of elements of the sequences
constructed in this induction step and adapt

and

for all

Since

for all *m* this implies

We fix *m* with these properties, we write

and

for all

Now we consider the process of constructing sequences as finished and proceed to prove properties of the whole set. By induction, (3.13) leads to

and hence

We claim that the
diagonal sequence

Note that by construction, for all

Hence we have the representation

First we show that

Fix

Then (3.19), (3.22), and the translation
invariance of the norm yield for

It is easy to see that the sequence

To finish the proof of (3.21), suppose for a
contradiction that

for every

as

with

is a subsequence of

We are now in the position to finish the proof of uniform
continuity of

and

By Lemma 3.1 and (3.21), we have

It only remains to prove BL-splitting for

as *v* in (3.26). Using this, a standard
reasoning by contradiction yields the claim.
∎

## 4 Construction of examples

We first treat the case

for *u* and

for each

as

For the other example,

for each

and hence

as

The construction of these counterexamples is closely related to
Example 1.2. First consider the function

introduce

and

as

and the claim follows. Note that the example above has no
simple analogue in the case

Now we treat the function

Define

and *n*, where *r* and center *z*. Again, it is straightforward to
check that

for all *n*. Hence by the calculation in (4.1), we have

and the claim follows. Note that this example has no simple
analogue in the case

I would like to thank Kyril Tintarev for an informative exchange on this subject. Moreover, I thank the referee for suggesting to analyze the limiting cases in Theorem 1.3.

## References

- [1]↑
N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), no. 2, 423–443.

- [2]↑
N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal. 234 (2006), no. 2, 277–320.

- [3]↑
N. Ackermann and T. Weth, Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Commun. Contemp. Math. 7 (2005), no. 3, 269–298.

- [4]↑
J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge Tracts in Math. 95, Cambridge University Press, Cambridge, 1990.

- [5]↑
Y. V. Bogdanskii, Laplacian with respect to a measure on a Hilbert space and an L 2 {L_{2}}-version of the Dirichlet problem for the Poisson equation, Ukrainian Math. J. 63 (2012), no. 9, 1336–1348.

- [6]↑
H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490.

- [7]↑
J. Chabrowski, Weak Convergence Methods for Semilinear Elliptic Equations, World Scientific, Singapore, 1999.

- [8]↑
G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Math. Soc. Lecture Note Ser. 293, Cambridge University Press, Cambridge, 2002.

- [9]↑
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd ed., Encyclopedia Math. Appl. 152, Cambridge University Press, Cambridge, 2014.

- [10]↑
Y. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations 29 (2007), no. 3, 397–419.

- [11]↑
Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal. 251 (2007), no. 2, 546–572.

- [13]↑
W. Kryszewski and A. Szulkin, Infinite-dimensional homology and multibump solutions, J. Fixed Point Theory Appl. 5 (2009), no. 1, 1–35.

- [14]↑
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145.

- [15]↑
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283.

- [16]↑
P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), no. 1, 33–97.

- [17]↑
E. Priola, On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions, Studia Math. 136 (1999), no. 3, 271–295.

- [18]↑
K. Tintarev and K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications, Imperial College Press, London, 2007.

- [19]↑
M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996.