Double phase anisotropic variational problems involving critical growth

In this paper, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions type concentration-compactness principle and its variant at infinity for the solution space, which are our independent interests. By employing these results, we obtain a nontrivial nonnegative solution to problems of generalized concave-convex type. We also obtain infinitely many solutions when the nonlinear term is symmetric. Our results are new even for the $p(\cdot)$-Laplace equations.


Introduction
In this paper, we investigate the existence of a nontrivial nonnegative solution and infinitely many solutions to the following double phase anisotropic problem of Schrödinger-Kirchhoff type: where a(x, ξ) = ∇ ξ A(x, ξ) behaves like |ξ| q(x)−2 ξ for small |ξ| and like |ξ| p(x)−2 ξ for large |ξ| with M : [0, ∞) → R is a real function, which is continuous and nondecreasing on some interval [0, τ 0 ); V : R N → R is measurable and positive a.e. in R N ; f : R N × R → R is a Carathéodory function and is of subcritical growth; b ∈ L ∞ (R N ) and b(x) > 0 for a.e.x ∈ R N ; and λ > 0 is a parameter.
Note that from (A1) we easily obtain a(x, −ξ) = −a(x, ξ), ∀(x, ξ) ∈ R N × R N . (1.3) Moreover, as shown in [61,Eqn. (3)] there exist positive constants M 1 , M 2 such that (1.4) The basic assumptions for the potential V and the variable exponent α to get the desired solution space are the following: For obtaining our concentration-compactness principle, we will need the following further assumption: (V 2) ess sup x∈BR V (x) < ∞ for any R > 0.Moreover, for covering some classes of subcritical terms we will include the following assumption on V in some situations: The studies of differential equations and variational problems with nonhomogeneous operators and non-standard growth conditions have attracted extensive attentions during the last decades.The interest in the equations associated by nonhomogeneous nonlinearities has consistently developed in light of the pure or applied mathematical perspective to illustrate some concrete phenomena arising from nonlinear elasticity, plasticity theory, and plasma physics.Let us recall some related results by way of motivation.Azzollini et al. in [2,3] introduced a new class of nonhomogeneous operators with a variational structure: − div(φ ′ (|∇u| 2 )∇u), where φ ∈ C 1 (R + , R + ) has a different growth near zero and infinity.Such a behaviour occurs if φ(t) = 2((1 + t) 1 2 − 1), which corresponds to the prescribed mean curvature operator defined by div ∇u In particular, Azzollini, d'Avenia and Pomponio in [2] proved the existence of a nontrivial nonnegative radially symmetric solution for the quasilinear elliptic problem where N ≥ 2, φ(t) behaves like t q/2 for small t and t p/2 for large t, and 1 < p < q < N, 1 < α ≤ p * q ′ p ′ , max{α, q} < s < p * (1.6) with p ′ = p/(p − 1) and q ′ = q/(q − 1).Under the above assumption (1.6), Chorfi and Rȃdulescu [18] considered the standing wave solutions for the following Schrödinger equation with unbounded potential: where the nonlinearity f : R N → R also satisfies the subcritical growth and a : R N → (0, ∞) is a singular potential satisfying some conditions.Anisotropic partial differential equations have recently gained significant attention, thanks to their applications in double and multiphase variational energies, along with their relevance in integral form anisotropic energies.For insights into this topic, we refer the reader to the survey paper [51] and the references therein.Problem (1.5) is relevant to double phase anisotropic phenomenon, in the sense that the differential operator has a different growth near zero and infinity.The double phase problems are described by the following functional ˆΩ H(x, ∇u) dx with the so-called (p, q)-growth conditions: c|ξ| p ≤ H(x, ξ) ≤ C(|ξ| q + 1).This (p, q)-growth condition was first treated by Marcellini [47][48][49][50] and it has been extensively studied in the last decades.Particularly double phase functionals have been introduced by Zhikov in the context of homogenization and Lavrentiev's phenomenon [64,65].After that it engages the enormous researchers' attention to the development of both theoretical and applications aspects of various double phase differential problems.For an overview of the subject, we refer the readers to the survey paper [52].Regularity theory for double phase functionals had been an unsolved issue for a while.However, we would like to mention a series of remarkable papers by Mingione et al. [5][6][7][21][22][23].Also we refer to the works of Bahrouni-Rȃdulescu-Repovš [4], Byun-Oh [15], Colasuonno-Squassina [20], Gasiński-Winkert [35,36], Liu-Dai [46], Papageorgiou-Rȃdulescu-Repovš [53,54], Perera-Squassina [55], Ragusa-Tachikawa [57], Zhang-Rȃdulescu [61], Zeng-Bai-Gasiński-Winkert [62,63].
In recent years the study on partial differential equations with variable exponent has received an increasing deal of attention because they can be perceived as their application in the mathematical modeling of many physical phenomena occurring in diverse studies related to electro-rheological fluids, image processing and the flow in porous media etc.There are many reference papers associated with the study of elliptic problems with variable exponent; see [25,27,58] and the references therein for more background on applications.Also we refer the readers to [24,41,57,59,61] for double phase differential problems with variable exponent.In this directions, Zhang and Rȃdulescu in a recent work [61] extended the results in [18] to the more general variable exponent case where the differential operator Φ(x, ξ) has behaviors like |ξ| q(x)−2 ξ for small ξ and like |ξ| p(x)−2 ξ for large ξ, 1 < α(•) ≤ p(•) < q(•) < N .In order to analyze problem (1.7), the authors gave useful elementary properties of a function space, called the variable exponent Orlicz-Sobolev space that is a generalization of Orlicz-Sobolev space setting in [2].In particular, they provided fundamental imbedding results in the solution space, such as the Sobolev imbedding and the compact imbedding when the potential V satisfies conditions (V 1) and (V 3).Also they obtained some topological properties for the energy functional corresponding to (1.7).Motivated by this work, the authors in [59] discussed the existence of multiple solutions to problem (1.7) with V ≡ 1 when nonlinear term f has concave-convex nonlinearities.Very recently, Cen et al. in [16] discussed the existence of multiple solutions to double phase anisotropic variational problems for the case of a combined effect of concave-convex nonlinearities: , a is an appropriate potential function defined in (0, ∞), and f : R N × R N → R is a Carathéodory function.Especially the superlinear (convex) term f substantially fulfils a weaker condition as well as Ambrosetti-Rabinowitz condition.The present paper can be seen as a continuation of the earlier works [16,59,61] to the case of problem (1.1) with critical growth and containing a Kirchhoff term.The critical problem was originally studied in the pioneer paper by Brezis-Nirenberg [13] dealing with Laplace equations.Since then many researchers have been interested in such problems and there have been extensions of [13] in many directions.As we know, one of the difficulties in studying elliptic equations in an unbounded domain involving critical growth is the absence of compactness arising in relation with the variational approach.To overcome this difficulty, the concentration-compactness principles (the CCPs, for short), which were initially provided by Lions [44,45], and its variant at infinity [8,10,17] have been used.In particular, these principles have played a crucial role in showing the precompactness of a minimizing sequence or a Palais-Smale sequence.By making use of these CCPs or extending them to the suitable solution spaces, many authors have been successful to deal with critical problems involving elliptic equations of various types, see e.g., [11,12,33,[37][38][39][40] and references therein.Regarding the nonlocal Kirchhoff term, it was first introduced by Kirchhoff [42] to study an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings.Elliptic problems of Kirchhoff type have a strong background in several applications in physics and have been intensively studied by many authors in recent years, for example [1,19,26,31,60] and the references therein.
The main feature of the present paper is to establish the existence and multiplicity of nontrivial nonnegative solutions to double phase anisotropic problem of Schrödinger-Kirchhoff type with critical growth.To do this, we first establish a Lions type concentration-compactness principle and its variant at infinity for the solution space of (1.1).As mentioned before, as compared with the p(•)-Laplacian, the solution space belongs to the variable exponent Orlicz-Sobolev spaces and the general variable exponent elliptic operator a has a different growth near zero and infinity.Hence this problem with nonhomogeneous operator has more complex nonlinearities than those of the previous works [33,34,[38][39][40], so more exquisite analysis has to be meticulously carried out when we obtain the CCPs for the solution space (see Theorems 3.1 and 3.2).As far as we are aware, there were no such results for the variable exponent double phase anisotropic problems in this situation.This is one of novelties of this paper.
As an application of this result, we will be concerned with the existence of a nontrivial nonnegative solution and infinitely many solutions to a class of double phase problems involving critical growth.Concerning this, on a new class of nonlinearities which is a concave-convex type of nonlinearities we get the existence result of a nontrivial nonnegative solution via applying the Ekeland variational principle (see Theorem 4.1).When the symmetry of the reaction term f is additionally assumed, by employing the genus theory we derive that problem (1.1) admits infinitely solutions (see Theorem 4.2).Although the proofs of Theorems 4.1 and 4.2 follow the basic idea of those in [19], we believe these consequences are new even in the constant exponent case as well as for the p(•)-Laplace equations.This is another novelties of the present paper.To the best of our knowledge, this study is the first effort to develop some existence and multiplicity results to the variable exponent double phase anisotropic problems with critical growth because we observe a new class of nonlinearities which is of a generalized concave-convex type.
Our paper is organized as follows.In Section 2 we review some properties of the variable exponent spaces.In Section 3 we establish the concentration-compactness principles for variable exponent Orlicz-Sobolev space, which plays as the solution space of problem (1.1).The final main section, Section 4, is devoted to the study of the existence and multiplicity of nontrivial solutions to problem (1.1).In the appendix, we provide a proof for a helpful inequality presented in Section 2.

Lebesgue-Sobolev spaces with variable exponent.
In this subsection, we briefly recall the definition and some basic properties of the Lebesgue-Sobolev spaces with variable exponent, which were systematically studied in [27].

Variable exponent Orlicz-Sobolev spaces.
In this subsection we review the definition and properties of the variable exponent Orlicz-Sobolev space introduced in [61], which is the solution space for problem (1.1) and the construction of energy functional.
Throughout this subsection, we assume (A4), (V 1) and (P 1).Let Ω be an open domain in R N .We define the following linear space: which is endowed with the norm: We define the variable exponent Orlicz-Sobolev space W(V, Ω) as which is equipped with the norm: We have the following (see [61,Proposition 3.11 and Theorem 3.14 (ii)]).
Proposition 2.6.The following conclusions hold: We also have the following compact imbedding, which is crucial to get our existence results; its proof can be found in [59,Theorem 3.10].
The next result is from [61,Corollary 3.13 (iii)] that shows the density of smooth functions with compact support in X.
Proposition 2.8.The space C ∞ c (R N ) is dense in X.Finally, we present a useful estimate regarding the main operator in (1.1), which is proved in Appendix and will be frequently employed in the next sections.Define and p α (x) := min{p(x), α(x)}, q α (x) := max{q(x), α(x)}, x ∈ R N .

The concentration-compactness principles
Let C c (R N ) be the set of all continuous functions u : R N → R whose support is compact, and let C 0 (R N ) be the completion of C c (R N ) relative to the supremum norm | • | L ∞ (R N ) .Let M(R N ) be the space of all signed finite Radon measures on R N with the total variation norm.We may identify M(R N ) with the dual of C 0 (R N ) via the Riesz representation theorem, that is, for each µ ∈ C 0 (R N ) * there is a unique element in M(R N ), still denoted by µ, such that (see, e.g., [32,Section 1.3.3]).We identify L 1 (R N ) with a subspace of M(R N ) through the imbedding We now state the main result of this section, that is, a concentration-compactness principle for the variable exponent Orlicz-Sobolev space X.
Then, there exist {x i } i∈I ⊂ C of distinct points and {ν i } i∈I , {µ i } i∈I ⊂ (0, ∞), where I is at most countable and C is given in (1.2), such that where S b and M 1 are given in (3.1) and (1.4), respectively.
Note that the preceding result does not provide any information about a possible loss of mass at infinity.The next theorem expresses this fact in quantitative terms.
Assume in addition that: Then Before giving a proof of Theorems 3.1 and 3.2, we review some auxiliary results for Radon measures.

Lemma 3.3 ( [39]
).Let ν, {ν n } n∈N be nonnegative and finite Radon measures on R N such that 37]).Let µ, ν be two nonnegative and finite Radon measures on R N such that for some constant C > 0 and for some p, q, r ∈ C + (R N ) satisfying max{p(•), q(•)} ≪ r(•).Then, there exist an at most countable set The following result is an extension of the Brezis-Lieb Lemma to weighted variable exponent Lebesgue spaces.
We are now in a position to prove Theorems 3.1 and 3.2.In the rest of this section, we denote the ball in R N centered at x 0 with radius ǫ by B ǫ (x 0 ) and simply write it as B ǫ when x 0 is the origin.We also denote Then, up to a subsequence, we have So, by Lemma 3.5, we deduce that From this and (3.11), we easily obtain 13) for some finite nonnegative Radon measure μ on R N .By the definition of X, it is easy to see that φv ∈ X for any φ ∈ C ∞ c (R N ) with supp(φ) ⊂ B R and for any v ∈ X.So, utilizing (3.1), for any We will prove that lim sup and lim sup where M 1 is given in (1.4).Assuming (3.15)-(3.17)for the moment, by taking the limit superior as n → ∞ in (3.14) and invoking Lemma 3.3 we obtain μ (R N ) ; hence, (3.5) follows in view of Lemma 3.4 and (3.12).
To see (3.15), we first notice that From this and the facts that Then we obtain (3.15) by invoking Proposition 2.5 and (3.11).
To see (3.16), we note that By this, we obtain (3.16) in view of Proposition 2.5.Finally, we prove (3.17).To this end, we first note that by (1.4), where From this and the fact that we get that where Up to a subsequence, we may assume that and We only prove (3.22) since (3.23) can be proved similarly.Clearly, (3.22) holds for the case λ 1 * = 0.For the case λ 1 * > 0, using (1.4) we have that for n large, Taking the limit as n → ∞ in the last equality and using (3.13) we obtain dμ. Equivalently, Thus, we have proved (3.22) (and similarly, (3.23)); hence, (3.17) follows due to (3.21).We claim that {x i } i∈I ⊂ C. Assume by contradiction that there is some Thus, invoking Lemma 3.5 again, we infer From this and the fact that ν(B) ≤ lim inf n→∞ ´B b|u n | t(x) dx (see [32,Proposition 1.203]), we obtain ν(B) ≤ ´B b|u| t(x) dx.Meanwhile, from (3.5), we have Thus by (3.1), we have Taking the limit superior as n → ∞ and invoking Lemma 3.3, then taking the limit superior as ǫ → 0 + in the last inequality we obtain From Proposition 2.1, we have Thus, we obtain On the other hand, arguing as that obtained (3.18) we get that Invoking Proposition 2.1, we have Similarly we have .
Thus, one has lim sup where µ i := lim ǫ→0 + µ(B ǫ (x i ).We have for all n ∈ N and all ǫ ∈ (0, 1).Thus, by invoking Proposition 2.5 we obtain lim sup (3.30) Now we analyze the last term in (3.24) to get (3.7).First we have That is, By Proposition 2.5 again, we have that , where c is a positive constant.Here in view of Proposition 2.1 we have used the fact that Hence, the estimate (3.34) infers lim Hence, one has Here and in the sequel, by •, • we denote the duality pairing between X and its dual X * .By (A6), Φ ′ : X → X * is strictly monotone and hence, Φ is weakly lower semicontinuous and therefore, x) .By extracting µ to its atoms, we deduce (3.6) and the proof is complete.
For the first term of the right-hand side of (3.36) we notice that Thus we obtain For the second term of the right-hand side of (3.36), we notice that 1 − φ R is a continuous function with compact support in R N .Hence, Clearly, lim R→∞ ´RN φ R dµ = 0 in view of the Lebesgue dominated convergence theorem.By this and (3.38), we deduce lim Using this and (3.37), we obtain (3.8) by taking the limit superior as n → ∞ and then letting R → ∞ in (3.36).In the same fashion, to obtain (3.9) we decompose Arguing as above, we obtain and using (3.40), we easily obtain (3.9) from (3.39).Moreover, by replacing φ R with φ in the above aguments we also have Next, we prove (3.10) when (E ∞ ) is additionally assumed.It is easy to see that φ R v ∈ X for all R > 0 and all v ∈ X.Thus by (3.1), we have Let ǫ be arbitrary in (0, 1).By (E ∞ ), there exists (3.43) For R > R 0 given, let {u n k } k∈N be a subsequence of {u n } n∈N such that Using Proposition 2.1 with (3.43), we have From this, (3.41) and (3.44) we deduce lim (3.45) On the other hand, arguing as that obtained (3.18) again with noticing supp(|∇u Invoking Proposition 2.1 and taking into account (3.43) again, we have Now we analyze the last term in (3.42) to show (3.10).First we have That is, Also we note that max Arguing as that leads to (3.35) we obtain lim R→∞ ˆB2R\BR u∇φ R p(x) dx = 0.

application
As an application of Theorems 3.1 and 3.2, in this section we will obtain the existence of a nontrivial nonnegative solution and infinitely many solutions for problem (1.1) when the reaction term is of generalized concave-convex type.
(F 1 ) There exist x ∈ R N and all τ ∈ R.
A typical example for f fulfilling ( . By a solution of problem (1.1), we mean a function u ∈ X such that This definition is clearly well defined under above assumptions thanks to the aforementioned imbeddings on X.
Our first existence result is the next theorem.
We will make use of critical points theory to determine solutions to problem (1.1).In order to get necessary properties regarding the Kirchhoff term, we truncate the function M (t) as follows.Let us fix τ0 ∈ (0, τ 0 ) such that and and Let λ > 0. We define modified energy functionals J λ , J λ : X → R as where u + := max{u, 0} and for a.e.x ∈ R N and all τ ∈ R. By a standard argument, we can show that J λ , J λ ∈ C 1 (X, R) and its Fréchet derivative J ′ λ , J ′ λ : X → X * are given by From the definition (4.4) of M 0 , it is clear that any critical point u of J λ (resp.J λ ) is a solution (resp.a nonnegative solution) to problem (1.1) provided ´RN A(x, ∇u) dx ≤ τ0 .
In the next two subsections, we always assume that (A1) − (A6), (P 1), (P 2), (V 1), (V 2), (E ∞ ), (M), (F 1 ) and (F 2 ) hold with We also make use of the following estimate that is easily derived from (3.1) and (4.1): Here and in the rest of this section, C i (i = 7, 8, • • • ) stand for positive constants depending only on the data, and we can take C 7 > 1.

Existence of a nontrivial nonnegative solution.
In this subsection, we will prove Theorem 4.1 via employing the Ekeland variational principle for J λ .For this purpose, we first obtain several auxiliary results.The next lemma provides a certain range of levels such that the local Palas-Smale condition for J λ is satisfied.In the following, by a (PS) c -sequence {u n } n∈N for a C 1 functional I : X → R we mean and we say that I satisfies the (PS) c condition if every (PS) c -sequence for I admits a convergent subsequence.Set where β and C 4 are given in (F 2 ).

14) and
for some positive constant C independent of n.From (4.11) and (4.13)-(4.15),we have that {u + n } n∈N is a (PS) c -sequence for J λ in X.
Set v n := u + n for n ∈ N.Then, v n ≥ 0 a.e. in R N and {v n } n∈N is a bounded (PS) c -sequence for J λ in X.By the reflexivity of X and Theorems 3.1-3.2,we find {x i } i∈I ⊂ C with I at most countable such that, up to a subsequence, we have v n ⇀ u in X, (4.17) We claim that I = ∅ and µ ∞ = ν ∞ = 0. To this end, let us suppose on the contrary that this does not hold.We first consider the case that there exists i ∈ I. Let ǫ > 0 and define φ i,ǫ as in the proof of Theorem 3.1.By (1.4) and (4.5) we have From this and (4.5), we have We will show that lim sup ǫ→0 + lim sup n→∞ T (v n , φ i,ǫ ) = 0, where T (v n , φ i,ǫ ) is each term in the righthand side of (4.24).To this end, we first note that by the boundedness of {v n } in X, we have Noticing the boundedness of {φ i,ǫ v n } in X, it follows from (4.11) that lim sup By (4.17) and (F 1 ), we have ) in view of (4.1).From this, (F 1 ) and (4.16), invoking the Lebesgue dominated convergence theorem, we easily see that lim sup Finally, invoking the Young inequality, (A3) and (4.25) we deduce that for an arbitrary δ > 0, for a positive constant C. By (4.17) and Proposition 2.5 we have as n → ∞.From this and (4.28) we obtain Arguing as that obtained (3.35) we get lim sup Thus, (4.29) implies that lim sup Since δ > 0 was taken arbitrarily, the preceding inequality leads to lim sup By passing to the limit superior as n → ∞ and taking into account (4.26), (4.27) and (4.30), we deduce from (4.24) that m 0 µ i ≤ ν i .
Combining this with (4.20) gives with h p (x) := p(x) t(x)−p(x) , h q (x) := q(x) t(x)−q(x) for x ∈ R N and Next, we consider the other case, namely, µ ∞ > 0. Let φ R be as in the proof of Theorem 3.2.Arguing as that obtained (4.24) we have By the same way of deriving (4.26), (4.27) and (4.30), we get Using this, (3.37) and (3.40) while passing to the limit superior as n → ∞ and then R → ∞ in (4.32), we derive Combining this with (3.10) gives We get from the last two inequalities that where .
From the estimate (4.31) for the case I = ∅ and the estimate (4.33) for the case µ ∞ > 0 we arrive at Arguing as those lead to (4.12) we have 2C4t − , we deduce from the last estimate that Passing to the limit as n → ∞ in the last inequality, invoking (4.19), (4.22) and (4.34), we obtain where Invoking Proposition 2.2, we have where l(•) := t(•) r(•) .Combining the last inequality with (4.35) gives where a > 0. Note that by Proposition 2.1 we have Using this fact, we consider the following two cases.
Therefore, in any case, we obtain where This contradicts with (4.10); that is, we have shown that I = ∅ and µ ∞ = ν ∞ = 0. Hence, (4.19) yields ´RN bv t(x) n dx → ´RN bu t(x) dx.From this and (4.16) we obtain in view of Lemma 3.5.Moreover, we also have in view of (4.1) and (4.17).Using (4.37), (4.38), (F 1 ) and invoking the Hölder type inequality (Proposition 2.2), we easily obtain On the other hand, by the monotonicity of ξ → a(x, ξ) (due to (A2)) and τ → |τ | α(x)−2 τ we have where By the boundedness of {v n } n∈N in X, (4.5), (4.11), and (4.17) we easily see that Invoking (4.17) again, we obtain from the last equality that From this and (4.17) we derive v n → u in X in view of [61,].Combining this with (4.13) gives u n = v n − u − n → u.Finally, to obtain the conclusion for J λ , we directly argue with {u n } n∈N instead of {v n } n∈N in the above arguments.The proof is complete.

Existence of infinitely many solutions.
In this subsection, we will prove Theorem 4.2 employing the genus theory for the truncated energy functionals.Our argument follows the proof of [19,Theorem 2.2].
Next, we will construct sequence of critical points {u n } n∈N of T λ such that A(u n ) < α 2 τ 1 (λ) p − α via genus theory.Let us denote by Σ the set of all closed subset E ⊂ X \ {0} such that E = −E, namely, u ∈ E implies −u ∈ E. For E ∈ Σ, let us denote the genus of a E by γ(E) (see [56] for the definition and properties of the genus).where T −ǫ λ := {u ∈ X : T λ (u) ≤ −ǫ}.From the assumption of A and f , T −ǫ λ is a closed subset of X \ {0} and is symmetric with respect to the origin; hence, γ(T −ǫ λ ) is well defined.Proof of Lemma 4.8.Let k ∈ N and let X k be a subspace of X of dimension k.Since all norms on X k are mutually equivalent, we find That is, Thus, by taking η with 0 < η < min As in [30, Lemma 3.9], we have c k < 0, ∀k ∈ N. (4.61) Let λ (2) * > 0 be such that where K 1 , K 2 and l are as in Lemma 4.4.Set where λ (1) * is given by (4.54).We have the following.
where K c := {u ∈ X \ {0} : T ′ λ (u) = 0 and T λ (u) = c}.Proof.Let λ ∈ (0, λ * ).Then, by (4.61) and the choice of λ * we have Thus, K c is a compact set in view of Lemmas 4.4 and 4.7.Using this fact and a standard argument for which the deformation lemma is applied, we derive the desired conclusion (see, for example, [9,Lemma 4.4] or [30,Lemma 3.10]. Proof of Theorem 4.2.Let λ * be defined as in (4.62).Let λ ∈ (0, λ * ).In view of Lemma 4.9, we find sequence {u n } n∈N of critical points of T λ with T λ (u n ) < 0 for all n ∈ N. By Lemma 4.7, {u n } are solutions to problem (1.1).Let us denote by u λ one of u n .By Lemma 4.7 again, we have From this and (2.3) we arrive at lim λ→0 + u λ = 0, and the proof is complete.
If x is the only element of {a 0 , b 0 , c 0 } that is greater than or equal 1 and the remaining two elements are denoted by y, z, then y, z ≤ 1 and it holds A(u) ≥ m 0 3 −q + α min u p − α , u q + α .