New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems

In this paper we present new embedding results for Musielak–Orlicz Sobolev spaces of double phase type. Based on the continuous embedding of W1,H(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,\mathcal {H}}(\Omega )$$\end{document} into LH∗(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\mathcal {H}_*}(\Omega )$$\end{document}, where H∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}_*$$\end{document} is the Sobolev conjugate function of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document}, we present much stronger embeddings as known in the literature. Based on these results, we consider generalized double phase problems involving such new type of growth with Dirichlet and nonlinear boundary condition and prove appropriate boundedness results of corresponding weak solutions based on the De Giorgi iteration along with localization arguments.


Introduction
Recently, Crespo-Blanco-Gasiński-Harjulehto-Winkert [16] studied the so-called double phase operator with variable exponents given by div |∇u| p(x)−2 ∇u + µ(x)|∇u| q(x)−2 ∇u , u ∈ W 1,H (Ω) ( with p, q ∈ C(Ω) such that 1 < p(x) < q(x) < N for all x ∈ Ω, 0 ≤ µ(•) ∈ L 1 (Ω) and W 1,H (Ω) is the corresponding Musielak-Orlicz Sobolev space being a uniformly convex space.Under the assumptions above, it is shown that the operator is continuous, bounded and strictly monotone.Moreover, under some Lipschitz continuity properties on the exponents and the weight function, we have the continuous embedding where the function H * is called the Sobolev conjugate function of H given by H(x, t) = t p(x) + µ(x)t q(x) for (x, t) ∈ Ω × [0, ∞), see Definition 3.1 for the precise characterization of H * .The proof of the embedding (1.2) is based on general embedding results of Musielak-Orlicz Sobolev spaces obtained by Fan [29] under the additional condition with q + and p − being the maximum and minimum of q and p on Ω, respectively.It is known that the embedding in (1.2) is not sharp, see Adams-Fournier [1], Donaldson-Trudinger [28] or Fan [29].
In the case of sharp embedding results from Orlicz Sobolev spaces into Orlicz spaces we refer to the work of Cianchi [14].So far, there does not exist any generalization of such sharp embeddings to Musielak-Orlicz Sobolev spaces.
The main objective of the paper is twofold.In the first part we want to discuss how we can obtain better embedding results from W 1,H (Ω) into L ϕ (Ω) by using the embedding (1.2).It will be seen that we get indeed a much better continuous embedding of the form with G * (x, t) := t p * (x) + µ(x) q * (x) q(x) t q * (x) , (x, t) ∈ Ω × [0, ∞), (1.4) where for a function r ∈ C(Ω) with 1 < r(x) < N for all x ∈ Ω, the critical exponent r * (•) is given by r * (x) := N r(x) N − r(x) , x ∈ Ω.
In addition we are able to prove that the exponent q * (x) q(x) in (1.4) of µ is optimal under all possible exponents so that (1.3) hold true.However, we do not know if the embedding in (1.3) is sharp under all generalized Φ-functions.In the first part we furthermore obtain trace embeddings from W 1,H (Ω) into L ϕ (∂Ω).Based on a general trace embedding result for Musielak-Orlicz Sobolev spaces obtained by Liu-Wang-Zhao [50], we show the following critical trace embedding with T * (x, t) := t p * (x) + µ(x) q * (x) q(x) t q * (x) , (x, t) where for a function r ∈ C(Ω) with 1 < r(x) < N for all x ∈ Ω, the critical exponent r * (•) is given by r * (x) := (N − 1)r(x) N − r(x) , x ∈ Ω.
We also prove corresponding "subcritical" embeddings related to (1.3) and (1.5) which turn out to be compact.
In the second part of the paper, based on the new embedding results in (1.3) and (1.5), we study the boundedness of weak solutions to Dirichlet and Neumann problems of the form − div A(x, u, ∇u) = B(x, u, ∇u) in Ω, and − div A(x, u, ∇u) = B(x, u, ∇u) in Ω, where Ω is a bounded domain in R N , N ≥ 2, with Lipschitz boundary Γ := ∂Ω, ν(x) denotes the outer unit normal of Ω at x ∈ Γ and the functions and C : Γ × R → R are Carathéodory functions that fulfill structure conditions as developed in (1.4) and (1.6), see hypotheses (D 1 ), (D 2 ), (N 1 ) and (N 2 ).Our results are based on the so-called De Giorgi-Nash-Moser theory, which provides iterative methods based on truncation techniques to get a priori bounds for certain equations, see the works of De Giorgi [25], Nash [57] and Moser [55].The techniques developed in these papers provided powerful tools to prove local and global boundedness, the Harnack and the weak Harnack inequality and the Hölder continuity of weak solutions.For more information we refer to the monographs of Gilbarg-Trudinger [36], Ladyženskaja-Ural ′ ceva [46], Ladyženskaja-Solonnikov-Ural ′ ceva [47] and Lieberman [48].Our proofs for L ∞ -bounds are mainly based on the papers of Ho-Kim [41], Ho-Kim-Winkert-Zhang [42] and Winkert-Zacher [69,70].We also mention the boundedness results in the works of Barletta-Cianchi-Marino [4] (for problems in Orlicz spaces), Gasiński-Winkert [33,35] (for double phase Dirichlet and Neumann problems), Ho-Sim [40] (for weighted problems), Kim-Kim-Oh-Zeng [44] (for variable exponent double phase problems with a growth less than p * (•)), Marino-Winkert [53,54] (for critical problems in W 1,p (Ω)) and Winkert [68] (for subcritical problems in W 1,p (Ω)).
The paper is organized as follows.In Section 2 we recall some properties of the double phase operator with variable exponents and present relevant embedding results.Section 3 is devoted to the study of the critical and subcritical embeddings mentioned above, see Propositions 3.4, 3.5 and 3.6 for the critical case and Propositions 3.7 and 3.8 for the subcritical case.In Section 4 we consider problems (1.7) and (1.8) where we suppose subcritical growth and prove a priori bounds for corresponding weak solutions, see Theorems 4.2 and 4.3.Finally, using the critical embedding in (1.3), we also develop boundedness results for weak solutions of (1.7) and (1.8) in this case, see Theorems 5.1 and 5.2 in Section 5.
Let Ω be a bounded domain in R N with Lipschitz boundary Γ := ∂Ω and let M (Ω) be the space of all measurable functions u : Ω → R.
We start with the following definition.
For a given ϕ ∈ Φ(Ω) we define the corresponding modular ρ ϕ by Then, the Musielak-Orlicz space L ϕ (Ω) is defined by Similarly, we define Musielak-Orlicz spaces L ϕ (Γ) on the boundary equipped with the norm • ϕ,Γ , where we use the (N − 1)-dimensional Hausdorff surface measure σ on R N .
The following proposition can be found in Musielak [56, Theorem 7.7 and Theorem 8.5].
(i) Let ϕ ∈ Φ(Ω).Then the Musielak-Orlicz space L ϕ (Ω) is complete with respect to the norm Next, we define Musielak-Orlicz Sobolev spaces.For this purpose, we need the following definition.
) is measurable for all t ∈ R and ϕ(x, •) is a N -function for a. a. x ∈ Ω.We denote the class of all generalized N -functions by N (Ω).
Let us now come to our special Musielak-Orlicz Sobolev space and its properties, which was introduced in [16].In the following, for h ∈ C(Ω) we denote We suppose the following assumptions: (H1) p, q ∈ C(Ω) such that 1 < p(x) < N and p(x) < q(x) for all x ∈ Ω and 0 ≤ µ(•) ∈ L 1 (Ω).Under hypothesis (H1), let H : Ω × [0, ∞) → [0, ∞) be the nonlinear function defined by H(x, t) := t p(x) + µ(x)t q(x) for all (x, t) ∈ Ω × [0, ∞). (2.1) Recall that the corresponding modular to H is given by Then, the corresponding Musielak-Orlicz space L H (Ω) is given by endowed with the norm Next, we can introduce the Musielak-Orlicz Sobolev space W 1,H (Ω) defined by equipped with the norm where u q + H . On W 1,H (Ω), we will also work with the following equivalent norm where the modular ρ1,H,Ω is given by ρ1,H,Ω (u) = The following results can be found in [ The following embedding results can be found in [16].
Proposition 2.7.Let hypotheses (H1) be satisfied .Then the following embeddings hold: Let us now suppose stronger conditions as in (H1): (H2) p, q ∈ C(Ω) such that 1 < p(x) < N and p(x) < q(x) < p * (x) for all x ∈ Ω and 0 ≤ µ(•) ∈ L ∞ (Ω).Under (H2) we have the following result, see [16,Proposition 2.18].Proposition 2.9.Let hypothesis (H2) be satisfied.Then the following hold: Next we mention the following lemma concerning the geometric convergence of sequences of numbers will be the key to our arguments to obtain the boundedness of solutions via the De Giorgi iteration.The proof of the lemma can be found in the paper of Ho-Sim [43,Lemma 4.3] Lemma 2.10.Let {Z n }, n = 0, 1, 2, . . ., be a sequence of positive numbers, satisfying the recursion inequality Furthermore, in the next sections we frequently use Young's inequality of the form Let us now fix our notation.We write N 0 := N ∪ {0} and for a real number t > 1 we denote by t ′ := t t−1 the conjugate number of t.For a measurable function v : Ω → R we set v + := max{v, 0} and v − := max{−v, 0}.
By |E| we denote the N -dimensional Lebesgue measure of E ⊂ R N and by |E| σ the (N −1)-dimensional surface measure of E ⊂ R N .For 1 ≤ ρ ≤ ∞, the space L ρ (E) is the usual Lebesgue space with norm • ρ,E .
3. New embedding results for Musielak-Orlicz Sobolev spaces W 1,H (Ω) In this section, we want to discuss new embedding results for the space W 1,H (Ω) into an Musielak-Orlicz space L ϕ (Ω) for a suitable Φ-function ϕ.These results extend those in Proposition 2.7.
First, we are going to introduce the Sobolev conjugate function of H.We define for all x ∈ Ω It is well known, since Ω is a bounded domain, that L H (Ω) = L H1 (Ω) and W 1,H (Ω) = W 1,H1 (Ω), see Musielak [56].Hence, for embedding results of W 1,H (Ω) we may use H 1 instead of H.For simplification, we write H instead of H 1 .
We start with the following definition.
Definition 3.1.We denote by where In order to have further properties on W 1,H (Ω) and W 1,H 0 (Ω), we suppose the following stronger assumptions as those in (H1) and (H2).(H3) p, q ∈ C 0,1 (Ω) such that 1 < p(x) < q(x) < N for all x ∈ Ω, The next proposition, obtained in [16,Proposition 2.21], provides fundamental embedding results on W 1,H (Ω) and W 1,H 0 (Ω).Proposition 3.2.Let hypotheses (H3) be satisfied.Then the following hold: It is worth pointing out that in [16, Proposition 2.21] the authors did not assume q + < N as well as they used a stronger condition of q(•) p(•) , namely q + p − < 1 + 1 N .However, from the proof of [16, Proposition 2.21], we can easily see that it indeed needs the condition q + < N and while one can relax the condition of q(•) p(•) as mentioned above.Note that thanks to the Poincaré-type inequality (3.2), the norm Regarding the critical boundary trace embedding, we have the following proposition.
continuously, where The proof of this result can be easily obtained by repeating the argument in the proof of [16,Proposition 2.19] to verify the conditions of Theorem 4.2 in Liu-Wang-Zhao [50].We leave the details for the reader.
In the following, we will give an exact form of the critical terms, which will help us to study double phase problems with a larger class of nonlinear terms than in previous works.
We have the following proposition.
Proposition 3.4.Let hypotheses (H3) be satisfied.Then we have the continuous embedding where G * is given by Proof.First, we are going to prove that where H * is the Sobolev conjugate function of H given in Definition 3.1.

It follows that
Similarly, we obtain for any (x, t) which implies (3.7) From (3.6) and (3.7) we get that Then, (3.5) follows.Invoking Propositions 2.2(ii) and 3.2 along with (3.5) we have the continuous embeddings which shows (3.3).The proof is complete.
Proof.For simplification, let Ω = B be the unit ball in R N and let p, q, α are constants satisfying α > 1, 1 < p < q < N and q p < 1 + 1 N .First, we will show that if (3.8) holds then r ≤ p * and s Clearly, v λ ∈ W 1,H 0 (B) for all λ ≥ 1.If the embedding (3.8) holds, then by Proposition 2.9 we find C > 0 such that v λ Br,s,α,Ω ≤ C ∇v λ H for all λ ≥ 1. (3.9) By the definition of the Musielak-Orlicz norm, we easily see that for ϕ(x, t) . By means of this fact, from (3.9) we find a constant C > 0 such that for all λ ≥ 1. (3.10) In order to see s ≤ q * , let µ(x) ≡ 1.From (3.10) we get By change of the variable y = λx we get from (3.11) that for all λ ≥ 1.
Since the preceding inequality holds for all λ ≥ 1, noticing p−N p < q−N q , we obtain Next, let µ(x) = |x|.Then, from (3.10) we have for all λ ≥ 1, and for all λ ≥ 1. (3.12) By change of the variable y = λx we get from (3.11) that for all λ ≥ 1.
Since the preceding inequality holds for all λ ≥ 1, noticing that q , we obtain Finally, we will show α ≥ q * q .For this purpose, let p, q, α > 1 be constants satisfying N +1 N < q < N and for ε ∈ (0, 1) small enough.From (3.12) we get for all λ ≥ 1.
Since the preceding inequality holds for all λ ≥ 1, noticing p−N p > q−N −1 q , we obtain From this and (3.13) we easily deduce that This means that if α < q * q , then by taking p ∈ (1, q) satisfying (3.13) with ε > 0 sufficiently small such that α < q * q − N 2 N −q ε we have that (3.14) cannot happen.Hence the embedding (3.8) does not hold.Thus, we have shown that the necessary condition for (3.8) to be valid for all p ∈ (1, q) and for all 0 Next, we will look for an explicit form for the critical boundary trace embedding.We have the following proposition.Proposition 3.6.Let hypotheses (H3) be satisfied.Then we have the continuous embedding where T * is given by with the critical exponents p * , q * given in (2.5).
Proof.From Jensen's inequality and (3.5), we have This implies that (3.15).We set λ = u W,Γ and assume first that λ > 0.Then, we obtain where Hence, we get From this and Proposition 3.3 we arrive at where C is a positive constant independent of u.The proof is complete.
In the last part of this section we prove new compact embedding results.Proposition 3.7.Let hypotheses (H3) be satisfied and let where r, s ∈ C(Ω) satisfy 1 < r(x) ≤ p * (x) and 1 < s(x) ≤ q * (x) for all x ∈ Ω.Then, we have the continuous embedding Furthermore, if r(x) < p * (x) and s(x) < q * (x) for all x ∈ Ω, then the embedding in (3.16) is compact.
Proof.First, it is clear that From this along with Propositions 2.2 and 3.4 we obtain (3.16).
Let us now suppose that r(x) < p * (x) and s(x) < q * (x) for all x ∈ Ω.In order to prove the compactness of the embedding in (3.16) it is sufficient to show that Ψ ≪ H * due to Proposition 3.2(ii).This means, for any k > 0, we need to show that lim t→∞ Ψ(x, kt) H * (x, t) = 0 uniformly for a. a. x ∈ Ω.
Similarly to Proposition 3.7, we have the following compact boundary trace embedding.
Proof.We have Then, the embedding (3.18) follows from Propositions 2.2 and 3.6 by taking (3.19) into account.
Next, suppose that ℓ(x) < p * (x) and m(x) < q * (x) for all x ∈ Ω and note that Let {u n } n∈N be a bounded sequence in W 1,H (Ω).From (3.20) we can suppose, up to a subsequence not relabeled, that u n → u in measure on Γ.Let ε > 0 be given and set From Proposition 3.6 we see that {v j,k } j,k∈N is bounded in L T * (Γ), say v j,k T * ,Γ ≤ k 0 for all j, k ∈ N.
Arguing as in the proof of Proposition 3.7, we find t 0 > 0 such that Then, arguing as in the proof of Theorem 8.24 of Adams-Fournier [1], we find N ε such that v j,k Υ,Γ < 1 for all j, k ≥ N ε .That is, we have shown that u j − u k Υ,Γ < ε for all j, k ≥ N ε .Hence, u n → u in L Υ (Γ).The proof is complete.

4.
A priori bounds for generalized double phase problems with subcritical growth In this section, we prove the boundedness of weak solutions to the problems (1.7) and (1.8) when the nonlinearities involved satisfy a subcritical growth as developed in Propositions 3.7 and 3.8.The proofs are using ideas from the papers of Ho-Kim [41], Ho-Kim-Winkert-Zhang [42] and Winkert-Zacher [69,70].
Let hypotheses (H3) be satisfied.We suppose the following structure conditions on A and B: for a. a. x ∈ Ω and for all (t, ξ) ∈ R × R N , where α 1 , α 2 , α 3 , β are positive constants and r, s ∈ C(Ω) satisfy p(x) < r(x) < p * (x) and q(x) < s(x) < q * (x) for all x ∈ Ω.For the second problem (1.8) with nonlinear boundary condition we need the additional assumption on C: q(x) |t| h(x)−1 + 1 for a. a. x ∈ Γ and for all t ∈ R, where γ is a positive constant and ℓ, h ∈ C(Ω) satisfy p(x) < ℓ(x) < p * (x) and q(x) < h(x) < q * (x) for all x ∈ Ω.The weak formulation of (1.7) and (1.8) read as follows.
Under the assumptions (D 1 ) and (N 1 ) we know that the terms in (4.1) and (4.2) are well-defined due to Propositions 3.7 and 3.8.
For the Dirichlet problem (1.7) we have the following result.(Ω) of problem (1.7) belongs to L ∞ (Ω) and satisfies the following a priori estimate where C, τ 1 , τ 2 are positive constants independent of u and Ψ(x, t) := t r(x) + µ(x) Proof.The proof is based on the ideas used in Ho-Kim [41], Winkert-Zacher [69,70] and will use Lemma 2.10.Let u be a weak solution of problem (1.7).
Step 1. Defining the recursion sequence and basic estimates.
For each n ∈ N 0 , we define where and with κ * > 0 to be specified later.Obviously,

It is clear that
Moreover, from the estimates and we obtain the following inequalities and By the assumptions on the exponents we have Combining this with (4.6) and (4.8) gives for all n ∈ N 0 .Next, we are going to show the following estimate for truncated energies Here and in the rest of the proof, C i (i ∈ N) are positive constants independent of u, n and κ * .To this end, testing (4.1) by ϕ = (u Since u ≥ u − κ n+1 > 0 on A κn+1 , using (D 1 )(ii) and (D 1 )(iii) along with Young's inequality and the fact that u ≤ u r(x) + 1 on A κn+1 , we obtain the estimates q(x) u s(x) + 1 dx.
Combining the last two estimates with (4.11) and then using (4.7) it follows From this and (4.8) we obtain (4.10).
In this step, we will obtain (4.3) by using an argument similar as in Ho-Kim [41, Proof of Theorem 4.2].From Lemma 2.10, we get using (4.27) that

.31)
In order to specify κ * satisfying (4.31), we first estimate is equivalent to .
On the other hand we have that , is equivalent to Therefore, by choosing This implies that and so ess sup Replacing u with −u in the arguments above, we obtain ess sup Therefore, where C is a positive constant independent of u.Note that by Proposition 2.5, we have the following relation Combining this and (4.35), we derive (4.3) and the proof is complete.
Next, we want to prove a priori bounds for problem (1.8).We have the following result.
Proof.The proof uses similar ideas as the proof of Theorem 4.2.
Step 1. Defining the recursion sequence {X n } n∈N0 and basic estimates.
Step 2. Estimating X n+1 by X n .
Let n ∈ N 0 .We are going to estimate X n+1 by X n through estimating Z n+1 by Z n and Y n+1 by X n .To this end, let {B i } m i=1 be a finite open cover of Ω as in Step 2 of the proof of Theorem 4.2 with the same notations.Denote by I the set of all i ∈ {1, • • • , m} such that Γ i := B i ∩ Γ = ∅.We may take R such that (4.12) and (4.13) hold and and From Step 2 of the proof of Theorem 4.2, we have where b, µ 1 , µ 2 are given by (4.28) and δ 1 , δ 2 are given by (4.29).
We will also apply Lemma 2.10 to the sequence {X n } n∈N0 with the recursion inequality (4.58).Note that By repeating the arguments in Step 3 of the proof of Theorem 4.2 we get that where C, τ1 and τ2 are positive constants independent of u and In view of Proposition 2.5 we easily derive (4.36) from (4.59) and this completes the proof.

The boundedness for generalized double phase problems with critical growth
The aim of this section is to discuss the boundedness of weak solutions to the problems (1.7) and (1.8) when we suppose a critical growth based on the Propositions 3.4 and 3.6 via the idea in Ho-Kim-Winkert-Zhang [42] using the De Giorgi iteration together with a localization method.Let hypotheses (H3) be satisfied and we state our hypotheses on the data.
(D 2 ): The functions for a. a. x ∈ Ω and for all (t, ξ) ∈ R × R N , where α 1 , α 2 , α 3 and β are positive constants.For problem (1.8) we need an additional assumption for the boundary term.
q * (x) q(x) |t| q * (x)−1 + 1 for a. a. x ∈ Γ and for all t ∈ R, where γ is a positive constant.The definitions of weak solutions to problems (1.7) and (1.8) are the same as that given in Definition 4.1.In view of Propositions 3.4 and 3.6, these definitions make sense under the above conditions (D 2 ) and (N 2 ).
We start with the Dirichlet problem (1.7) and have the following result.Proof.As before, we can cover Ω by balls {B i } m i=1 with radius R such that each Ω i := B i ∩ Ω (i = 1, • • • , m) is a Lipschitz domain.Note that by (H3), it holds q(x) < p * (x) for all x ∈ Ω.Thus, we may take the radius R sufficiently small such that Let u be a weak solution to problem (1.7) and let κ * ≥ 1 be sufficiently large such that where A κ for κ ∈ R is defined in (4.4) and recall that, for all (x, t) ∈ Ω × [0, ∞), H(x, t) := t p(x) + µ(x)t q(x) and G * (x, t) := t p * (x) + µ(x) q * (x) q(x) t q * (x) , see (2.1) and (3.4).Then, let {κ n } n∈N0 be as in (4.5) and define v n := (u − κ n+1 ) + for each n ∈ N 0 .Moreover, we define Similarly to the previous Section 4, we easily obtain and (5.5) In order to apply Lemma 2.10, in the following we will establish recursion inequalities for {Z n } n∈N0 .
In the rest of the proof, as before, C i (i ∈ N) stand for positive constants independent of n and κ * .

Claim 2: It holds that
Note that u ≥ u − κ n+1 > 0 and u > κ n+1 ≥ 1 on A κn+1 .Applying this along with the structure conditions in (D 2 )(ii), (D 2 )(iii) along with Young's inequality, we reach the following estimates Combining the last two estimates with (5.7) and then using (4.7), we obtain dx.

This yields
Thus, from Claim 1 and the last inequality we obtain the assertion in Claim 2.
Using the Claims 1 and 2 along with (5.4) gives us where b := 2 So, writing Z n := Z 2n and b := b 2 , we have (5.9) Now we can apply Lemma 2.10 to (5.9).This yields (5.12) Lemma 2.10 applied to (5.12) now leads to . (5.14) We point out that Hence, if we choose κ * > 1 sufficiently large, we obtain .
Therefore, (5.2), (5.11) and (5.14) are satisfied and we then get (5.10) and (5.13) which says that This implies that Thus, (u − 2κ * ) + = 0 a. e. in Ω and so ess sup Replacing u by −u in the above arguments we also get that ess sup From the last two estimates we obtain This shows the assertion of the theorem.
Next, we are going to study the boundedness of weak solutions of (1.8) under critical growth and the additional structure condition (N 2 ).This result reads as follows.
Combining the last three estimates, we obtain This finishes the proof.