In this work, we demonstrate the existence of distributional solutions for a specific class of anisotropic nonlinear elliptic partial differential equations with variable exponents, of the type :

(1.1)-∑i=1N∂i(∣∂iu∣pi(x)-2∂iu)=∑i=1N(∣f∣+∣u∣)pi(x)-1, in Ω,u=0, on ∂Ω,

where Ω⊂RN (N≥2) is an open bounded Lipschitz domain (having a Lipschitz boundary ∂Ω), ∂iu=∂u∂xi,i=1,…,N, and the datum f belongs to the variable exponents anisotropic Lebesgue space Lp→(⋅)(Ω), which is defined as follows:

Lp→(⋅)(Ω)=∩ i=1NLpi(⋅)(Ω).

Here, problem (1.1) is p→(x)-Laplacian operator equations, which involve the anisotropic operator with variable exponents defined between the space W∘1,p→(⋅)(Ω)(which will be further discussed in Section 2) and its dual, as follows:

u↦-∑i=1N∂i(∣∂iu∣pi(x)-2∂iu).

It is important to note that operators of this type have numerous applications in applied sciences. For instance, they are commonly used in electro-rheological fluids and image processing, as seen in references [6, 7, 2]. The noveltly of our work is that we take the right-hand side as a compound nonlinearity G(x,u)=∑i=1N(∣f∣+∣u∣)pi(x)-1 that links the unknown u and the datum f∈Lp→(⋅)(Ω). One cannot seperate these to reduce the problem to the classical case where the right-hand side is in certain Sobolev spaces.

The proof is based on the usual method, which requires proving the existence of a sequence of suitable approximate solutions (un) using the Leray-Schauder's fixed point Theorem. Prior estimates are then used to show the boundedness of the solutions un and the almost everywhere convergence of their partial derivatives ∂iun,i=1,…,N, which can be converted into strong L1-convergence. With this convergence, we can pass to the limit by L1-strongly sense in ∣∂iun∣pi(x)-2∂iun, and in (∣fn∣+∣un∣)pi(x)-1, and finally we conclude the convergence of un to the solution of (1.1).

The paper is divided into several sections, with Section 2 covering mathematical preliminaries. In this section, we discuss variable exponents anisotropic Lebesgue-Sobolev spaces and their key characteristics, as well as mentioning some embedding theorems. The main theorem and its proof can be found in Section 3.

In this section, we will provide a brief reminder about variable exponent anisotropic Lebesgue and Sobolev spaces. We will mention their most important properties and facts that are relevant to this paper. For further information, please refer to sources [5, 1, 3]).

Let Ω⊂RN>(N≥2) be a bounded open subset, we define the set

C+(Ω¯)={p→(⋅)∈C(Ω¯,R), 1<p-≤p+<∞},

where, p+=maxx∈Ω¯p→(x),   and   p−=minx∈Ω¯p→(x).

Let p→(⋅)∈C+(Ω¯). Then the following Young's inequality holds true for all a,b∈R and all ε>0,

(2.1)∣ab∣≤ε∣a∣p→(x)+c(ε)∣b∣p'(x),

where, p'(·) denotes the Sobolev conjugate of p(⋅) (i.e. 1p(⋅)+1p'(·)=1 in Ω¯).

In addition, for any two real a,b((a,b)≠(0,0)) :

(2.2)(|a|p(x)−2a−|b|p(x)−2b)(a−b)≥22−p+ |a−b|p(x) ,if p(x)≥2,(p- 1)|a−b|2 (|a|+|b|)2−p(x) ,if 1<p(x)<2.

Also, we will recall this elementary inequality:

(2.3)(a1+…+am)r≤max{1,mr−1}(a1r+…+amr),

which is valid for ai≥0,i=1,…,m and r≥0.

The variable exponent Lebesgue space Lp(⋅)(Ω) defined by

Lp(⋅)(Ω):={measurable functionsu:Ω↦R; ρp(⋅)(u)<∞},

where,

ρp(⋅)(u):=∫Ω∣u(x)∣p(x)dx, the convex modular of u.

It is a Banach space, and reflexive if p->1, under the norm

∥u∥p(⋅):=∥u∥Lp(⋅)(Ω)=infγ>0∣ρp(⋅)(u/γ)≤1.

The Hölder type inequality:

∣∫Ω uvdx∣≤ 1 p−+1 p′ −‖u‖ p(⋅) ‖v‖ p′(⋅) ≤2‖u‖ p(⋅) ‖v‖ p′(⋅) ,

holds true.

The variable exponents Sobolev space W1,p(⋅)(Ω) defined as fellows

W1,p(⋅)(Ω):=u∈Lp(⋅)(Ω):∣Du∣∈Lp(⋅)(Ω),

it becomes a Banach space when equipped with the norm

(2.4)u↦∥u∥W1,p(⋅)(Ω):=∥Du∥p(⋅).

We define also the Banach space W01,p(⋅)(Ω) by

W01,p(⋅)(Ω):=C0∞(Ω)¯W1,p(⋅)(Ω),

endowed with the norm (2.4). Moreover, is reflexive and separable if p(⋅)∈C+(Ω¯).

The following results came in [1, 3].

If (un),u∈Lp(⋅)(Ω), then we have

(2.5)minρp(⋅)(u)1p+,ρp(⋅)(u)1p−≤‖u‖p(⋅)≤maxρp(⋅)(u)1p+,ρp(⋅)(u)1p−,
(2.6)min‖u‖p(⋅)p−,‖u‖p(⋅)p+≤ρp(⋅)(u)≤max‖u‖p(⋅)p−,‖u‖p(⋅)p+.

Now, we will introduce the concept of anisotropic Sobolev spaces with variable exponents W1,p→(⋅)(Ω), as we need them to solve our problem (1.1).

Let pi(·)∈C(Ω¯,[1,+∞)),i=1,…,N, and we set for every x in Ω¯

p→(x)=(p1(x),…,pN(x)), p+(x)=max1≤i≤Npi(x),p−(x)=min1≤i≤Npi(x),p¯(x)=N∑ i=1 N1 pi (x), p+(x)=max1≤i≤Npi(x),p++=maxx∈Ω¯p+(x), p−(x)=min1≤i≤Npi(x),p−−=minx∈Ω¯p−(x),  p ¯ ⋆(x)=Np¯ (x)N−p¯ (x), for p¯ (x)<N, +∞, for p¯ (x)≥N.

The Banach space W1,p→(⋅)(Ω) is defined by

W1,p→(⋅)(Ω)=braceu∈Lp+(·)(Ω),Diu∈Lpi(·)(Ω), i=1,…,Nbrace,

under the norm

‖u‖W1,p→(⋅)(Ω)=‖u‖p+(⋅)+∑ i=1NDi upi (⋅).

The spaces W01,p→(⋅)(Ω) and W∘1,p→(⋅)(Ω) are defined as follow

W01,p→(⋅)(Ω)=C0∞(Ω)¯W1,p→(⋅)(Ω), W∘1,p→(⋅)(Ω)=W1,p→(⋅)(Ω)∩W01,1(Ω).

The following embedding results given in [4, 5].

Let Ω⊂RN be a bounded domain and p→(⋅)∈(C+(Ω¯))N.

Lemma 2.1. If r∈C+(Ω¯) and (p+(x), p¯⋆(x)). Then the embedding

(2.7)W∘1,p→(⋅)(Ω)↪Lr(·)(Ω) is compact.

Lemma 2.2. If we have

(2.8)∀x∈Ω¯, p+(x)<p¯⋆(x).

Then the following inequality holds

(2.9)∥u∥Lp+(·)(Ω)≤C∑i=1N∣Diu∣Lpi(·)(Ω), ∀u∈W∘1,p→(⋅)(Ω),

where C>0 independent of u.Thus,

(2.10)u↦∑i=1N∣Diu∣Lpi(·)(Ω) is an equivalent norm on W∘1,p→(⋅)(Ω).

Definition 3.1. The function u is a solution of the problem (1.1) in the sense of distributions if and only if u∈W01,1(Ω), and for all φ∈Cc∞(Ω),

∑i=1N∫Ω∣∂iu∣pi(x)-2∂iu∂iφdx=∑i=1N∫Ω(∣f∣+∣u∣)pi(x)-1φdx.

Our main result is the following.

Theorem 3.1. Let p→(⋅)∈(C+(Ω¯))N such that p¯<N and (2.8) holds, and assume that f∈Lp→(⋅)(Ω). Then the problem (1.1) has at least one distributional solution u∈W∘1,p→(⋅)(Ω).

3.1. Existence of approximate solutions

Let (fn) be a sequence of bounded functions defined in Ω which converges to f in Lp→(⋅)(Ω). Since fn∈Lp→(⋅)(Ω), from (2.5) we obtain

‖fn‖pi(⋅)≤1+ρpi(⋅)1pi−(⋅)(fn)≤2+ρpi1p−−(fn)<∞.

Through this, we conclude that

(3.1)fn is bounded in Lpi(·)(Ω),i=1,…,N.

Lemma 3.1. Let p→(⋅)∈(C+(Ω¯))N such that p¯<N and (2.8) holds, and assume that f∈Lp→(⋅)(Ω). Then, there exists at least one weak solution un∈W∘1,p→(⋅)(Ω) to the approximated problems

(3.2)-∑i=1N∂i(∣∂iun∣pi(x)-2∂iun)=∑i=1N(∣fn∣+∣un∣)pi(x)-1, in Ω,un=0, on ∂Ω,

in the following sense

(3.3)∑ i=1N∫Ω∣∂iun∣pi(x)−2∂iun∂iφdx=∑ i=1N∫Ω (∣f n∣+∣ u n∣) p i(x)−1 ,

for every φ∈W∘1,p→(⋅)(Ω)∩L∞(Ω).

Before proving Lemma 3.1 we must prove the following lemma:

Lemma 3.2. Let n∈N* fixed, and for all (v,θ)∈X×[0,1] where X=Lp+(·)(Ω) we consider the problem

(3.4)−∑ i=1N∂i∣∂iu∣pi(x)−2∂iu=θ∑ i=1N(∣fn∣+∣v∣)pi(x)−1, in Ω,u=0 on ∂Ω.

For all (v,θ)∈X×[0,1] the problem (3.4) has only the weak solution u satisfying for all φ∈W∘1,p→(⋅)(Ω), the weak formulation

(3.5)∑ i=1N∫Ω∣∂iu∣pi(x)−2∂iu∂iφdx=θ∑ i=1N∫Ω(∣fn ∣+∣v∣) p i (x)−1 φdx.

Moreover, the operator Ψ:X×[0,1]⟶X defined by :

Ψ(v,θ)=u⇔(u is the only weak solution of the problem (3.4)),

is continuous and compact.

Proof Using (2.3) and the fact that fn,v∈Lpi(·)(Ω) we get for all (v,θ)∈X×[0,1] that

(3.6)∫Ω(∣fn∣+∣v∣)pi(x)-1p'i(x)dx≤c∫Ω(∣fn∣pi(x)+∣v∣pi(x))dx≤C.

Therefore, from (3.6) we obtain for all i=1,…,N,

(∣fn∣+∣v∣)pi(x)-1∈Lp'i(·)(Ω),

and this implies that

θ∑ i=1N(∣fn∣+∣v∣)pi(x)−1∈Lp→′(⋅)(Ω)=∩i=1NL p ′i(⋅)(Ω).

Here, p′→(⋅)=(p′1(⋅),…,p′N(⋅)) where p'i(·) denotes the Sobolev congugate of pi(·).

The existence of the weak solution u of the problem (3.4) in Lp+(·)(Ω) is directly produced by the main Theorem on monotone operators, and the uniqueness of this solution is a direct result of the homogeneous problem and this by assuming the existence of two weak solutions.

Now we give an estimate of the solution u of the problem (3.4). Taking φ=u as test function, and using (2.8), (2.3), (2.5), (2.6), the fact that pi(·)≤p¯*(·) (from (2.8)), Lemma 2.1, boundedness of fn∈Lp→(⋅)(Ω), and Hölder inequality, we have

(3.7)∑i=1N ∫Ω∣∂i u∣ pi (x)dx≤∑i=1N ∫Ω(∣fn ∣+∣v∣) pi (x)−1∣un∣dx≤2‖∑i=1N (∣fn∣+∣v∣) pi(x)−1‖ p′ i(x)‖u‖pi(⋅)≤c∑i=1N ‖∣f∣ p i(x)−1‖ p′ i(x) +∑i=1N ‖∣v∣ p i(x)−1‖ p′ i(x) ‖u‖p→(x)≤c2N+∑i=1N ∫Ω ∣ f n ∣ p i (x) dx 1 p i− +∑i=1N ∫Ω ∣ v ∣ p i (x) dx 1 p i− ‖u‖p→(x)≤cc′+∑i=1N ‖v‖ p i(⋅) p i+ p i− ‖u‖p→(x)≤C1+‖v‖ p+(⋅) p++ p−− ‖u‖p→(x),

where for the last equality we used that

c′+∑ i=1N‖v‖pi(⋅) pi+ pi−≤c′+∑ i=1N‖v‖pi(⋅) p++ p−−≤c″+c‴‖v‖p+(⋅)p++p−−

for appropriate constants c” and c”'.

On the other hand, by (2.6), we get

1+∫Ω∣∂iu∣pi(x)dx≥∥∂iu∥pi(·)pi-,i=1,…,N,

and we have

1+∥∂iu∥pi(·)pi-≥∥∂iu∥pi(·)p--,i=1,…,N.

Through this, we find that

2+∫Ω∣∂iu∣pi(x)dx≥∥∂iu∥pi(·)p--,i=1,…,N.

Then, we conclude

2N∣Ω∣+∑i=1N∫Ω∣∂iu∣pi(x)dx≥∑i=1N∥∂iu∥pi(·)p--.

So, we get

(3.8)∑i=1N∫Ω∣∂iu∣pi(x)dx≥1N∑i=1N∣∂iu∣pi(·)p---2N∣Ω∣.

By combining (3.7) and (3.8), we obtain

(3.9)‖u‖p→(⋅)p−−≤c1+‖v‖p+(⋅)p++ p−− ‖u‖p→(⋅)+c′,

where, c>0 and c'>0. Since ∥u∥p→(⋅)>1, from (3.9) we have

(3.10)∥u∥p→≤c1+∥v∥p+(·)p++p--+c'1p---1.

Since ∥u∥p→(⋅)≤1, we find that (3.10) only holds in this case with under certain conditions, such as c≥1 or c'≥1. The goal is to combine the two cases ∥u∥p→(⋅)>1, and ∥u∥p→(⋅)≤1 into same result (3.10).

We will now prove the continuity of Ψ. Let (vm,θm) be a sequence of Lp+(·)(Ω)×[0,1] converging to (v,θ) in this space. Then,

(3.11)vm→v, Strongly in Lp+(⋅)(Ω),
(3.12)θm→θ, in ℝ.

After considering the sequence (um) defined by um=Ψ(vm,θm),m∈N*, we obtain for n fixed in N* and all φ∈W∘1,p→(⋅)(Ω)

(3.13)∑i=1N∫Ω∣∂ium∣pi(x)-2∂ium∂iφdx=θm∑i=1N∫Ω(∣fn∣+∣vm∣)pi(x)-1φdx.

For v,θ defined in (3.11), (3.12), we put u=Ψ(v,θ), then we have for n fixed in N* and all φ∈W∘1,p→(⋅)(Ω)

(3.14)∑i=1N∫Ω∣∂iu∣pi(x)-2∂iu∂iφdx=θ∑i=1N∫Ω(∣fn∣+∣v∣)pi(x)-1φdx.

By(3.10) and the boundedness of (vm) in Lp+(·)(Ω) (from (3.11)):

(3.15)∥um∥p→(⋅)=∥Ψ(vm,θm)∥p→(⋅)≤c1+∥vm∥p+(·)p++p--+c'1p---1≤ϱ,

with ϱ>0 independent of m.

From (3.15) we conclude that the sequence (um) is bounded in W∘1,p→(⋅)(Ω). So, there exists a function w∈W∘1,p→(⋅)(Ω) and a subsequence (still denoted by (um)) such that

(3.16)um⇀w weakly in W∘1,p→(⋅)(Ω).

By (3.16), (2.8), and Lemma 2.1, we obtain that

(3.17)um⟶w Strongly in Lp+(·)(Ω).

Since the function s↦(∣fn∣+∣s∣)pi(x)-1 is continuous on Lp+(·)(Ω), we can pass to the limit in (3.13) as m⟶+∞, then we get for all φ∈W∘1,p→(⋅)(Ω),

(3.18)∑i=1N∫Ω∣∂iw∣pi(x)-2∂iw∂iφdx=θ∑i=1N∫Ω(∣fn∣+∣v∣)pi(x)-1φdx,

and this implies that w=Ψ(v,θ).

The uniqueness of the weak solution of problem (3.4) then shows that w=u=Ψ(v,θ). So,

Ψ(vm,θm)=um⟶u=Ψ(v,θ) Strongly in Lp+(·)(Ω).

Which shows the continuity of Ψ.

We now move on to prove the compactness of Ψ. Let B˜ be a bounded of Lp+(·)(Ω)×[0,1]. Thus B˜ is contained in a product of the type B×[0,1] with B a bounded set of Lp+(·)(Ω), which can be assumed to be a ball of center O and of radius r>0.

For u∈Ψ(B˜), thanks to (3.10), we get

∥u∥p→(⋅)≤c1+rp++p--+c'1p---1=ρ.

For u=Ψ(v,θ) with (v,θ)∈B×[0,1] ( ∥v∥p+(·)≤r). This proves that Ψ applies B˜ in the closed ball in W∘1,p→(⋅)(Ω)⊂Lp+(·)(Ω) of center O and radius ρ. Let un be a sequence of elements of Ψ(B˜), therefore un=Ψ(vn,θn) with (vn,θn)∈B˜. Since un remains in a bounded of W∘1,p→(⋅)(Ω), it is possible to extract a subsequence which converges strongly to an element u of Lp+(·)(Ω). This proves that Ψ(B˜)¯Lp+(·)(Ω) is compact. So Ψ is compact.

Proof (of the Lemma 3.1):

It is clear that

(3.19)Ψ(v,0)=0, ∀v∈X,

because u=0∈Lp+(·)(Ω) the only weak solution of the problem (3.4) in the case θ=0.

Now we show that there is an M>0 such that

(3.20)∀(v,θ)∈X×[0,1]:v=Ψ(v,θ)⇒∥v∥X≤M.

For that, we give the estimate for v∈Lp+(·)(Ω) such that v=Ψ(v,θ), then we have for all φ∈W∘1,p→(⋅)(Ω),

(3.21)∑i=1N∫Ω∣∂iv∣pi(x)-2∂iv∂iφdx=θ∑i=1N∫Ω(∣fn∣+∣v∣)pi(x)-1φdx.

After choosing φ=v in (3.21), and using (2.3), the fact that fn∈Lp→(⋅)(Ω),v∈Lp+(·)(Ω), and Young's inequality, we obtain

(3.22)∑i=1N ∫Ω∣∂iv∣pi(x)dx≤c∑i=1N ∫Ω(∣fn ∣ p i (x)−1∣v∣+∣v∣ p i (x))dx≤c∑i=1N ∫Ω∣v∣pi(x)dx+cC(ε)∑i=1N ∫Ω∣fn∣ pi(x)dx+ε∑i=1N ∫Ω∣v∣ pi(x)dx=c(1+ε)∑i=1N ∫Ω∣v∣pi(x)dx+cC(ε)∑i=1N ∫Ω∣fn∣pi(x)dx≤c(1+ε)N∣Ω∣+∑i=1N ∫Ω∣v∣ p+(x)dx+C′(ε)≤C″(ε).

So, for any fixed choice of ε in (3.22), we obtain

(3.23)∑i=1N∫Ω∣∂iv∣pi(x)dx≤C.

Using similar arguments to those used for (3.8), we get

(3.24)∑i=1N∫Ω∣∂iv∣pi(x)dx≥1N∑i=1N∥∂iv∥pi(·)p---2N∣Ω∣.

By combining (3.23) and (3.24) with using (2.8), we obtain that

(3.25)C'Np--∥v∥p+(·)p--≤C''.

From this, we conclude that, there exist c>0, such that

(3.26)∥v∥p+(·)≤c.

This implies (3.20). Through, (3.19), (3.20), and Lemma 3.2, we can apply the Leray-Schauder Theorem. So, the operator Ψ1:X⟶X defined by Ψ1(u)=Ψ(u,1) has a fixed point, which shows the existence of a solution of the approximated problems (3.2) in the sense of (3.3).

3.1.1. A Priori Estimates

Lemma 3.3. Let f,a and pi,i=1,…,N be restricted as in Theorem 3.1. Then there exist C>0 independent of n, such that

(3.27)∥un∥p→(⋅)≤C.

Proof After choosing φ=un in (3.3), and using (2.3), the fact that fn∈Lp→(⋅)(Ω),un∈W∘1,p→(⋅)(Ω), and Young's inequality, we obtain

(3.28)∑i=1N ∫Ω∣∂iun∣pi(x)dx≤c∑i=1N ∫Ω(∣f n ∣ p i (x)−1∣ u n ∣+∣ u n ∣ p i (x))dx≤c∑i=1N ∫Ω∣un∣pi(x)dx+cC(ε)∑i=1N ∫Ω∣fn∣ pi(x)dx+ε∑i=1N ∫Ω∣un∣ pi(x)dx=c(1+ε)∑i=1N ∫Ω∣un∣pi(x)dx+cC(ε)∑i=1N ∫Ω∣fn∣pi(x)dx≤c(1+ε)N∣Ω∣+∑i=1N ∫Ω∣un∣ p+(x)dx+C′(ε)≤C″(ε).

So, for any fixed choice of ε in (3.28), we obtain

(3.29)∑i=1N∫Ω∣∂iun∣pi(x)dx≤C.

By following the same arguments like in (3.8), we get

(3.30)∑i=1N∫Ω∣∂iun∣pi(x)dx≥1N∑i=1N∥∂iun∥pi(·)p---2N∣Ω∣.

By combining (3.29) and (3.30) with using (2.8), we obtain that

(3.31)C'Np--∥un∥p→(⋅)p--≤C''.

From this, we conclude that, there exist c>0 independent of n, such that

(3.32)∥un∥p→(⋅)≤c.

Therefore, (3.27) has been proven.

Lemma 3.4. There exists a subsequence (still denoted (un)) such that, for all i=1,…,N

(3.33)∂iun⟶∂iua.e. in Ω.

Proof From (3.27) the sequence (un) is bounded in W∘1,p→(⋅)(Ω).

So, there exists a function u∈W∘1,p→(⋅)(Ω) and a subsequence (still denoted by (un)) such that

(3.34)un⇀u weakly in W∘1,p→(⋅)(Ω) and a.e in Ω.

We consider the function

Θn=∑i=1N∫Ω∣∂iun∣pi(x)-2∂iun-∣∂iu∣pi(x)-2∂iu(∂iun-∂iu)dx,

and let's prove that,

(3.35)limn→+∞Θn=0.

We can write Θn in the following form

Θn=∑i=1N ∫Ω∣∂iun∣pi(x)−2∂iun(∂iun−∂iu)dx−∑i=1N ∫Ω∣∂iu∣pi(x)−2∂iu(∂iun−∂iu)dx=In−Jn,

where,

In=∑i=1N ∫Ω∣∂iun∣pi(x)−2∂iun(∂iun−∂iu)dx,Jn=∑i=1N ∫Ω∣∂iu∣pi(x)−2∂iu(∂iun−∂iu)dx.

After choosing φ=un-u in (3.3), with the use of (3.34), and boundedness of (∣fn∣+∣un∣)pi(x)-1 in Lp'i(·) (( p'i(·) is the Sobolev conjugate of pi(·))), we can obtain

(3.36)limn→+∞In=0.

Since (∂iun) is bounded in Lpi(·) (due (3.27)), then there exists a function w∈Lpi(·) and a subsequence (still denoted by (∂iun)) such that

(3.37)∂iun⇀w weakly in Lpi(x).

Through (3.37) and the boundedness of ∣∂iun∣pi(x)-2∂iun in Lp'i(x) we conclude that

(3.38)limn→+∞∑i=1N∫Ω∣∂iun∣pi(x)-2∂iun(∂iun-w)dx=0.

Combining (3.36) and (3.38), we get

(3.39)limn→+∞∑i=1N∫Ω∣∂iun∣pi(x)-2∂iun(∂iu-w)dx=0.

Equation (3.39) implies that w=∂iu, and so

(3.40)∂iun⇀∂iu weakly in Lpi(x).

From (3.40) and the boundedness of ∣∂iu∣pi(x)-2∂iu in Lp'i(x) we conclude that

(3.41)limn→+∞Jn=0.

From (3.36) and (3.41) we get (3.35).

We put for all i=1,…,N

λi,n(x)=∣∂iun∣pi(x)-2∂iun-∣∂iu∣pi(x)-2∂iu(∂iun-∂iu).

Through (2.2) we conclude that, for all i=1,…,N

(3.42)λi,n(x)>0.

Then, (3.42) and (3.35) gives us, for all i=1,…,N

(3.43)λi,n(x)⟶0, strongly in L1(Ω).

Therefore, for a subsequence (still denoted by (un) ), we get for every i=1,…,N

(3.44)λi,n(x)⟶0 a.e. in Ω.

Then there exists a subset Ω0⊂Ω, such that, ∣Ω0∣=0 and for all x∈Ω\Ω0

∣∂iu(x)∣<∞ ,and λi,n(x)⟶0.

From (3.44), we have for some functions k

λi,n(x)≤k(x).

Let us prove that, there exists a function g such that

(3.45)∣∂iun(x)∣≤g(x).

From (2.2), we obtain

(3.46)k(x)≥c(∣∂iun∣−∣∂iu∣)p−−−1,if pi(x)≥2,c′∣∂iun∣−∣∂iu∣1+∣∂iun∣+∣∂iu∣2,if 1<pi(x)<2.

Through (3.46), we obtain (3.45). Now, we proceed by contradiction to prove that

(3.47)∂iun(x)⟶∂iu(x) in Ω\Ω0.

For this reason, we assume that there exists x0∈Ω\Ω0 such that ∂iun(x0) does not converge to ∂iu(x0). The Bolzano Weierstrass theorem implies that

∂iun(x0)⟶b∈R.

By the passage to the limit in λi,n(x0) when n⟶+∞, we obtain

∣b∣pi(x0)-2b-∣∂iu(x0)∣pi(x0)-2∂iu(x0)(b-∂iu(x0))=0.

From (2.2), we get that b=∂iu(x0). Therefore, we find that (3.33) has been proven.

3.2. Proof of the Theorem 3.1

From (3.33) and (3.27), Vitali's theorem gives , for all i=1,…,N

(3.48)∂iun⟶∂iu in L1(Ω) and a.e. in Ω.

So, we have

(3.49)∣∂iun∣pi(x)-2∂iun⟶∣∂iu∣pi(x)-2∂iu​ a.e. in​Ω.

By (3.27) we can get, for all i=1,…,N

(3.50)∫Ω∣∣∂iun∣pi(x)-2∂iun∣pi'(x)dx=∫Ω∣∂iun∣pi(x)dx≤c, pi'(·)=pi(·)pi(·)-1.

Equation (3.50) implies that for all i=1,…,N

(3.51)∣∂iun∣pi(x)-2∂iun uniformly bounded in Lpi'(·)(Ω).

By Young's inequality and since ∂iun∈Lpi(·)(Ω), we get for all ε>0

(3.52)∫Ω∣∣∂iun∣pi(x)−2∂iun∣dx=∫Ω∣∂iun∣pi(x)−1dx≤C(ε)+ε∫Ω∣∂iun∣pi(x)dx≤C(ε)+εc=C′(ε).

For any fixed choice for ε, we conclude that, for all i=1,…,N

(3.53)∣∂iun∣pi(x)-2∂iun∈L1(Ω).

So by (3.53), (3.49), (3.51), and Vitali's theorem, we derive, for all i=1,…,N

(3.54)∣∂iun∣pi(x)-2∂iun⟶∣∂iu∣pi(x)-2∂iu​ strongly in​L1(Ω).

Now from (3.34) we conclude that

(3.55)(∣fn∣+∣un∣)pi(x)-1⟶(∣f∣+∣u∣)pi(x)-1 a.e. in Ω.

On the other hand, by (2.3) and since fn,un∈Lpi(·)(Ω), we obtain for all i=1,…,N

(3.56)∫Ω∣(∣fn∣+∣un∣)pi(x)-1∣pi'(x)dx=∫Ω(∣fn∣+∣un∣)pi(x)dx≤c∫Ω∣fn∣pi(x)+∣un∣pi(x)dx≤C.

Equation (3.56) implies that, for all i=1,…,N

(3.57)(∣fn∣+∣un∣)pi(x)-1 uniformly bounded in Lpi'(·)(Ω).

Like in the proof of (3.53), using the inequality (2.3) and that fn,un∈Lpi(·)(Ω), we can obtain for all i=1,…,N

(3.58)(∣fn∣+∣un∣)pi(x)-1∈L1(Ω).

So by (3.58), (3.55), (3.57), and Vitali's theorem, we derive, for all i=1,…,N

(3.59)(∣fn∣+∣un∣)pi(x)-1⟶(∣f∣+∣u∣)pi(x)-1​ strongly in​L1(Ω).

So we can pass to the limit in (3.3). Thus, we have proven the theorem 3.1.