Existence of <i>T</i>-<i>ν</i>-<i>p</i>(<i>x</i>)-Solution of a Nonhomogeneous Elliptic Problem with Right Hand Side Measure

Using the theory of weighted Sobolev spaces with variable exponent and the L ¹-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure. Among the interest of this article is the given of a very important approach to ensure the existence of a weak solution of this type of problem and of generalization to a system with the minimum of conditions.

• Nonhomogeneous Elliptic Equations
• Dirichlet Problems
• Weighted Sobolev Spaces with Variable Exponent
• Minty’s Lemma
• <i>T</i>-<i>ν</i>-<i>p</i>(<i>x</i>)-Solutions

1. Introduction

Consider the nonhomogeneous and nonlinear Dirichlet boundary value problem:

( P )     { − div ( a ( x , u , ∇ u ) ) = μ   in     Ω u = 0   on     ∂ Ω ,

where Ω is a bounded open domain of I R N ( N ≥ 2 ) and

A u = − div ( a ( x , u , ∇ u ) ) is a Leray-Lions operator defined from the weighted Sobolev spaces with variable exponent W 0 1, p ( x ) ( Ω , ν ) into its dual W − 1, p ′ ( x ) ( Ω , ν ∗ ) with ν ∗ = ν 1 − p ′ ( x ) and 1 p ( x ) + 1 p ′ ( x ) = 1 . The datum μ is a measure that admits an L¹-dual composition.

Throughout the paper, we suppose that the exponent p ( ⋅ ) is an element of C + ( Ω ¯ ) = {log-Hölder continuous function p ( ⋅ ) : Ω ¯ → I R such that 1 < p − ≤ p ( x ) ≤ p + < N } (where for all h ∈ C + ( Ω ¯ ) , we denote h + and h − by h + = sup x ∈ Ω h ( x ) and h − = inf x ∈ Ω h ( x ) ) and that ν is a weight function defined on Ω (i.e., ν is a measurable function which is strictly positive a.e. in Ω ) satisfying:

ν ∈ L l o c 1 ( Ω ) , (1.1)

ν − 1 p ( x ) − 1 ∈ L l o c 1 ( Ω ) , (1.2)

ν − s ( x ) ∈ L 1 ( Ω )       for   some     s ( x ) ∈ ( N p ( x ) , ∞ ) ∩ ( 1 p ( x ) − 1 , ∞ ) . (1.3)

The problem ( P ) is studied where the following assumptions are satisfied:

(H₁) a is a Carathéodory function satisfying:

| a ( x , r , ξ ) | ≤ β ν 1 p ( x ) [ b ( x ) + | r | p ( x ) − 1 + ν 1 p ′ ( x ) ( γ ( r ) | ξ | ) p ( x ) − 1 ] (1.4)

[ a ( x , r , ξ ) − a ( x , r , η ) ] ( ξ − η ) ≥ 0   ∀ ξ , η ∈ I R N (1.5)

a ( x , r , ξ ) ξ ≥ α ν | ξ | p ( x ) , (1.6)

where b ( ⋅ ) is a positive function in L p ′ ( x ) ( Ω ) , γ ( r ) is a continuous function and α , β are strictly positive constants.

(H₂) The second member μ is supposed of the form:

μ = f − div   F , (1.7)

where f ∈ L 1 ( Ω ) and F ∈ ( L p ′ ( x ) ( Ω , ν ∗ ) ) N .

A typical example of the problem ( P ) is the following involving the so-called p ( x ) -Laplacian operator with weight:

Δ ν , p ( x ) u = div ( ν ( x ) | ∇ u | p ( x ) − 2 ∇ u ) .

The operator Δ ν , p ( x ) becomes p-Laplacian when p ( x ) ≡ p (a constant) and ν ( x ) ≡ 1 . The p ( x ) -Laplacian operator with weight possesses more complicated nonlinearities than the classical p-Laplacian, for example, it is inhomogeneous with some degeneracy or singularity. For the applied background of p ( x ) -Laplacian, we refer to (see [1] ). The study of differential equations with variable exponents has been a very active field in recent years, we find applications in electro-rheological fluids (see [1] and [2] ) and in image processing (see [3] ).

Under our assumptions (in particular (1.5), the problem ( P ) does not admit, in general, a weak solution since the term a ( x , u , ∇ u ) may not belong to ( L l o c 1 ( Ω ) ) N . To overcome this difficulty we use in this paper the framework of L¹-version of Minty’s lemma (similar to the one used in [4] ). And due to the assumption (1.6) it may be a degenerated or singular problem. Note also that, since the datum is a measure, then the notion of a weak solution cannot be used, hence it is replaced by another approach of solution calling T - ν - p ( x ) -solution (see definition 3.1 below).

Dirichlet problem of type ( P ) was considered in ( [5] [6] ), where in the first work the case of p ( x ) ≡ p (a constant) and ν ( x ) ≡ 1 is treated, while the second work concerns the degenerated case with p ( x ) ≡ p (a constant). Hence our present paper can be seen as a generalization of the two works ( [5] [6] ). We also point out that the existence of solutions for elliptic equations with variable exponents can be found in [7] [8] and [9] and.

This paper is divided into three sections, organized as follows: In Section 2, we introduce and prove some properties of the weighted Sobolev spaces with variable exponent and in Section 3, we prove the existence of T - ν - p ( x ) -solutions of our problem ( P ) . Among the research objectives of this article is to introduce it for applications in physics and also will be a platform for the problem systems of Dirichlet and others.

2. Weighted Sobolev Spaces with Variable Exponent

Let p ∈ C + ( Ω ¯ ) and ν be a weighted function in Ω .

We define the weighted Lebesgue space with variable exponents L p ( x ) ( Ω , ν ) as the set of all measurable functions u : Ω → I R for which the convex weight-modular

ρ ν , p ( x ) ( u ) = ∫ Ω       ν ( x ) | u | p ( x ) d x

is finite. The expression

‖ u ‖ p ( x ) , ν = inf { μ > 0 : ∫ Ω     ν ( x ) | u μ | p ( x ) d x ≤ 1 }

defines a norm in L p ( x ) ( Ω , ν ) , called the Luxemburg norm.

Proposition 2.1. The space ( L p ( x ) ( Ω , ν ) , ‖   .   ‖ p ( x ) , ν ) is a Banach space.

Proof. By considering the operator M ν 1 p ( x ) : L p ( x ) ( Ω , ν ) → L p ( x ) ( Ω ) defined by

M ν 1 p ( x ) ( f ) = f ν 1 p ( x ) ,

for all f ∈ L p ( x ) ( Ω , ν ) , it’s easy to show that M ν 1 p ( x ) is an isomorphism and hence we can deduce.

Remark 2.1. When ν ( x ) ≡ 1 , the weighted Lebesgue spaces with variable exponent L p ( x ) ( Ω , ν ) coincides with the Lebesgue space with variable exponent L p ( x ) ( Ω ) .

The weight-modular ρ ν , p ( x ) coincides with the modular ρ p ( x ) defined on L p ( x ) ( Ω ) by ρ p ( x ) ( u ) : = ∫ Ω | u | p ( x ) d x (for more details see [10] [11] [12] and [13] ).

Lemma 2.1. For all function u ∈ L p ( x ) ( Ω , ν ) , the following assertions are satisfied:

1) ρ ν , p ( x ) ( u ) > 1   ( = 1 ; < 1 ) ⇔ ‖ u ‖ p ( x ) , ν > 1   ( = 1 ; < 1 ) , respectively.

2) If ‖ u ‖ p ( x ) , ν > 1 , then ‖ u ‖ p ( x ) , ν p − ≤ ρ ν , p ( x ) ( u ) ≤ ‖ u ‖ p ( x ) , ν p + .

3) If ‖ u ‖ p ( x ) , ν < 1 , then ‖ u ‖ p ( x ) , ν p + ≤ ρ ν , p ( x ) ( u ) ≤ ‖ u ‖ p ( x ) , ν p − .

Proof. It suffices to remark that ρ ν , p ( x ) ( u ) = ρ p ( x ) ( ν 1 p ( x ) u ) and ‖ ν 1 p ( x ) u ‖ = ‖ u ‖ p ( x ) , ν , and using the analogous result in [13].

Proposition 2.2. Let Ω be a bounded open domain of I R N and ν be a weight function on Ω satifying the integrability condtions (1.1) and (1.2). Then L p ( x ) ( Ω , ν ) ↪ L l o c 1 ( Ω ) .

Proof.

Let K be an included compact on Ω . By vertue of Hölder inequality we have,

∫ K | u | d x = ∫ K | u | ν 1 p ( x ) ν − 1 p ( x ) d x ≤ 2 ‖ | u | ν 1 p ( x ) ‖ L p ( x ) ( K ) ‖ ν − 1 p ( x ) ‖ L p ′ ( x ) ( K ) ≤ 2 ‖ u ‖ p ( x ) , ν ( ∫ K     ν − p ′ ( x ) p ( x ) d x + 1 ) 1 p ′ − ≤ 2 ‖ u ‖ p ( x ) , ν ( ∫ K     ν − 1 p ( x ) − 1 d x + 1 ) 1 p ′ − .

Hence, the conditions (1.1) and (1.2) allow to conclude.

We define the weighted Sobolev space with variable exponents denoted W 1, p ( x ) ( Ω , ν ) , by

W 1, p ( x ) ( Ω , ν ) = { u ∈ L p ( x ) ( Ω ) : ∂ u ∂ x i ∈ L p ( x ) ( Ω , ν ) , i = 1, ⋯ , N } ,

equipped with the norm

‖ u ‖ 1, p ( x ) , ν = ‖ u ‖ p ( x ) + ∑ i = 1 N ‖ ∂ u ∂ x i ‖ p ( x ) , ν

which is equivalent to the Luxemburg norm

| | | u | | | = inf { μ > 0 : ∫ Ω ( | u μ | p ( x ) + ν ( x ) ∑ i = 1 N | ∂ u ∂ x i μ | p ( x ) ) d x ≤ 1 } .

Proposition 2.3. Let ν be a weight function on Ω satisfying the conditions (1.1) and (1.2). Then the space ( W 1, p ( x ) ( Ω , ν ) , ‖   .   ‖ 1, p ( x ) , ν ) is a Banach space.

Proof. Let ( u n ) n be a Cauchy sequence in ( W 1, p ( x ) ( Ω , ν ) , ‖   .   ‖ 1, p ( x ) , ν ) . Then ( u n ) n is a Cauchy sequence in L p ( x ) ( Ω ) and ( ∂ u n ∂ x i ) n is also a Cauchy sequence in L p ( x ) ( Ω , ν ) for all i = 1 , ⋯ , N . By vertue of proposition 2.1, we can deduce that there exist u ∈ L p ( x ) ( Ω ) and v i ∈ L p ( x ) ( Ω , ν ) such that:

u n → u       in     L p ( x ) ( Ω )

and

∂ u n ∂ x i → v i       in     L p ( x ) ( Ω , ν )       for   all     i = 1, ⋯ , N .

Moreover, by using proposition 2.2, we have L p ( x ) ( Ω , ν ) ⊂ L l o c 1 ( Ω ) ⊂ D ′ ( Ω ) . Thus, for all φ ∈ D ( Ω ) one has,

〈 T v i , φ 〉 = lim n → ∞ 〈 T ∂ u n ∂ x i , φ 〉 = − lim n → ∞ 〈 T u n , ∂ φ ∂ x i 〉 = − 〈 T u , ∂ φ ∂ x i 〉 = 〈 T ∂ u ∂ x i , φ 〉 .

Hence T v i = T ∂ u ∂ x i , i.e. v i = ∂ u ∂ x i .

Consequently,

u ∈ W 1, p ( x ) ( Ω , ν )

and

u n → u       in     W 1, p ( x ) ( Ω , ν ) .

Remark 2.2. Since ν satisfies the conditions (1.1) and (1.2), it’s easy to prove that C 0 ∞ ( Ω ) is included in W 1, p ( x ) ( Ω , ν ) ; then we can define the following space

W 0 1, p ( x ) ( Ω , ν ) = C 0 ∞ ( Ω ) ¯ ‖   .   ‖ 1, p ( x ) , ν ,

which is also a Banach space under the norm ‖   .   ‖ 1, p ( x ) , ν .

Proposition 2.4. (Characterization of the dual space).

Let p ( . ) ∈ C + ( Ω ¯ ) and ν be a weight function on Ω satisfying the conditions (1.1) and (1.2). Then for all G ∈ ( W 0 1, p ( x ) ( Ω , ν ) ) ∗ , there exists a unique system of functions ( g 0 , g 1 , ⋯ , g N ) ∈ L p ′ ( x ) ( Ω ) × ( L p ′ ( x ) ( Ω , ν 1 − p ′ ( x ) ) ) N such that,

G ( f ) = ∫ Ω     f ( x ) g 0 ( x ) d x + ∑ i = 1 N     ∫ Ω     ∂ f ∂ x i g i ( x ) d x ,     ∀ f ∈ W 0 1, p ( x ) ( Ω , ν ) .

Proof. The proof of this proposition is similar to that used in [12] (theorem3.16).

Now, let us introduce the function p s defined by

p s ( x ) = p ( x ) s ( x ) s ( x ) + 1 .

We have

p s ( x ) < p ( x )       a .e .   in     Ω

and

{ p s ∗ ( x ) = N p s ( x ) N − p s ( x ) = N p ( x ) s ( x ) N ( s ( x ) + 1 ) − p ( x ) s ( x )       if     p ( x ) s ( x ) < N ( s ( x ) + 1 ) , p s ∗ ( x )   is   arbitrary   ,     otherwise   .

Proposition 2.5. Let p , s ∈ C + ( Ω ¯ ) and ν be a weight function on Ω which satisfies the conditions (1.1), (1.2) and (1.3). Then W 1, p ( x ) ( Ω , ν ) ↪ W 1, p s ( x ) ( Ω ) .

Proof. According to the Hölder inequality and the condition (1.3), one has

∫ Ω | v ( x ) | p s ( x ) d x = ∫ Ω | v ( x ) | p s ( x ) ν p s ( x ) p ( x ) ν − p s ( x ) p ( x ) d x ≤ ( 1 ( p p s ) − + 1 ( s + 1 ) − ) ‖ | v ( x ) | p s ( x ) ν p s ( x ) p ( x ) ‖ p ( x ) p s ( x ) ‖ ν − p s ( x ) p ( x ) ‖ s ( x ) + 1 ≤ ( 1 ( p p s ) − + 1 ( s + 1 ) − ) ( ∫ Ω | v ( x ) | p ( x ) ν ( x ) d x ) 1 γ 1 ( ∫ Ω     ν ( x ) − s ( x ) d x ) 1 γ 1 ¯ ≤ C ( ∫ Ω | v ( x ) | p ( x ) ν ( x ) d x ) 1 γ 1 ( ∫ Ω     ν ( x ) − s ( x ) d x ) 1 γ 1 ¯ ≤ C ( ∫ Ω | v ( x ) | p ( x ) ν ( x ) d x ) 1 γ 1 .

If we take v = ∂ u ∂ x i , we then obtain

∫ Ω | ∂ u ∂ x i | p s ( x ) d x ≤ C ( ∫ Ω | ∂ u ∂ x i | p ( x ) ν ( x ) d x ) 1 γ 1

where

γ 1 = { ( p p s ) −   if       ‖ | ∂ u ∂ x i ( x ) | p s ( x ) ν p s ( x ) p ( x ) ‖ p ( x ) p s ( x ) ≥ 1 , ( p p s ) +   if       ‖ | ∂ u ∂ x i ( x ) | p s ( x ) ν p s ( x ) p ( x ) ‖ p ( x ) p s ( x ) < 1.

Consequently, we can write

‖ ∂ u ∂ x i ( x ) ‖ p s ( x ) γ 2 ≤ C ( ∫ Ω | ∂ u ∂ x i | p ( x ) ν ( x ) d x ) 1 γ 1 ≤ C 0 C 1 ‖ ∂ u ∂ x i ( x ) ‖ p ( x ) , ν γ 3 γ 1

where

γ 2 = { ( p s ) −   if     ‖ ∂ u ∂ x i ( x ) ‖ p s ( x ) ≥ 1 , ( p s ) +   if     ‖ ∂ u ∂ x i ( x ) ‖ p s ( x ) < 1 ,

and

γ 3 = { p +   si       ‖ ∂ u ∂ x i ( x ) ‖ p ( x ) , ν ≥ 1 , p −   si       ‖ ∂ u ∂ x i ( x ) ‖ p ( x ) , ν < 1.

Thus

‖ ∂ u ∂ x i ‖ p s ( x ) ≤ C ‖ ∂ u ∂ x i ‖ p ( x ) , ν γ 3 γ 1 γ 2 ,     i = 1,2, ⋯ , N . (2.1)

Note that C = c ( γ 1 , γ 2 , γ 3 ) denotes some positive constant which may be changing step by step.

Since p s ( x ) < p ( x ) p.p. in Ω , then, there exists a positive constant C such that

‖ u ‖ L p s ( x ) ( Ω ) ≤ C ‖ u ‖ L p ( x ) ( Ω ) .

Thus, we conclude that

W 1, p ( x ) ( Ω , ν ) ↪ W 1, p s ( x ) ( Ω ) .

Corollary 2.1. Let p , s ∈ C + ( Ω ¯ ) and ν be a weight on Ω which satisfies the conditions (1.1), (1.2) and (1.3). Then W 1, p ( x ) ( Ω , ν ) ↪↪ L r ( x ) ( Ω ) , for 1 ≤ r ( x ) < p s ∗ ( x ) .

Corollary 2.2. Let p ∈ C + ( Ω ¯ ) and ν be a weight function on Ω which satisfies the conditions (1.1), (1.2) and (1.3). Then

‖ u ‖ L p ( x ) ( Ω ) ≤ C ‖ ∇ u ‖ L p ( x ) ( Ω ; ν ) ,     ∀ u ∈ C 0 ∞ ( Ω ) .

Proof. Let u ∈ C 0 ∞ ( Ω ) . Since 1 ≤ p ( x ) < p s ∗ ( x ) , we deduce by vertue of the embedding W 1, p s ( x ) ( Ω ) ↪ L p ( x ) ( Ω ) that,

‖ u ‖ L p ( x ) ( Ω ) ≤ C 1 ( ‖ u ‖ L p s ( x ) ( Ω ) + ‖ ∇ u ‖ ( L p s ( x ) ( Ω ) ) N ) .

Thus, in view of the proposition 2.5, we obtain

‖ u ‖ L p ( x ) ( Ω ) ≤ C 2 ‖ ∇ u ‖ L p s ( Ω ) ≤ C 3 ‖ ∇ u ‖ L p ( x ) ( Ω ; ν ) ,

which allows to conclude that

‖ u ‖ L p ( x ) ( Ω ) ≤ C ‖ ∇ u ‖ L p ( x ) ( Ω ; ν ) .

3. Existence Result

Consider the nonhomogeneous nonlinear Dirichlet boundary problem:

( P )     { − div ( a ( x , u , ∇ u ) ) = − div     F     in     Ω u = 0     on     ∂ Ω .

Definition 3.1. A function u is called a T - ν - p ( x ) -solution of problem ( P ) if:

{ u ∈ W 0 1, p ( x ) ( Ω , ν ) , ∫ Ω     a ( x , u , ∇ u ) ∇ T k ( u − φ ) d x = ∫ Ω     f T k ( u − φ ) d x + ∫ Ω     F ∇ T k ( u − φ ) d x ,     ∀ φ ∈ W 0 1, p ( x ) ( Ω , ν ) ∩ L ∞ ( Ω ) .

Theorem 3.1. Let suppose that the assumptions (1.1)-(1.7) are satisfied. Then the problem ( P ) has at least one T - ν - p ( x ) -solution.

Remark 3.1. Note that in the particular case where p ( . ) ≡ p (constant), γ ( r ) = 1 and ν = 1 , the same result is proved in [14] by using the approach of pseudo-monotonicity.

3.1. Approximate Problem

Let ( f n ) n be a sequence of functions in L ∞ ( Ω ) which converges strongly to f in L 1 ( Ω ) such that ‖ f n ‖ L ∞ ( Ω ) ≤ ‖ f ‖ L ∞ ( Ω ) . For n ≥ 1 , we consider the approximate problem of ( P )

( P n )   { u n ∈ W 0 1, p ( x ) ( Ω , ν ) − div ( a ( x , T n ( u n ) , ∇ u n ) ) = f n − div     F     in     Ω .

This section is devoted to establishing the existing solution for the approximate problem ( P n ) .

Theorem 3.2. The operator A k defined by,

A k : W 0 1, p ( x ) ( Ω , ν ) → W − 1, p ′ ( x ) ( Ω , ν ∗ ) u ↦ A k u = − div ( a ( x , T k ( u ) , ∇ u ) )

is bounded, coercive, hemicontinuous and pseudo-monotone.

Proof of Theorem 3.2

● The operator A k is bounded. Indeed for all u , v ∈ W 0 1, p ( x ) ( Ω , ν ) , one has

| 〈 A k u , v 〉 | = | ∫ Ω     a ( x , T k ( u ) , ∇ u ) ∇ v d x | = | ∫ Ω     a ( x , T k ( u ) , ∇ u ) ν − 1 p ( x ) ∇ v ν 1 p ( x ) d x | ≤ ( 1 p − + 1 p ′ − ) ‖ a ( x , T k ( u ) , ∇ u ) ν − 1 p ( x ) ‖ L p ′ ( x ) ( Ω ) ‖ ∇ v ν 1 p ( x ) ‖ L p ( x ) ( Ω ) ≤ 2 ( ∫ Ω | a ( x , T k ( u ) , ∇ u ) ν − 1 p ( x ) | p ′ ( x ) d x ) 1 p ′ − ‖ ∇ v ‖ L p ( x ) ( Ω , ν ) ≤ 2 ( ∫ Ω ( b ( x ) + | T k ( u ) | p ( x ) − 1 + ν 1 p ′ ( x ) ( γ ( T k ( u ) ) | ∇ u | ) p ( x ) − 1 ) p ′ ( x ) d x ) 1 p ′ − ‖ ∇ v ‖ L p ( x ) ( Ω , ν ) ≤ C 1 ( ∫ Ω ( b ( x ) p ′ ( x ) + | T k ( u ) | p ( x ) + ν ( x ) ( γ ( T k ( u ) ) | ∇ u | ) p ( x ) ) d x ) 1 p ′ − ‖ v ‖ W 0 1, p ( x ) ( Ω , ν )

≤ ( C 1 + C 2 + C 3 ( ∫ Ω | T k ( u ) | p ( x ) + ν ( x ) ( γ ( T k ( u ) ) | ∇ u | ) p ( x ) d x ) 1 p ′ − ) ‖ v ‖ W 0 1, p ( x ) ( Ω , ν ) .

Since γ ( . ) is continuous and | T k ( u ) | ≤ k a.e. in Ω , then γ ( T k ( u ) ) | ∇ u | is bounded in W 0 1, p ( x ) ( Ω , ν ) ; hence the operator A k is bounded.

● The operator A k is hemicontinuous. Indeed, let t be a reality that tends to t 0 . We have

a ( x , T k ( u + t v ) , ∇ T k ( u + t v ) ) → a ( x , T k ( u + t 0 v ) , ∇ T k ( u + t 0 v ) ) ,       a .e .   in     Ω .

Since ( a ( x , T k ( u + t v ) , ∇ T k ( u + t v ) ) ) t is bounded in ( L p ′ ( Ω ) ) N , we deduce that A k ( u + t v ) converges to A k ( u + t 0 v ) weakly in W − 1, p ′ ( x ) ( Ω , ν ∗ ) as t tends to t 0 .

● The operator A k is coercive. Indeed, for all u ∈ W 0 1, p ( x ) ( Ω , ν ) , we have

〈 A k u , u 〉 ‖ u ‖ W 0 1, p ( x ) ( Ω , ν ) ≥ ∫ Ω     ν ( x ) | ∇ u | p ( x ) d x ‖ u ‖ W 0 1, p ( x ) ( Ω , ν ) ≥ ‖ u ‖ W 0 1, p ( x ) ( Ω , ν ) δ ‖ u ‖ W 0 1, p ( x ) ( Ω , ν ) ≥ ‖ u ‖ W 0 1, p ( x ) ( Ω , ν ) δ − 1 ,

where

δ = { p −   if     ‖ u ‖ W 0 1, p ( x ) ( Ω , ν ) ≤ 1, p +   if     ‖ u ‖ W 0 1, p ( x ) ( Ω , ν ) > 1,

Obviously, we have ‖ u ‖ W 0 1, p ( x ) ( Ω , ν ) δ − 1 tends to infinity, when ‖ u ‖ W 0 1, p ( x ) ( Ω , ν ) → ∞ , hence we conclude.

● It remains to show that A k is pseudo-monotone: Let ( u j ) j be a sequence in W 0 1, p ( x ) ( Ω , ν ) such that

u j ⇀ u       in     W 0 1, p ( x ) ( Ω , ν )     and     lim sup j 〈 A k u j , u j − u 〉 ≤ 0. (3.1)

Firstly, we prove that A k u j converges to A k u weakly in W − 1, p ′ ( x ) ( Ω , ν ∗ ) . Indeed, since ( u j ) j is a bounded sequence in W 0 1, p ( x ) ( Ω , ν ) , then by the growth condition, ( A k u j ) j is bounded in W − 1, p ′ ( x ) ( Ω , ν ∗ ) , therefore there exists a function h k = ( h k i ) such that,

A k u j ⇀ h k       dans       W − 1, p ′ ( x ) ( Ω , ν ∗ ) , a i ( x , T k ( u j ) , ∇ u j ) ⇀ h k i       in       L p ′ ( x ) ( Ω , ν ∗ ) ,       for     i = 1, ⋯ , N . (3.2)

Hence, we can write

lim sup j 〈 A k u j , u j 〉 ≤ 〈 h k , u 〉 . (3.3)

On the one hand, by (1.5), we have

∑ i = 1 N     ∫ Ω ( a i ( x , T k ( u j ) , ∇ v ) − a i ( x , T k ( u j ) , ∇ u j ) ) ( ∂ v ∂ x i − ∂ u j ∂ x i ) d x ≥ 0, ∀ v ∈ W 0 1, p ( x ) ( Ω , ν ) .

Then

∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u j ) ∂ u j ∂ x i d x ≥ ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u j ) ∂ v ∂ x i d x − ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ v ) ∂ v ∂ x i d x     + ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ v ) ∂ u j ∂ x i d x . (3.4)

Since u j → u strongly in L p ( x ) ( Ω ) and a.e. in Ω , then

a i ( x , T k ( u j ) , ∇ v ) → a i ( x , T k ( u ) , ∇ v )     strongly   in     L p ′ ( x ) ( Ω , ν ∗ )     for     i = 1, ⋯ , N . (3.5)

Therefore,

∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ v ) ∂ v ∂ x i d x → ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u ) , ∇ v ) ∂ v ∂ x i d x (3.6)

and

∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ v ) ∂ u j ∂ x i d x → ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u ) , ∇ v ) ∂ u ∂ x i d x . (3.7)

By vertue of (3.2), we have

∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u j ) ∂ v ∂ x i d x → ∑ i = 1 N     ∫ Ω     h k i ∂ v ∂ x i d x . (3.8)

Now, combining (3.4)-(3.6) and (3.7), we obtain

lim j → ∞ ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u j ) ∂ u j ∂ x i d x ≥ ∑ i = 1 N     ∫ Ω     h k i ∂ v ∂ x i d x + ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u ) , ∇ v ) ∂ u ∂ x i d x     − ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u ) , ∇ v ) ∂ v ∂ x i d x .

Due to (3.3), we deduce that

∑ i = 1 N     ∫ Ω     h k i ∂ u ∂ x i d x ≥ ∑ i = 1 N     ∫ Ω     h k i ∂ v ∂ x i d x + ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u ) , ∇ v ) ∂ u ∂ x i d x     − ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u ) , ∇ v ) ∂ v ∂ x i d x .

This implies that,

∑ i = 1 N     ∫ Ω ( a i ( x , T k ( u j ) , ∇ v ) − h k i ) ( ∂ v ∂ x i − ∂ u j ∂ x i ) d x ≥ 0 ,   ∀ v ∈ W 0 1 , p ( x ) ( Ω , ν ) . (3.9)

On the other hand, choose v = u + t w in (3.9) (with t ∈ ] − 1,1 [ ). It’s easy to see that

∫ Ω ( a ( x , T k ( u ) , ∇ ( u + t w ) ) − h k ) ∇ w d x = 0,   ∀ w ∈ W 0 1, p ( x ) ( Ω , ν ) ,   ∀   t ∈ ] − 1,1 [ .

Hence A k u = h k ∈ W − 1 , p ′ ( x ) ( Ω , ν ∗ ) , and we deduce that A k u j weakly converges to A k u in W − 1 , p ′ ( x ) ( Ω , ν ∗ ) .

Secondly, we prove that 〈 A k u j , u j 〉 → 〈 A k u , u 〉 . Indeed, in view of (3.2) and (3.3), we have

lim sup 〈 A k u j , u j 〉 ≤ 〈 A k u , u 〉 = 〈 h k , u 〉 .

It remains to show that,

lim inf 〈 A k u j , u j 〉 ≥ 〈 A k u , u 〉 = 〈 h k , u 〉 .

For that, we have

〈 A k u j , u j 〉 = ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u j ) ∂ u j ∂ x i d x = ∑ i = 1 N     ∫ Ω ( a i ( x , T k ( u j ) , ∇ u j ) − a i ( x , T k ( u j ) , ∇ u ) ) ( ∂ u j ∂ x i − ∂ u ∂ x i ) d x   + ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u ) ( ∂ u j ∂ x i − ∂ u ∂ x i ) d x   + ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u j ) ∂ u ∂ x i d x .

Since ∑ i = 1 N     ∫ Ω ( a i ( x , T k ( u j ) , ∇ u j ) − a i ( x , T k ( u j ) , ∇ u ) ) ( ∂ u j ∂ x i − ∂ u ∂ x i ) d x ≥ 0 , we deduce that

〈 A k u j , u j 〉 ≥ ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u ) ) ( ∂ u j ∂ x i − ∂ u ∂ x i ) d x   + ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u j ) ∂ u ∂ x i d x .

Therefore,

lim inf 〈 A k u j , u j 〉 ≥ lim inf ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u ) ( ∂ u j ∂ x i − ∂ u ∂ x i ) d x   + lim inf ∑ i = 1 N     ∫ Ω     a i ( x , T k ( u j ) , ∇ u j ) ∂ u ∂ x i d x .

Hence, lim inf 〈 A k u j , u j 〉 ≥ ∑ i = 1 N     ∫ Ω     h i ∂ u ∂ x i d x ≥ 〈 A k u , u 〉 . This achieved the proof.

3.2. Proof of Theorem 3.1

The proof is divided into 4 steps.

Step 1: We will show that ( u n ) n is a Cauchy sequence in measure. Using T k ( u n ) as a test function in ( P n ) leads to,

∫ Ω     a ( x , T k ( u n ) , ∇ u n ) ∇ T k ( u n ) d x = ∫ Ω     f n T k ( u n ) d x + ∫ Ω     F ⋅ ∇ T k ( u n ) d x .

From (1.6) and (1.7), we deduce for all k > 1 that,

α ∑ i = 1 N     ∫ Ω | ∂ T k ( u n ) ∂ x i | p ( x ) ν ( x ) d x ≤ k ‖ f ‖ L 1 + ∑ i = 1 N     ∫ Ω | F i | ν ( x ) − 1 p ( x ) | ∂ T k ( u n ) ∂ x i | ν ( x ) 1 p ( x ) d x ≤ k ‖ f ‖ L 1 + ∑ i = 1 N     ∫ Ω | F i | ν ( x ) − 1 p ( x ) ( α 2 ) − 1 p ( x ) | ∂ T k ( u n ) ∂ x i | ν ( x ) 1 p ( x ) ( α 2 ) 1 p ( x ) d x .

Now, by Young’s inequality, we obtain

α ∑ i = 1 N     ∫ Ω | ∂ T k ( u n ) ∂ x i | p ( x ) ν ( x ) d x ≤ k ‖ f ‖ L 1 + ∑ i = 1 N     ∫ Ω | F i | p ′ ( x ) ν ( x ) − p ′ ( x ) p ( x ) C ( α ) p ′ ( x ) d x + ∑ i = 1 N ∫ Ω | ∂ T k ( u n ) ∂ x i | p ( x ) ν ( x ) α 2 p ( x ) d x ≤ k ‖ f ‖ L 1 + ∑ i = 1 N     ∫ Ω | F i | p ′ ( x ) ν ( x ) − p ′ ( x ) p ( x ) C ( α , p ′ − ) d x + ∑ i = 1 N     ∫ Ω | ∂ T k ( u n ) ∂ x i | p ( x ) ν ( x ) α 2 p − d x . (3.10)

Then, one has

( 1 − 1 2 p − ) α ∑ i = 1 N     ∫ Ω | ∂ T k ( u n ) ∂ x i | p ( x ) ν ( x ) d x ≤ k ‖ f ‖ L 1 + C ( α , p ′ − ) k + ∑ i = 1 N     ∫ Ω | F i | p ′ ( x ) ν ( x ) − p ′ ( x ) p ( x ) d x ,

for k ≥ 1 , which implies that

∑ i = 1 N     ∫ Ω | ∂ T k ( u n ) ∂ x i | p ( x ) ν ( x ) d x ≤ C k       for   all     k > 1. (3.11)

Let k > 0 large enough and B R be a ball of Ω . Using (3.11) and applying Hölder’s inequality and Poincaré’s inequality, we obtain

k   m e a s ( { | u n | > k } ∩ B R ) = ∫ { | u n | > k } ∩ B R | T k ( u n ) | d x ≤ ‖ T k ( u n ) ‖ L 1 ( Ω ) ≤ C ‖ T k ( u n ) ‖ L p ( x ) ( Ω )

≤ C ‖ ∇ T k ( u n ) ‖ p ( x ) , ν (by vertue of Corollary 2.2) (3.12)

≤ C ( ∫ Ω     ∑ i = 1 N | ∂ T k ( u n ) ∂ x i | p ( x ) ν ( x ) d x ) 1 κ (by vertue of Lemma 2.1)

≤ C k 1 κ ,

where

κ = { p −   if     ‖ ∇ T k ( u n ) ‖ p ( x ) , ν ≤ 1 , p +   if     ‖ ∇ T k ( u n ) ‖ p ( x ) , ν > 1 ,

which implies that,

m e a s ( { | u n | > k } ∩ B R ) ≤ C k 1 − 1 κ ,     ∀ k > 1. (3.13)

So, we have, for all δ > 0 ,

m e a s ( { | u n − u m | > δ } ∩ B R ) ≤ m e a s ( { | u n | > k } ∩ B R ) + m e a s ( { | u m | > k } ∩ B R )     + m e a s ( { | T k ( u n ) − T k ( u m ) | > δ } ) . (3.14)

Since ( T k ( u n ) ) n is bounded in W 0 1, p ( x ) ( Ω , ν ) , there exists a subsequence, still denoted by T k ( u n ) and a measurable function v k ∈ W 0 1, p ( x ) ( Ω , ν ) such that T k ( u n ) converges to v k weakly in W 0 1, p ( x ) ( Ω , ν ) , strongly in L p ( x ) ( Ω ) and

almost everywhere in Ω . Hence ( T k ( u n ) ) n is a Cauchy sequence in measure in Ω .

Let ε > 0 . Then by (3.13), there exists k ( ε ) > 0 such that,

m e a s ( { | u n − u m | > δ } ∩ B R ) < ε ,     ∀   n ,   m ≥ n 0 ( k ( ε ) , δ , R ) .

This proves that ( u n ) n is a Cauchy sequence in measure in B R , thus converges almost everywhere to some measurable function u. Hence

T k ( u n ) ⇀ T k ( u )     weakly   in     W 0 1, p ( x ) ( Ω , ν ) ,   strongly   in     W p ( x ) ( Ω ) ,       and   a .e .   in     Ω . (3.15)

Step 2: We shall prove that

∫ Ω     a ( x , u n , ∇ φ ) ∇ T k ( u n − φ ) d x ≤ ∫ Ω     f n T k ( u n − φ ) d x + ∫ Ω     F ∇ T k ( u n − φ ) d x ∀ φ ∈ W 0 1, p ( x ) ( Ω , ν ) ∩ L ∞ ( Ω ) . (3.16)

Let φ ∈ W 0 1, p ( x ) ( Ω , ν ) ∩ L ∞ ( Ω ) and let n be large enough ( n ≥ k + ‖ φ ‖ ∞ ). Using the admissible test function T k ( u n − φ ) in ( P n ) leads to

∫ Ω     a ( x , u n , ∇ u n ) ∇ ( T k ( u n − φ ) ) d x = ∫ Ω     f n T k ( u n − φ ) d x + ∫ Ω     F ∇ T k ( u n − φ ) d x , (3.17)

i.e.,

∫ Ω     a ( x , u n , ∇ u n ) ∇ T k ( u n − φ ) d x + ∫ Ω     a ( x , u n , ∇ φ ) ∇ T k ( u n − φ ) d x   − ∫ Ω     a ( x , u n , ∇ φ ) ∇ T k ( u n − φ ) d x = ∫ Ω     f n T k ( u n − φ ) d x + ∫ Ω     F ∇ T k ( u n − φ ) d x , (3.18)

which implies that

∫ Ω ( a ( x , u n , ∇ u n ) − a ( x , u n , ∇ φ ) ) ∇ T k ( u n − φ ) d x   + ∫ Ω     a ( x , u n , ∇ φ ) ∇ T k ( u n − φ ) d x = ∫ Ω     f n T k ( u n − φ ) d x + ∫ Ω     F ∇ T k ( u n − φ ) d x . (3.19)

Thanks to assumption (1.5) and the definition of truncation function, we have

∫ Ω ( a ( x , u n , ∇ u n ) − a ( x , u n , ∇ φ ) ) ∇ T k ( u n − φ ) d x ≥ 0. (3.20)

Combining (3.19) and (3.20), we obtain (3.16).

Step 3: We claim that

∫ Ω     a ( x , u , ∇ φ ) ∇ T k ( u − φ ) d x ≤ ∫ Ω     f T k ( u − φ ) d x + ∫ Ω     F ∇ T k ( u − φ ) d x   ∀ φ ∈ W 0 1, p ( x ) ( Ω , ν ) ∩ L ∞ ( Ω ) . (3.21)

Let M = k + ‖ φ ‖ ∞ . Since T M ( u n ) converges to T M ( u ) weakly in W 0 1, p ( x ) ( Ω , ν ) , then

T k ( u n − φ ) ⇀ T k ( u − φ )     weakly   in     W 0 1, p ( x ) ( Ω , ν ) . (3.22)

Thanks to assumption (1.4), we have

| a ( x , T M ( u n ) , ∇ φ ) | p ′ ( x ) ν p ′ ( x ) p ( x ) ≤ β [ b ( x ) + | T M ( u n ) | p ( x ) − 1 + ν 1 p ′ ( x ) ( γ ( T M ( u n ) ) | ∇ φ | ) p ( x ) − 1 ] p ′ ( x ) ≤ C [ b ( x ) p ′ ( x ) + | T M ( u n ) | p ( x ) + ν ( x ) γ 0 p ( x ) | ∇ φ | p ( x ) ] , (3.23)

where γ 0 = sup { | γ ( s ) | : | s | ≤ k + ‖ φ ‖ ∞ } and C is a positive constant. Since T M ( u n ) converges to T M ( u ) weakly in W 0 1, p ( x ) ( Ω , ν ) , strongly in L p ( x ) ( Ω ) and a.e. in Ω , thus

| a ( x , T M ( u n ) , ∇ φ ) | p ′ ( x ) ν p ′ ( x ) p ( x ) → | a ( x , T M ( u ) , ∇ φ ) | p ′ ( x ) ν p ′ ( x ) p ( x )       a .e   in     Ω

and

C [ b ( x ) p ′ ( x ) + | T M ( u n ) | p ( x ) + ν ( x ) γ 0 p ( x ) | ∇ φ | p ( x ) ] → C [ b ( x ) p ′ ( x ) + | T M ( u ) | p ( x ) + ν ( x ) γ 0 p ( x ) | ∇ φ | p ( x ) ] .

Combining (3.21), (3.22) and using Vitali’s theorem, we obtain

∫ Ω     a ( x , u n , ∇ φ ) ∇ T k ( u n − φ ) d x → ∫ Ω     a ( x , u , ∇ φ ) ∇ T k ( u − φ ) d x . (3.24)

Now, we show that

∫ Ω     f n T k ( u n − φ ) d x → ∫ Ω     f T k ( u − φ ) d x . (3.25)

In the first time, we have f n T k ( u n − φ ) → f T k ( u − φ ) a.e in Ω , | f n T k ( u n − φ ) | ≤ k | f n | and k | f n | → k | f | in L 1 ( Ω ) . In the second time, by using Vitali’s theorem we obtain (3.25).

Since F ∈ ( L p ′ ( x ) ( Ω , ν ∗ ) ) N , one has

∫ Ω     F ∇ T k ( u n − φ ) d x → ∫ Ω     F ∇ T k ( u − φ ) d x . (3.26)

Thanks to (3.24), (3.25) and (3.26), we obtain (3.21).

Step 4: In this step, we introduce the following generalization of Minty’s lemma in weighted Sobolev space with variable exponents W 1, p ( x ) ( Ω , ν ) (which is proved in [15] ).

Lemma 3.1. ( [15] ) Let u be a measurable function such that T k ( u ) ∈ W 0 1, p ( x ) ( Ω , ν ) for every k > 0 . Then the following statements are equivalent:

1) ∫ Ω     a ( x , u , ∇ φ ) ∇ T k ( u − φ ) d x ≤ ∫ Ω     f T k ( u − φ ) d x + ∫ Ω     F ∇ T k ( u − φ ) d x ,

2) ∫ Ω     a ( x , u , ∇ u ) ∇ T k ( u − φ ) d x = ∫ Ω     f T k ( u − φ ) d x + ∫ Ω     F ∇ T k ( u − φ ) d x ,

for every φ ∈ W 0 1, p ( x ) ( Ω , ν ) ∩ L ∞ ( Ω ) and for every k > 0 .

Finally, the result (3.21) and the lemma 3.1 lead to the completion of the proof of theorem 3.1.

4. Conclusion

In this article, we have demonstrated the existence of a solution of a problem with a second measure member and in the space of Sobolev with variable exponent using Minty’s lemma. It is a very important technique in which we use the notions of hemicontinuous and pseudo-monotonic instead of broad or strict monotony.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

1. Ru&#382;i&#269;ka, M. (2000) Electrorheological Fluids: Modeling and Mathematical Theory (Lecture Notes in Mathematics, Vol. 1748). Springer, Berlin.

2. Zhikov V.V. ,et al. (2004)On the Density of Smooth Functions in Sobolev-Orlicz Spaces Zapiski Nauchnykh Seminarov POMI 310, 67-81.

3. Chen, Y., Levine, S. and Rao, M. (2006) Variable Exponent, Linear Growth Functionals in Image Restoration. SIAM Journal on Applied Mathematics, 66, 1383-1406. https://doi.org/10.1137/050624522

4. Boccardo, L. and Orsina. L. (1999) Existence Resultats for Direchlet Problem in L¹ via Minty’s Lemma. Applicable Analysis, 76, 309-313. https://doi.org/10.1080/00036810008840887

5. Boccardo L. ,et al. (2002)A Remark on Some Nonlinear Elliptic Problem Electronic Journal of Differential Equations 8, 47-52.

6. Akdim, Y., Azroul, E. and Rhoudaf, M. (2008) Existence of T-Solution for Degenerated Problem via Minty’s Lemma. Acta Mathematica Sinica, English Series, 24, 431-438. https://doi.org/10.1007/s10114-007-0970-4

7. Akdim, Y., Azroul, E. and Benkirane, A. (2001) Existence Solutions for Quasilinear Degenerated Elliptic Equations. Electronic Journal of Differential Equations, 2001, 1-19.

8. Abdellaoui, B. and Peral, I. (2003) Existence and Nonexistence Results for Quasilinear Elliptic Equations Involving the p-Laplaces with a Critical Potential. Annali di Matematica Pura ed Applicata, 182, 247-270. https://doi.org/10.1007/s10231-002-0064-y

9. Wen, G., Zhang, Y. and Chen, D. (2014) Approximate Solutions to the Discontinuous Riemann-Hilbert Problem of Elliptic Systems of First Order Complex Equations. Applied Mathematics, 5, 1546-1556. https://doi.org/10.4236/am.2014.510148

10. Fan, X.L. and Zhang, Q.H. (2003) Existence for p(x)-Laplacien Direchlet Problem. Nonlinear Analysis, 52, 1843-1852. https://doi.org/10.1016/S0362-546X(02)00150-5

11. Fan, X.L. and Zhao, D. (1998) On the Generalized Orlicz-Sobolev Space W^k,p(x)(Ω). Journal of Gansu Education College, 12, 1-6.

12. Kovácik, O. and Rákosnik, J. (1991) On Spaces L^p(x) and W^k,p(x). Czechoslovak Mathematical Journal, 41, 592-618. https://doi.org/10.21136/CMJ.1991.102493

13. Zhao, D., Qiang, W.J. and Fan, X.L. (1997) On Generalized Orlicz Spaces L^p(x)(Ω). Journal of Gansu Sciences, 9, 1-7.

14. Drabek, P., Kufner, A. and Mustonen, V. (1998) Pseudo-Monotonicity and Degenerated or Singular Elliptic Operators. Bulletin of the Australian Mathematical Society, 58, 213-221. https://doi.org/10.1017/S0004972700032184

15. Azroul, E., Barbara, A., Elekhlifi, M. and Rhoudaf, M. (2012) T-p(x)-Solutions for Nonlinear Elliptic Equations with an L¹-Dual Datum. Applicationes Mathematicae, 39, 339-364. https://doi.org/10.4064/am39-3-8