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Existence of solutions for quasilinear elliptic systems in divergence form with variable growth
Journal of Inequalities and Applications volume 2014, Article number: 23 (2014)
This paper is concerned with the following Dirichlet problem for a quasilinear elliptic system with variable growth: in Ω, on ∂ Ω, where is a bounded domain. By means of the Young measure and the theory of variable exponent Sobolev spaces, we obtain the existence of solutions in for each .
1 Introduction and main result
In this paper, we consider the Dirichlet problem for the quasilinear elliptic system
where () is a bounded domain. Here, , is Lipschitz continuous and , and σ satisfies the conditions (H1)-(H3) below. In the following, let denote the real vector space of matrices equipped with the inner product (with the usual summation convention).
(H1) (Continuity) is a Carathéodory function, i.e., is measurable for every and is continuous for almost every .
(H2) (Growth and coercivity) There exist , , , and , such that
(H3) (Monotonicity) σ satisfies one of the following conditions:
For any and , is and monotone, i.e.
for any , and .
There exists a function such that and is convex and .
σ is strictly monotone, i.e., σ is monotone and
Our main result is as follows.
Theorem 1.1 If σ satisfies conditions (H1)-(H3), then the Dirichlet problem (1.1) has a weak solution for every .
After Kovác̆ik and Rákosník first discussed and spaces in , a lot of research has been done concerning these kinds of variable exponent spaces, for example, see [2–6] for the properties of such spaces and [7–9] for the applications of variable exponent spaces on partial differential equations. These problems with variable exponent growth possess very complicated nonlinearities, for instance, the -Laplacian operator is inhomogeneous. In recent years, these problems have received considerable attention and raised many difficult mathematical problems. The theory as regards various mathematical problems with -growth conditions has important applications in nonlinear elastic mechanics, imaging processing, electrorheological fluids, and other physics phenomena [10–15].
Condition (H2) states the variable growth and coercivity condition. For the case that is a constant function, Norbert Hungerbühler  studied the problem above. The classical monotone operator methods developed by [17–20] cannot be applied in functions only satisfying the condition in . Inspired by the works mentioned above, we want to extend the result of  to the case that σ satisfies variable growth conditions. To our knowledge, problem (1.1) with variable growth conditions has never been studied by others.
The classical result of Leray and Lions and other typical monotone operator methods require strict monotonicity or monotonicity in the variables (see [17–20] and the references therein). In (H3), it is not required that σ is strict monotone or monotone in the variables as had usually been assumed in previous work. We only require that is monotone. Here is an example: ,
For , . For , set . We can take . If , ; if , . (H1) and (H2) are satisfied. is monotone and , but not strictly monotone. If (H3)(ii) is assumed, this problem cannot be treated by conventional methods since the gradients of the approximating solutions do not necessarily converge pointwise where W is not strictly convex (and thus σ is not strictly monotone). Technically, this can be achieved by considering the Young measure generated by the sequence of gradients of approximating solutions which is inspired by .
This paper is organized as follows: In Section 2, several important properties on variable exponent spaces are presented; in Section 3, we give some conclusions concerned with the Young measure in a variable exponent space; in Section 4, we construct the Galerkin approximation sequence; in Section 5, the proof of Theorem 1.1 is given.
In this section, we first recall some facts on variable exponent spaces and . See [1, 4–6] for details.
Let be the set of all Lebesgue measurable functions , where () is a nonempty open subset. Denote
where . The variable exponent Lebesgue space is the class of all functions u such that for some . is a Banach space endowed with the norm (2.2). Equation (2.1) is called the modular of u in .
For a given , we define the conjugate function as
Lemma 2.1 ()
Let , then the inequality
holds for every , .
In the following of this section, for every , we assume .
Lemma 2.2 ()
For any , we have:
If , then .
If , then .
Lemma 2.3 ()
If , is reflexive, and the dual space of is .
Lemma 2.4 ()
Let , where denotes the Lebesgue measure of Ω, , then the necessary and sufficient condition for is that for almost every , and in this case the embedding is continuous.
Next k is a given positive integer. Given a multi-index , we set and , where is the generalized derivative operator.
The generalized Sobolev space is the class of functions u on Ω such that for every multi-index α with . is a Banach space endowed with the norm
By we denote the subspace of which is the closure of with respect to the norm (2.3).
For any , define
then is an equivalent norm of . If Ω is a bounded domain, is an equivalent norm of .
Lemma 2.5 ()
The spaces and are separable. Furthermore they are reflexive if .
We denote the dual space of by , then we have
Lemma 2.6 ()
Let . Then for every , there exists such that
The norm of is defined as
Lemma 2.7 ()
Let Ω be a domain in with cone property. If is Lipschitz continuous and , is measurable and satisfies for almost every , then there is a continuous embedding .
Lemma 2.8 ()
Let Ω be a domain in with cone property. If is continuous and , then for any measurable function defined in Ω with for almost every , and , there is a continuous compact embedding .
Lemma 2.9 ()
Let , , satisfy the Carathéodory conditions, , . If there exist a nonnegative function and a constant such that
for every and almost every , then the Nemyckii operator , defined by
is continuous and bounded.
3 The Young measure generated by sequences in variable exponent space
Weak convergence is a basic tool of modern nonlinear analysis, because it has the same compactness properties as the convergence in finite dimensional spaces (see ). But this notion does not behave as we desire with respect to nonlinear functionals and operators. The Young measure is a device to overcome these difficulties. For the details we refer to [21–24]. Inspired by these works, we will show our conclusions on Young measures in variable exponent space. In what follows, we denote as the negative part of . First, we recall the definition of Young measures and some lemmas.
Definition 3.1 ()
Assume that the sequence is bounded in . Then there exist a subsequence and a Borel probability measure on for a.e. , such that for each we have
We call the family of Young measure associated with the subsequence .
Lemma 3.2 ()
Let be Lebesgue measurable (not necessarily bounded) and , , be a sequence of Lebesgue measurable functions. Then there exist a subsequence and a family of nonnegative Radon measures on , such that
for almost every .
weakly∗ in for any , where and .
If for any(3.1)
then for almost every , and for any measurable we have weakly in for continuous φ provided the sequence is weakly precompact in .
Lemma 3.2 is the fundamental theorem of the Young measure. A family satisfying (i)-(ii) always exists and is a probability measure if equation (3.1) holds. Lemma 3.2 has useful applications in nonlinear PDE theory. The following lemmas are useful for us.
Lemma 3.3 ()
If and is the Young measure generated by the sequence , then we have in measure if and only if for almost every .
Lemma 3.4 ()
Let . If the sequences and generate the Young measures and , respectively, where and , then generates the Young measure .
Lemma 3.5 ()
Let be a Carathéodory function and be a sequence of measurable functions, where , such that in measure and generates the Young measure . Then
provided that the negative part is equiintegrable.
Theorem 3.6 If the sequence is bounded in , then there is a Young measure generated by satisfying and the weak -limit of is .
Proof It suffices to prove that satisfies equation (3.1) in Lemma 3.2. By Lemma 2.2, there is , for any ,
According to Lemma 3.2(iii), . By Lemma 2.3, is reflexive, then there is a subsequence of (still denoted by ) weakly convergent in . Moreover weakly converges in . By Lemma 3.2(iii), taking φ as the identity mapping I, we have weakly in . □
Theorem 3.7 Let . If in , then the sequence generates the Young measure . Moreover, for almost every , is a probability measure and satisfies .
Proof Since in , is bounded in . By Lemma 2.8,
According to Theorem 3.6 and Lemma 3.3, generates the Young measure and generates the Young measure such that is a probability measure. By Lemma 3.4, the sequence generates the Young measure .
Since in and in , we have
By Theorem 3.6, we can infer that . □
4 Galerkin approximation
Let be a sequence of finite dimensional subspaces with the property that is dense in . We define the operator
where denotes the dual pairing of and , and σ satisfies (H1)-(H3).
Lemma 4.1 For every , the functional is linear and bounded.
Proof It is easy to see that is linear. By the growth condition in (H2) and Lemma 2.7,
By Lemma 2.1 and Lemma 2.2, for each
This implies that is bounded. □
Lemma 4.2 The restriction of J to a finite linear subspace of is continuous.
Proof By the continuity assumption (H1) and the growth condition in (H2),
Thus by Lemma 2.7 and Lemma 2.9, we can get the conclusion. □
Let us fix some k and assume that the dimension of is r and is a basis of . For simplicity, we write . Then we define
Lemma 4.3 G is continuous and
where ⋅ denotes the inner product of two vectors in .
Proof In order to prove that G is continuous, it is sufficient to show that in as in . Let , . Then is equivalent to and is equivalent to . We have
Since J is continuous on finite dimensional subspaces, we can get the continuity of G.
Moreover taking , is equivalent to , thus
as . □
Lemma 4.4 For any , there exists such that
Proof By Lemma 4.3, there exists such that for any we have and the topological argument  shows that has a solution . Hence, for each k there exists such that our conclusion holds. □
5 Proof of Theorem 1.1
In this section, first we give some lemmas for σ satisfying (H1)-(H3). Then we prove Theorem 1.1.
Lemma 5.1 If in , σ satisfies (H1)-(H3) and generates the Young measure , then the following inequality holds:
Furthermore is equiintegrable.
Proof Let us consider the sequence
We will use Lemma 3.5 giving
So we have to establish the equiintegrability of negative part of . We write in the form
To get the equiintegrability of the sequence , we take a measurable subset and by Lemma 2.1
Since is bounded in , by the growth condition in (H2) and Lemma 2.7,
Thus is bounded by Lemma 2.2. is arbitrarily small if the measure of is chosen small enough, and so is by Lemma 2.2. The equiintegrability of the sequence can be obtained also by the boundedness of . We can get the equiintegrability of by observing that
By equation (3.2) we can infer from Lemma 3.5 that equation (5.1) holds.
Next we will prove that . Define and fix . Then, there exists such that for any , or equivalently,
for any . Consequently, for , by Lemma 4.4 we may estimate X as follows:
The term is bounded by the growth condition (H2). On the other hand, by choosing in such a way that for any , the term is bounded by 2ε. Moreover, we have
Since is arbitrary, this proves . We conclude from equation (5.2) that equation (5.1) holds.
At last we get the equiintegrability of from
where is bounded by equation (5.3) and the term is arbitrarily small if the measure of is chosen small enough. □
Lemma 5.2 If equation (5.1) holds, is a probability measure for almost every , and , we find that for almost every
Proof Notice that
We infer from equation (5.1) that
By the monotonicity of σ, the integrand in the above inequality is nonnegative. It follows that for almost every
Lemma 5.3 If in and σ satisfies (H1)-(H3), then for any , we have
Proof By Theorem 3.7, the sequence generates the Young measure and for almost every , is a probability measure such that . The proof will be divided into three cases. In the following, cases (i)-(iii) correspond to the three cases of (H3).
Case (i): We claim that in this case for almost every and for every the following identity holds on :
where ∇ is the derivative with respect to the third variable of σ. Indeed, by the monotonicity of σ we have for each
and, by Lemma 5.2,
Then we get
Equation (5.4) follows from this inequality since the sign of t is arbitrary. Take , where is the matrix whose entry in the i th row and j th column is 1 and others are 0. Then by equation (5.4),
further we can get
Notice that , thus
Since equation (5.3) and Lemma 5.1 imply that the sequence is bounded and equiintegrable, by the Dunford-Pettis criterion and Lemma 3.2 its weak -limit is given by
By Lemma 2.3, the sequence converges weakly in . Hence its weak -limit is also . Then we conclude that
Case (ii): We start by showing that for almost every ,
If , by Lemma 5.2
On the other hand, by monotonicity, for we have
Subtracting equation (5.5) from equation (5.6), we get
for any . By monotonicity,
Since , we have
Then we have
for any , whenever . Now, it follows from equation (5.8) that
Thus we can conclude that , i.e. .
By the convexity of W we have for any . For every , we set , . Since the mapping is continuously differentiable, for every , ,
Hence and we obtain
Now consider the Carathéodory function
The sequence is equiintegrable, so
and the weak limit is
by equations (5.9) and (5.10). Then
Therefore by Vitali’s theorem
Case (iii): By strict monotonicity, it follows from Lemma 5.2 that , thus for almost every . By Lemma 3.3 and in the measure and by equation (3.2) in the measure. After extracting a suitable subsequence if necessary, we can infer that for almost every and for almost every . Then for almost every , moreover we have in the measure. By the equiintegrability of , already discussed above, the Vitali theorem implies
The proof of Lemma 5.3 is completed. □
Proof of Theorem 1.1 It is sufficient to prove that for any there is such that .
From the coercivity assumption in (H2) and Lemma 4.3, it follows that there exists such that whenever . Thus, for the sequence of Galerkin approximations constructed in Lemma 4.4, we have
Then we may extract a subsequence (still denoted by ) such that
For any , since is dense in , there is a sequence such that in as . By Lemma 5.3, we have
as . Lemma 4.4 implies that for all . □
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This work was supported by NSFC-11371110.
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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Fu, Y., Yang, M. Existence of solutions for quasilinear elliptic systems in divergence form with variable growth. J Inequal Appl 2014, 23 (2014). https://doi.org/10.1186/1029-242X-2014-23
- variable exponent
- quasilinear elliptic system
- monotone operator
- Young measure