Existence of solutions for quasilinear elliptic systems in divergence form with variable growth
© Fu and Yang; licensee Springer. 2014
Received: 17 May 2013
Accepted: 17 December 2013
Published: 14 January 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
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 .
- (i)For any and , is and monotone, i.e.
There exists a function such that and is convex and .
- (iii)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.
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.
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 .
Lemma 2.1 ()
holds for every , .
In the following of this section, for every , we assume .
Lemma 2.2 ()
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.
By we denote the subspace of which is the closure of with respect to the norm (2.3).
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 ()
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 ()
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 ()
We call the family of Young measure associated with the subsequence .
Lemma 3.2 ()
for almost every .
weakly∗ in for any , where and .
- (iii)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 ()
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 .
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 .
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 .
By Theorem 3.6, we can infer that . □
4 Galerkin approximation
where denotes the dual pairing of and , and σ satisfies (H1)-(H3).
Lemma 4.1 For every , the functional is linear and bounded.
This implies that is bounded. □
Lemma 4.2 The restriction of J to a finite linear subspace of is continuous.
Thus by Lemma 2.7 and Lemma 2.9, we can get the conclusion. □
where ⋅ denotes the inner product of two vectors in .
Since J is continuous on finite dimensional subspaces, we can get the continuity of G.
as . □
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.
Furthermore is equiintegrable.
where is bounded by equation (5.3) and the term is arbitrarily small if the measure of is chosen small enough. □
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).
Thus we can conclude that , i.e. .
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 .
as . Lemma 4.4 implies that for all . □
This work was supported by NSFC-11371110.
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