In this section, the existence of weak solutions is studied. Firstly, we introduce some Banach spaces
Lemma 2.1 
, the following relations hold
Lemma 2.2 [15, 16]
, if p satisfies condition
holds, where the positive constant C depends on p and Ω.
Note that the following inequality
does not in general hold.
Lemma 2.3 
Ω be an open domain
(that may be unbounded
) in with cone property
. If is a Lipschitz continuous function satisfying and is measurable and satisfies
then there is a continuous embedding .
The main theorem in this section is the following.
Theorem 2.1 Let
, the exponents
, satisfy conditions
(1.2)-(1.4). Then Problem
(1.1) has at least one weak solution in the class
And one of the following conditions holds:
(A1) , ;
(A2) , .
be an orthogonal basis of
is the subspace generated by the first k
vectors of the basis
. By normalization, we have
. Let us define the operator
For any given integer k
, we consider the approximate solution
here , and , in .
Here we denote by the inner product in .
Problem (1.1) generates the system of k
ordinary differential equations
By the standard theory of the ODE system, we infer that problem (2.2) admits a unique solution
. Then we can obtain an approximate solution
for (1.1), in
. And the solution can be extended to
, for any given
, by the estimate below. Multiplying (2.1)
and summing with respect to i
, we conclude that
By simple calculation, we have
Combining (2.3)-(2.4) and (H1)-(H2), we get
Integrating (2.5) over
, and using assumptions (1.2)-(1.4), we have
where C 1 is a positive constant depending only on , .
Hence, by Lemma 2.1, we also have
In view of (H1)-(H2) and (A1)-(A2), we also have
2 is a positive constant depending only on
. It follows from (2.7) that
Next, multiplying (1.1) by
and then summing with respect to i
, we get that the following holds:
From Lemma 2.2, we have
are embedding constants. From (2.10)-(2.14), we obtain that
Integrating (2.15) over
and using (2.7), Lemma 2.3, we get
where C 3 is a positive constant depending only on .
small enough in (2.16), we obtain the estimate
Hence, by Lemma 2.1, we have
where C 4 is a positive constant depending only on , , l, , T.
From estimate (2.17), we get
By (2.7)-(2.9) and (2.18), we infer that there exist a subsequence
and a function u
Next, we will deal with the nonlinear term. From the Aubin-Lions theorem, see Lions [[18
], pp.57-58], it follows from (2.21) and (2.22) that there exists a subsequence of
, still represented by the same notation, such that
almost everywhere in
. Hence, by (2.19)-(2.22),
Multiplying (2.2) by
is the space of
function with compact support in
) and integrating the obtained result over
, we obtain that
is a basis of
. Convergence (2.19)-(2.24) is sufficient to pass to the limit in (2.25) in order to get
. In view of (2.19)-(2.22) and Lemma 3.3.17 in [19
], we obtain
Hence, we get , . Then, the existence of weak solutions is established. □