First of all, the following modified De Giorgi iteration lemma will be useful (we give a proof in the Appendix).
Lemma 3.1 (Iteration lemma)
Suppose is a nonnegative and nonincreasing function on , it satisfies
(3.1)
for any , and for some constants , , , . Then
where , and .
Next, we prove a crucial a priori bound for via a De Giorgi iteration technique as in [15].
Proposition 3.2 Let and assume that is a nonnegative T-periodic continuous function such that
(3.2)
Then there exists a constant , such that , where R is independent of ϵ and η.
Proof Step 1. Multiplying (3.2) by , with any . Integrating over Ω and noticing that , we have
(3.4)
Since , we deal with the second term on the left-hand side of (3.4) as follows.
(3.5)
Combining (3.4) and (3.5), we have
(3.6)
We estimate the right-hand side of (3.6) by Hölder’s inequality, the embedding theorem and Young’s inequality with ϵ to deduce
(3.7)
Choosing and appropriately, we have from (3.6) and (3.7)
(3.8)
for any , where depends on q, , m, and Ω.
Integrating (3.6) over and using the T-periodicity of , we have
(3.9)
Similarly to (3.7), we obtain
(3.10)
where depends on q, , m, T and Ω. By Poincaré’s inequality, we have
(3.11)
Recall our assumption that , , and thus . Consequently, considering (3.11), we obtain
(3.12)
which implies that there exists a such that
(3.13)
From (3.8) and (3.13), we conclude
(3.14)
for any . In view of the T-periodicity of , (3.14) shows
We finally arrive at
(3.15)
for any , where C depends on q, , m, T and Ω.
Step 2. Let
where is the Lebesgue measure of the set . Multiplying (3.2) by on both sides, where represents the characteristic function of the interval , and integrating over , we have
Let . We assume that the absolutely continuous function attains its maximum at . Take , and θ small enough so that . (In fact, this is always possible because of the periodicity of ; for example, if , we take , then and .) Then we have and
(3.16)
Letting yields
(3.17)
After a direct computation, we obtain an estimate for the left-hand side of (3.17) as follows:
(3.18)
Substituting (3.18) into (3.17), we have
(3.19)
We now deal with (3.19). On one hand, by the embedding theorem
(3.20)
where S is the Sobolev embedding constant, and
On the other hand, from (3.15), where we may fix a special q, using Hölder’s inequality, we obtain
(3.21)
Let . Then (3.19), (3.20), and (3.21) imply
(3.22)
Utilizing Young’s inequality with ϵ, we obtain from (3.22)
Upon choosing ϵ appropriately, one obtains
(3.23)
For any , it is easy to see
(3.24)
The relationships (3.23) and (3.24) above imply that
(3.25)
Noticing that and , by the iteration Lemma 3.1, we obtain and thus , where
with
□
Theorem 3.3 Assume , for a.e. . Then there exists a positive constant R such that
where .
Proof From Proposition 3.2, we take , it implies that there exists a positive constant independent of ϵ and η, such that , for any , . Hence the topological degree is well defined in . Thanks to the homotopy invariance property of the Leray-Schauder degree, we have
(3.26)
Using the fact that , one has
(3.27)
From (3.26) and (3.27), we get . □
Using the standard method, similar to that in [3] or [13], one can prove the following.
Proposition 3.4 Assume that , . If solves , for some and , then for any . Moreover, if , then in .
In what follows, we prove a lower bound for the regularized problem.
Proposition 3.5
Let
be the first eigenvalue of
and let be the associated positive eigenfunction such that . Assume that , and . If satisfies for some , then , where
is the embedding constant of into , and is the Lebesgue measure of the domain Ω.
Proof We argue by contradiction. If not, then for each and , there exists a such that , with . For clarity, we divide the proof into four steps.
Step 1. Note that, by Proposition 3.4, in . Taking and multiplying
(3.28)
by , integrating over and using the T-periodicity of , we obtain
(3.29)
Step 2. Using , we have
Since and , we have . Hence
(3.30)
Thanks to the -Hölder’s inequality in variable exponent space, we have
(3.31)
Noting that and using Hölder’s inequality, we have
(3.32)
Integrating (3.31) over and noting (3.32), we get
(3.33)
Step 3. Multiplying (3.28) by , integrating over , noticing the T-periodicity of and , we deduce
Substituting this inequality into (3.33), we have
(3.34)
Substituting (3.34) into (3.30) and noticing that , we get
(3.35)
Considering that , from (3.29) and (3.35), we have
(3.36)
Step 4. We claim
(3.37)
from which we will derive a contradiction. First, to show (3.37), let in (3.36). Using the fact that and noting and , we get
Now the definition of and (3.37) yield
(3.38)
which is clearly a contradiction to the assumption that . This completes the proof. □
Theorem 3.6 Let be as given in Proposition 3.5. Then for all .
Proof In view of Proposition 3.5, for any fixed , we have proved that for all , . So the Leray-Schauder topological degree is well defined for all . Thanks to the homotopy invariance of the topological degree, we have
(3.39)
Also, from Proposition 3.5, we infer that admits no nontrivial solution in . Moreover, is not a solution of . So . Together with (3.39), we have . □