In this section we use the previous result to prove a bound for the solution of our main problem.
Consider in Ω the differential operator L defined by
and put
Suppose that the leading coefficients of operator L satisfy the assumption (h1) while the lower order coefficients verify the following condition:
where r and p are as in hypothesis (i1). Moreover, assume that the following condition on ρ holds:
where is defined in (2.10). For an example of function ρ whose regularizing function σ satisfy (h3) we can refer to [17].
We introduce now a class of mappings needed in the sequel. Let us fix a function which is equivalent to (for more details on the existence of such an α see, for instance, Theorem 2, Chapter IV in [18] and Lemma 3.6.1 in [19]). Hence, for any we define the functions
where verifies (2.9). It is easy to prove that each belongs to and
where
Remark 4.1 From hypothesis (h1) and Lemma 4.2 in [12] it follows that for any the functions (obtained as extensions of to with zero values out of Ω) belong to and
for t small enough.
Now we are able to prove our main result.
Theorem 4.2 Suppose that conditions (h1), (h2), (h3) hold. Fixing , let u be a solution of the problem
(4.1)
Then there exist an open ball
and a constant
such that
(4.2)
where c depends on n, p, r, ρ, ν, , , (), , , , .
Proof Without loss of generality it can be assumed that . For any , we put
(4.3)
Thus, from the last two conditions of (4.1) and from (2.11), (2.12) and (4.3) it follows that there exists such that . Moreover, taking into account the classical Sobolev embedding theorem (see Theorem 5.4 in [15]), there exists such that for all .
Let , with (which will be suitably chosen later), such that
(4.4)
For simplicity of notation, for each , we denote by the open ball .
Let us set
(4.5)
It is easily seen that
Moreover, for
(4.7)
(4.8)
Consider now the function defined by
(4.9)
Clearly
(4.10)
The first step of the proof is to show that there exists such that, for any , each function is a solution of a problem of type (3.1), where the coefficients of associated differential operator verify the assumptions of Lemma 3.1.
For any , it is easy to prove
(4.11)
Since
(4.12)
and u is a solution of problem (4.1), from (4.11) we deduce
(4.13)
where we have put
(4.14)
(4.15)
(4.16)
We observe that using the hypotheses (h0), (h1), (h2), the equivalence between ρ and σ, and (2.11)-(2.16), we easily get
(4.17)
Using now the estimate (4.13), it is easily seen that
(4.18)
This last inequality can be rewritten as
(4.19)
where we have set
(4.20)
(4.21)
Hence, putting together (4.9) with (4.19) we get
(4.22)
where
(4.23)
Observe that using the hypotheses (h1), (h2), and (4.17), (4.5)-(4.8), it is easy to prove that, for any , the coefficients (for ) and satisfy the first two conditions of assumption (i1) and the function . We show now that, for a suitable choice of the constant , there exists such that for any the coefficients verify also the last condition of (i1). To this aim, we firstly observe that using again hypotheses (h1), (h2), and (2.15), (2.16), from (4.15) we obtain
(4.24)
where depends on , n and s.
Thus, from (4.4), (2.11)-(2.14) and hypothesis (h3) it follows that there exists such that for any we get
(4.25)
Now, for , putting together (4.25) with (4.21) and using the assumption (h1), the properties (4.6)-(4.8) and (4.4), we obtain
(4.26)
Hence, fixing such that
(4.27)
from (4.26) it follows that for each
(4.28)
Putting together (4.28) with (4.25) and observing that in , we deduce that a.e. in Ω. The above considerations together with (4.4), (4.9), (4.10), and (4.22) show that for any the problem
(4.29)
satisfy the assumptions of Lemma 3.1. Therefore, there exists a constant depending on n, p, r, ρ, ν, , , , , , such that
(4.30)
By (4.10), the last bound with becomes
(4.31)
Now, in order to obtain the estimate (4.2), we have to provide a lower bound for the function in terms of the data f. First of all, we observe that, using the definitions (4.14) and (4.16), we can rewrite (4.23) as
(4.32)
On the other hand, by assumption (h1), and by (2.15), (4.7), and (4.4) we easily obtain
(4.33)
where depends on , n, s, ρ, . Thus, using (2.13) and hypothesis (h3) it follows that there exists , with , such that for any
(4.34)
Putting together (4.34) and (4.28) with (4.32) we obtain
(4.35)
Taking into account (4.35), from (4.31) we get
(4.36)
where we have put
(4.37)
and
(4.38)
To end the proof, we give some upper bounds for the functions and (with ). First of all, observe that using (4.7) and Hölder’s inequality in (4.37) we obtain
(4.39)
where depends on the same parameters as . Using now (4.4), the equivalence on ρ and σ, we get
(4.40)
where depends on the same parameters as . If we choose λ such that
(4.41)
from (4.40), for , we get
(4.42)
Arguing similarly we obtain, for each , the following bound on the function :
(4.43)
where depends on the same parameters as and on s. Thus, using again (2.13) and assumption (h3), we see that there exists , with , such that for we get
(4.44)
Finally, chosen , putting together (4.42) and (4.44) with (4.36) and using (4.4), (2.11), and (2.12) it follows that
(4.45)
where depends on the same parameters as and on . Taking into account (4.3) and using again (2.11) and (2.12), from (4.45) we get
(4.46)
where depends on the same parameters as and on .
Finally, if we choose
(4.47)
where is such that , the estimate (4.2) follows from (4.46), (4.47), and Remark 4.1. □