Entire Solutions for a Quasilinear Problem in the Presence of Sublinear and Super-Linear Terms
© C. A. Santos 2009
Received: 31 May 2009
Accepted: 2 October 2009
Published: 13 October 2009
We establish new results concerning existence and asymptotic behavior of entire, positive, and bounded solutions which converge to zero at infinite for the quasilinear equation where are suitable functions and are not identically zero continuous functions. We show that there exists at least one solution for the above-mentioned problem for each for some . Penalty arguments, variational principles, lower-upper solutions, and an approximation procedure will be explored.
In this paper we establish new results concerning existence and behavior at infinity of solutions for the nonlinear quasilinear problem
The class of problems (1.1) appears in many nonlinear phenomena, for instance, in the theory of quasiregular and quasiconformal mappings [1–3], in the generalized reaction-diffusion theory , in the turbulent flow of a gas in porous medium and in the non-Newtonian fluid theory . In the non-Newtonian fluid theory, the quantity is the characteristic of the medium. If , the fluids are called pseudoplastics; if Newtonian and if the fluids are called dilatants.
It follows by the nonnegativity of functions of parameter and a strong maximum principle that all non-negative and nontrivial solutions of (1.1) must be strictly positive (see Serrin and Zou ). So, again of , it follows that (1.1) admits one solution if and only if .
The main objective of this paper is to improve the principal result of Yang and Xu  and to complement other works (see, e.g., [8–20] and references therein) for more general nonlinearities in the terms and which include the cases considered by them.
In order to establish our results some notations will be introduced. We set
Additionally, we consider
In the sequel, we will establish some results concerning to quasilinear problems which are relevant in itself and will play a key role in the proof of Theorem 1.1.
We begin with the problem of finding classical solutions for the differential inequality
Our result is.
The next result improves one result of Goncalves and Santos  because it guarantees the existence of radially symmetric solutions in for the problem
The proof of principal theorem (Theorem 1.1) relies mainly on the technics of lower and upper solutions. First, we will prove Theorem 1.3 by defining several auxiliary functions until we get appropriate conditions to define one positive number and a particular upper solution of (1.1) for each .
After this, we will prove Theorem 1.6, motivated by arguments in , which will permit us to get a lower solution for (1.1). Finally, we will obtain a solution of (1.1) applying the lemma below due to Yin and Yang .
In the two next sections we will prove Theorems 1.3 and 1.6.
2. Proof of Theorem (1.4)
It is easy to check that
and, as a consequence,
Moreover, it is also easy to verify.
is well defined and continuous. Again, by using Lemma 2.1(i) and (ii),
More specifically, by DiBenedetto , , for some . In fact, satisfies
that is, by using (2.2), we have
To prove (1.9), first we observe, using Lemma 2.2(i) and (2.15), that
3. Proof of Theorem (1.5)
For this purpose, we observe that
By items (i)–(iii) above, and fulfill the assumptions of Theorem 1.3 in . Thus (3.1) admits one solution , for each Moreover, with satisfying
Adapting the arguments of the proof of Theorem 1.3 in , we show
Hence, by (3.3),
4. Proof of Main Result: Theorem 1.1
To complete the proof of Theorem 1.1, we will first obtain a classical and positive lower solution for problem (1.1), say , such that , where is given by Theorem 1.3. After this, the existence of a solution for the problem (1.1) will be obtained applying Lemma 1.7.
To get a lower solution for (1.1), we will proceed with a limit process in , where is a classical solution of problem (1.10) (given by Theorem 1.6) with , is a suitable function and for and is such that in .
Thus, it is easy to check the following lemma.
Indeed, first we observe that satisfies Lemma 4.1(ii). So, with similar arguments to those of , we show .
and from Lemma 4.1(i),
Following the same above arguments, we obtain
This is impossible again. Thus, we completed the proof of claim (4.4). Setting
it follows by claim (4.4) that
Hence, after some calculations, we obtain and setting it follows, by DiBenedetto , that for some . Recalling that and using Lemma 4.1(i), it follows that is a lower solution of (1.1) with
So, by Lemma 1.7, we conclude that problem (1.1) admits a solution. Besides this, the inequality (1.4) is a consequence of a result in . This completes the proof of Theorem 1.1.
Proof of Lemma 2.4.
The proof of item (iv) is an immediate consequence of Lemma 2.3(i). The item (v) follows by Lemma 2.3(i) and (ii) using Lebesgue's Theorem.
then the claim (i) of Lemma 2.4 follows from (A.2).
This ends the proof of Lemma 2.4.
The next lemma, proved in , was used in the proofs of Theorems 1.1 and 1.6. To enunciate it, we will consider , for some , satisfying
Computing the above integral, we obtain
Similar calculations show that
This research was supported by FEMAT-DF, DPP-UnB.
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