Open Access

Entire Solutions for a Quasilinear Problem in the Presence of Sublinear and Super-Linear Terms

Boundary Value Problems20092009:845946

https://doi.org/10.1155/2009/845946

Received: 31 May 2009

Accepted: 2 October 2009

Published: 13 October 2009

Abstract

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.

1. Introduction

In this paper we establish new results concerning existence and behavior at infinity of solutions for the nonlinear quasilinear problem

(1.1)

where , with , denotes the -Laplacian operator; and are continuous functions not identically zero and is a real parameter.

A solution of (1.1) is meant as a positive function with as and

(1.2)

The class of problems (1.1) appears in many nonlinear phenomena, for instance, in the theory of quasiregular and quasiconformal mappings [13], in the generalized reaction-diffusion theory [4], in the turbulent flow of a gas in porous medium and in the non-Newtonian fluid theory [5]. 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 [6]). So, again of [6], 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 [7] and to complement other works (see, e.g., [820] and references therein) for more general nonlinearities in the terms and which include the cases considered by them.

The principal theorem in [7] considered, in problem (1.1), and with . Another important fact is that, in our result, we consider different coefficients, while in [7] problem (1.1) was studied with .

In order to establish our results some notations will be introduced. We set

(1.3)

Additionally, we consider

(H1) (i)

(ii)

(H2) (i)

(ii)

Concerning the coefficients and ,

(H3) (i)

(ii)

Our results will be established below under the hypothesis .

Theorem 1.1.

Consider , then there exists one such that for each there exists at least one solution of problem (1.1). Moreover,
(1.4)
for some constant . If additionally
(1.5)
then there is a positive constant such that
(1.6)

Remark 1.2.

If we assume (1.5) with , , where , then (1.6) becomes
(1.7)

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

(1.8)

Our result is.

Theorem 1.3.

Consider , then there exists one such that problem (1.8) admits, for each , at least one radially symmetric solution , for some . Moreover, if in additionally one assumes (1.5), then there is a positive constant such that
(1.9)

Remark 1.4.

Theorems 1.1 and 1.3 are still true with if ( ) hypothesis is replaced by

In fact, ( ) implies ( ) , if . (see sketch of the proof in the appendix).

Remark 1.5.

In Theorem 1.3, it is not necessary to assume that and are continuous up to . It is sufficient to know that are continuous. This includes terms singular in .

The next result improves one result of Goncalves and Santos [21] because it guarantees the existence of radially symmetric solutions in for the problem

(1.10)

where , are continuous and suitable functions and is the ball in centered in the origin with radius .

Theorem 1.6.

Assume where , with , is continuous. Suppose that satisfies ( and additionally
(1.11)
then (1.10) admits at least one radially symmetric solution . Besides this, and satisfies
(1.12)

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 [21], 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 [22].

Lemma 1.7.

Suppose that is defined on and is locally Hölder continuous (with ) in . Assume also that there exist functions such that
(1.13)
and is locally Lipschitz continuous in on the set
(1.14)
Then there exists with satisfying
(1.15)

In the two next sections we will prove Theorems 1.3 and 1.6.

2. Proof of Theorem (1.4)

First, inspired by Zhang [20] and Santos [16], we will define functions and by

(2.1)

So, for each , let given by

(2.2)

where

(2.3)

It is easy to check that

(2.4)

and, as a consequence,

(2.5)

Moreover, it is also easy to verify.

Lemma 2.1.

Suppose that and hold. Then, for each ,

(i)

(ii) ,

(iii)

(iv) ,

(v) ,

(vi)

By Lemma 2.1(iii), (iv), and (2.2), the function , given by

(2.6)

is well defined and continuous. Again, by using Lemma 2.1(i) and (ii),

(2.7)

Besides this, , for each , and using Lemma 2.1, it follows that satisfies, for each , the following.

Lemma 2.2.

Suppose that and hold. Then, for each ,

(i) ,

(ii)

(iii)

(iv)

And, in relation to , we have the folowing.

Lemma 2.3.

Suppose that and hold. Then, for each ,

(i) ,

(ii)

Finally, we will define, for each , , by

(2.8)

So, is a continuous function and we have (see proof in the appendix).

Lemma 2.4.

Suppose that and hold. Then,

(i)

(ii)

(iii)

(iv)

(v)

By Lemma 2.4(ii), there exists a such that , where by either ( ) or ( ) , we have

(2.9)

So, by Lemma 2.4(v), there exists a such that . That is,

(2.10)

Let by

(2.11)

where , is given by where is the unique positive and radially symmetric solution of problem

(2.12)

More specifically, by DiBenedetto [23], , for some . In fact, satisfies

(2.13)

So, by (2.10), (2.11), and (2.13), we have for each ,

(2.14)

Hence, after some pattern calculations, we show that there is a such that and

(2.15)

As consequences of (2.9), (2.13) and (2.15), we have and

(2.16)

and hence, by Lemma 2.2 (i), (2.7) and , we obtain

(2.17)

that is, by using (2.2), we have

(2.18)

In particular, making , we get from (2.15), Lemma 2.2(i) and that and satisfies (1.8), for each . That is, is an upper solution to (1.1).

To prove (1.9), first we observe, using Lemma 2.2(i) and (2.15), that

(2.19)

So, by definition of , and hypothesis (1.5), we have

(2.20)

Thus,

(2.21)

Recalling that and using (1.5) again, we obtain

(2.22)

Thus by (2.9), (2.13), and , there is one positive constant such that (1.9) holds. This ends the proof of Theorem 1.3.

3. Proof of Theorem (1.5)

To prove Theorem (1.5), we will first show the existence of a solution, say , for each for the auxiliary problem

(3.1)

where In next, to get a solution for problem (1.10), we will use a limit process in .

For this purpose, we observe that

(i) ,

(ii) , by ( ) and by (1.11), it follows that

(iii) is non-increasing, for each

By items (i)–(iii) above, and fulfill the assumptions of Theorem  1.3 in [21]. Thus (3.1) admits one solution , for each Moreover, with satisfying

(3.2)

Adapting the arguments of the proof of Theorem  1.3 in [21], we show

(3.3)

where is the positive first eigenfunction of problem

(3.4)

and , independent of , is chosen (using ( )) such that

(3.5)

with denoting the first eigenvalue of problem (3.4) associated to the .

Hence, by (3.3),

(3.6)

Using ( ), (3.3), the above convergence and Lebesgue's theorem, we have, making in (3.2), that

(3.7)

So, making , after some calculations, we obtain that . This completes the proof of Theorem 1.6.

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 .

Let

(4.1)

Thus, it is easy to check the following lemma.

Lemma 4.1.

Suppose that and hold. Then,

(i)

(ii) is non-increasing,

(iii) and

Hence, Lemma 4.1 shows that fulfills all assumptions of Theorem 1.6. Thus, for each such that there exists one with and satisfying

(4.2)

equivalently,

(4.3)

Consider extended on by . We claim that

(4.4)

Indeed, first we observe that satisfies Lemma 4.1(ii). So, with similar arguments to those of [21], we show .

To prove , first we will prove that . In fact, if for some , then there is one such that

(4.5)

because and with as .

So, using Lemma A.1 (see the appendix) with , and , we obtain

(4.6)

and from Lemma 4.1(i),

(4.7)

As a consequence of the contradiction hypothesis and the definition of , we get

(4.8)

Recalling that , it follows that

(4.9)

So,

(4.10)

However, this is impossible. To end the proof of claim (4.4), we will suppose that there exist an and such that . Hence, there are with such that , and for all .

Following the same above arguments, we obtain

(4.11)

This is impossible again. Thus, we completed the proof of claim (4.4). Setting

(4.12)

it follows by claim (4.4) that

(4.13)

Moreover, making in (4.3), we use Lebesgue's theorem that

(4.14)

Hence, after some calculations, we obtain and setting it follows, by DiBenedetto [23], that for some . Recalling that and using Lemma 4.1(i), it follows that is a lower solution of (1.1) with

(4.15)

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 [6]. This completes the proof of Theorem 1.1.

Appendix

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.

Proof.

By Lemma 2.2(i),
(A.1)
So, using (2.2), (2.5), and Lemma 2.1(i) and (ii), we get
(A.2)
Since, by Lemma 2.1(iv),
(A.3)

then the claim (i) of Lemma 2.4 follows from (A.2).

On the other hand, for all , it follows from Lemma 2.1(vi) that

(A.4)

where the last equality is obtained by using ( )-(ii). Hence, using (A.2), the proof of Lemma 2.4(iii) is concluded.

Proof.

In this case ( ),
(A.5)
That is, does not depend on . So, by L'Hopital and Lemma 2.2(iv),
(A.6)

This ends the proof of Lemma 2.4.

The next lemma, proved in [21], was used in the proofs of Theorems 1.1 and 1.6. To enunciate it, we will consider , for some , satisfying

(A.7)

and we define the continuous function by

(A.8)

So, we have and

Lemma A.1.

If , then
(A.9)

Finally, we will sketch the proof of claim ( ), implies ( ) , if .

Below, will denote several positive constants and , the function

(A.10)

If , by estimating the integral in (A.10), we obtain

(A.11)

Using the assumption in the computation of the first integral above and Jensen's inequality to estimate the last one, we have

(A.12)

Computing the above integral, we obtain

(A.13)

Similar calculations show that

(A.14)

So, by ( ),

(A.15)

On the other hand, if , set

(A.16)

and note that either for all or for some . In the first case, for all . Hence

(A.17)

So has a finite limit as , because . In the second case, for and hence,

(A.18)

Integrating by parts and estimating using , we obtain

(A.19)

Again by ( ), we obtain that is a finite number. This shows the claim.

Declarations

Acknowledgment

This research was supported by FEMAT-DF, DPP-UnB.

Authors’ Affiliations

(1)
Department of Mathematics, University of Brasília

References

  1. Mikljukov VM: Asymptotic properties of subsolutions of quasilinear equations of elliptic type and mappings with bounded distortion. Matematicheskiĭ Sbornik. Novaya Seriya 1980, 111(1):42-66.MathSciNetGoogle Scholar
  2. Reshetnyak YuG: Index boundedness condition for mappings with bounded distortion. Siberian Mathematical Journal 1968, 9(2):281-285. 10.1007/BF02204791MATHMathSciNetView ArticleGoogle Scholar
  3. Uhlenbeck K: Regularity for a class of non-linear elliptic systems. Acta Mathematica 1977, 138(3-4):219-240.MATHMathSciNetView ArticleGoogle Scholar
  4. Herrero MA, Vázquez JL: On the propagation properties of a nonlinear degenerate parabolic equation. Communications in Partial Differential Equations 1982, 7(12):1381-1402. 10.1080/03605308208820255MATHMathSciNetView ArticleGoogle Scholar
  5. Esteban JR, Vázquez JL: On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Analysis: Theory, Methods & Applications 1986, 10(11):1303-1325. 10.1016/0362-546X(86)90068-4MATHMathSciNetView ArticleGoogle Scholar
  6. Serrin J, Zou H: Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Mathematica 2002, 189(1):79-142. 10.1007/BF02392645MATHMathSciNetView ArticleGoogle Scholar
  7. Yang Z, Xu B: Entire bounded solutions for a class of quasilinear elliptic equations. Boundary Value Problems 2007, 2007:-8.Google Scholar
  8. Ambrosetti A, Brezis H, Cerami G: Combined effects of concave and convex nonlinearities in some elliptic problems. Journal of Functional Analysis 1994, 122(2):519-543. 10.1006/jfan.1994.1078MATHMathSciNetView ArticleGoogle Scholar
  9. Bartsch T, Willem M: On an elliptic equation with concave and convex nonlinearities. Proceedings of the American Mathematical Society 1995, 123(11):3555-3561. 10.1090/S0002-9939-1995-1301008-2MATHMathSciNetView ArticleGoogle Scholar
  10. Brezis H, Kamin S:Sublinear elliptic equations in . Manuscripta Mathematica 1992, 74(1):87-106. 10.1007/BF02567660MATHMathSciNetView ArticleGoogle Scholar
  11. Brezis H, Oswald L: Remarks on sublinear elliptic equations. Nonlinear Analysis: Theory, Methods & Applications 1986, 10(1):55-64. 10.1016/0362-546X(86)90011-8MATHMathSciNetView ArticleGoogle Scholar
  12. Lair AV, Shaker AW: Entire solution of a singular semilinear elliptic problem. Journal of Mathematical Analysis and Applications 1996, 200(2):498-505. 10.1006/jmaa.1996.0218MATHMathSciNetView ArticleGoogle Scholar
  13. Lair AV, Shaker AW: Classical and weak solutions of a singular semilinear elliptic problem. Journal of Mathematical Analysis and Applications 1997, 211(2):371-385. 10.1006/jmaa.1997.5470MATHMathSciNetView ArticleGoogle Scholar
  14. Goncalves JV, Melo AL, Santos CA:On existence of -ground states for singular elliptic equations in the presence of a strongly nonlinear term. Advanced Nonlinear Studies 2007, 7(3):475-490.MATHMathSciNetGoogle Scholar
  15. Goncalves JV, Santos CA: Existence and asymptotic behavior of non-radially symmetric ground states of semilinear singular elliptic equations. Nonlinear Analysis: Theory, Methods & Applications 2006, 65(4):719-727. 10.1016/j.na.2005.09.036MATHMathSciNetView ArticleGoogle Scholar
  16. Santos CA: On ground state solutions for singular and semi-linear problems including super-linear terms at infinity. Nonlinear Analysis: Theory, Methods & Applications. In pressGoogle Scholar
  17. Yang Z: Existence of positive bounded entire solutions for quasilinear elliptic equations. Applied Mathematics and Computation 2004, 156(3):743-754. 10.1016/j.amc.2003.06.024MATHMathSciNetView ArticleGoogle Scholar
  18. Ye D, Zhou F: Invariant criteria for existence of bounded positive solutions. Discrete and Continuous Dynamical Systems. Series A 2005, 12(3):413-424.MATHMathSciNetGoogle Scholar
  19. Zhang Z: A remark on the existence of entire solutions of a singular semilinear elliptic problem. Journal of Mathematical Analysis and Applications 1997, 215(2):579-582. 10.1006/jmaa.1997.5655MATHMathSciNetView ArticleGoogle Scholar
  20. Zhang Z: A remark on the existence of positive entire solutions of a sublinear elliptic problem. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(3-4):727-734, 719–727.MATHView ArticleGoogle Scholar
  21. Goncalves JV, Santos CAP: Positive solutions for a class of quasilinear singular equations. Electronic Journal of Differential Equations 2004, 56: 1-15.MathSciNetGoogle Scholar
  22. Yin H, Yang Z: Some new results on the existence of bounded positive entire solutions for quasilinear elliptic equations. Applied Mathematics and Computation 2006, 177(2):606-613. 10.1016/j.amc.2005.09.091MATHMathSciNetView ArticleGoogle Scholar
  23. DiBenedetto E: local regularity of weak solutions of degenerate elliptic equations. Nonlinear Analysis: Theory, Methods & Applications 1983, 7(8):827-850. 10.1016/0362-546X(83)90061-5MATHMathSciNetView ArticleGoogle Scholar

Copyright

© C. A. Santos 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.