- Research Article
- Open access
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Entire Solutions for a Quasilinear Problem in the Presence of Sublinear and Super-Linear Terms
Boundary Value Problems volume 2009, Article number: 845946 (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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ2_HTML.gif)
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 [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., [8–20] 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ3_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ4_HTML.gif)
for some constant . If additionally
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ5_HTML.gif)
then there is a positive constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ6_HTML.gif)
Remark 1.2.
If we assume (1.5) with ,
, where
, then (1.6) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ9_HTML.gif)
Remark 1.4.
Theorems 1.1 and 1.3 are still true with if (
) hypothesis is replaced by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq59_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ11_HTML.gif)
then (1.10) admits at least one radially symmetric solution . Besides this,
and
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ13_HTML.gif)
and is locally Lipschitz continuous in
on the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ14_HTML.gif)
Then there exists with
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ16_HTML.gif)
So, for each , let
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ17_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ18_HTML.gif)
It is easy to check that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ19_HTML.gif)
and, as a consequence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ20_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ21_HTML.gif)
is well defined and continuous. Again, by using Lemma 2.1(i) and (ii),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ24_HTML.gif)
So, by Lemma 2.4(v), there exists a such that
. That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ25_HTML.gif)
Let by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ26_HTML.gif)
where ,
is given by
where
is the unique positive and radially symmetric solution of problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ27_HTML.gif)
More specifically, by DiBenedetto [23], , for some
. In fact,
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ28_HTML.gif)
So, by (2.10), (2.11), and (2.13), we have for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ29_HTML.gif)
Hence, after some pattern calculations, we show that there is a such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ30_HTML.gif)
As consequences of (2.9), (2.13) and (2.15), we have and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ31_HTML.gif)
and hence, by Lemma 2.2 (i), (2.7) and , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ32_HTML.gif)
that is, by using (2.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ34_HTML.gif)
So, by definition of ,
and hypothesis (1.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ35_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ36_HTML.gif)
Recalling that and using (1.5) again, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ37_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ39_HTML.gif)
Adapting the arguments of the proof of Theorem 1.3 in [21], we show
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ40_HTML.gif)
where is the positive first eigenfunction of problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ41_HTML.gif)
and , independent of
, is chosen (using (
)) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ42_HTML.gif)
with denoting the first eigenvalue of problem (3.4) associated to the
.
Hence, by (3.3),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ43_HTML.gif)
Using (), (3.3), the above convergence and Lebesgue's theorem, we have, making
in (3.2), that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ45_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ46_HTML.gif)
equivalently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ47_HTML.gif)
Consider extended on
by
. We claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ49_HTML.gif)
because and
with
as
.
So, using Lemma A.1 (see the appendix) with , and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ50_HTML.gif)
and from Lemma 4.1(i),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ51_HTML.gif)
As a consequence of the contradiction hypothesis and the definition of , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ52_HTML.gif)
Recalling that , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ53_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ54_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ55_HTML.gif)
This is impossible again. Thus, we completed the proof of claim (4.4). Setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ56_HTML.gif)
it follows by claim (4.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ57_HTML.gif)
Moreover, making in (4.3), we use Lebesgue's theorem that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ58_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ59_HTML.gif)
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ60_HTML.gif)
So, using (2.2), (2.5), and Lemma 2.1(i) and (ii), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ61_HTML.gif)
Since, by Lemma 2.1(iv),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ62_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ63_HTML.gif)
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 (),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ64_HTML.gif)
That is, does not depend on
. So, by L'Hopital and Lemma 2.2(iv),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ65_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ66_HTML.gif)
and we define the continuous function by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ67_HTML.gif)
So, we have and
Lemma A.1.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ68_HTML.gif)
Finally, we will sketch the proof of claim (), implies (
)
, if
.
Below, will denote several positive constants and
, the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ69_HTML.gif)
If , by estimating the integral in (A.10), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ70_HTML.gif)
Using the assumption in the computation of the first integral above and Jensen's inequality to estimate the last one, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ71_HTML.gif)
Computing the above integral, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ72_HTML.gif)
Similar calculations show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ73_HTML.gif)
So, by (),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ74_HTML.gif)
On the other hand, if , set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ75_HTML.gif)
and note that either for all
or
for some
. In the first case,
for all
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ76_HTML.gif)
So has a finite limit as
, because
. In the second case,
for
and hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ77_HTML.gif)
Integrating by parts and estimating using , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ78_HTML.gif)
Again by (), we obtain that
is a finite number. This shows the claim.
References
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.
Reshetnyak YuG: Index boundedness condition for mappings with bounded distortion. Siberian Mathematical Journal 1968, 9(2):281-285. 10.1007/BF02204791
Uhlenbeck K: Regularity for a class of non-linear elliptic systems. Acta Mathematica 1977, 138(3-4):219-240.
Herrero MA, Vázquez JL: On the propagation properties of a nonlinear degenerate parabolic equation. Communications in Partial Differential Equations 1982, 7(12):1381-1402. 10.1080/03605308208820255
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)90068-4
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/BF02392645
Yang Z, Xu B: Entire bounded solutions for a class of quasilinear elliptic equations. Boundary Value Problems 2007, 2007:-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.1994.1078
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/S0002-9939-1995-1301008-2
Brezis H, Kamin S:Sublinear elliptic equations in
. Manuscripta Mathematica 1992, 74(1):87-106. 10.1007/BF02567660
Brezis H, Oswald L: Remarks on sublinear elliptic equations. Nonlinear Analysis: Theory, Methods & Applications 1986, 10(1):55-64. 10.1016/0362-546X(86)90011-8
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.1996.0218
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.1997.5470
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.
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.2005.09.036
Santos CA: On ground state solutions for singular and semi-linear problems including super-linear terms at infinity. Nonlinear Analysis: Theory, Methods & Applications. In press
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.2003.06.024
Ye D, Zhou F: Invariant criteria for existence of bounded positive solutions. Discrete and Continuous Dynamical Systems. Series A 2005, 12(3):413-424.
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.1997.5655
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.
Goncalves JV, Santos CAP: Positive solutions for a class of quasilinear singular equations. Electronic Journal of Differential Equations 2004, 56: 1-15.
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.2005.09.091
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)90061-5
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This research was supported by FEMAT-DF, DPP-UnB.
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Santos, C. Entire Solutions for a Quasilinear Problem in the Presence of Sublinear and Super-Linear Terms. Bound Value Probl 2009, 845946 (2009). https://doi.org/10.1155/2009/845946
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DOI: https://doi.org/10.1155/2009/845946