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

Boundary Value Problems20092009:845946

DOI: 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq1_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq2_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq3_HTML.gif are suitable functions and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq4_HTML.gif are not identically zero continuous functions. We show that there exists at least one solution for the above-mentioned problem for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq5_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq6_HTML.gif . 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://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq7_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq8_HTML.gif , denotes the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq9_HTML.gif -Laplacian operator; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq11_HTML.gif are continuous functions not identically zero and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq12_HTML.gif is a real parameter.

A solution of (1.1) is meant as a positive function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq13_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq14_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq15_HTML.gif and

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ2_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq16_HTML.gif is the characteristic of the medium. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq17_HTML.gif , the fluids are called pseudoplastics; if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq18_HTML.gif Newtonian and if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq19_HTML.gif the fluids are called dilatants.

It follows by the nonnegativity of functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq20_HTML.gif of parameter http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq21_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq22_HTML.gif .

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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq23_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq24_HTML.gif which include the cases considered by them.

The principal theorem in [7] considered, in problem (1.1), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq26_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq27_HTML.gif . Another important fact is that, in our result, we consider different coefficients, while in [7] problem (1.1) was studied with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq28_HTML.gif .

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

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ3_HTML.gif
(1.3)

Additionally, we consider

(H1) (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq30_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq31_HTML.gif

(H2) (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq33_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq34_HTML.gif

Concerning the coefficients http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq36_HTML.gif ,

(H3) (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq38_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq39_HTML.gif

Our results will be established below under the hypothesis http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq40_HTML.gif .

Theorem 1.1.

Consider http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq41_HTML.gif , then there exists one http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq42_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq43_HTML.gif there exists at least one http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq44_HTML.gif solution of problem (1.1). Moreover,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ4_HTML.gif
(1.4)
for some constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq45_HTML.gif . If additionally
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ5_HTML.gif
(1.5)
then there is a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq46_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ6_HTML.gif
(1.6)

Remark 1.2.

If we assume (1.5) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq48_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq49_HTML.gif , then (1.6) becomes
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ7_HTML.gif
(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

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ8_HTML.gif
(1.8)

Our result is.

Theorem 1.3.

Consider http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq50_HTML.gif , then there exists one http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq51_HTML.gif such that problem (1.8) admits, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq52_HTML.gif , at least one radially symmetric solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq53_HTML.gif , for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq54_HTML.gif . Moreover, if in additionally one assumes (1.5), then there is a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq55_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ9_HTML.gif
(1.9)

Remark 1.4.

Theorems 1.1 and 1.3 are still true with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq56_HTML.gif if ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq57_HTML.gif ) hypothesis is replaced by

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq59_HTML.gif

In fact, ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq60_HTML.gif ) implies ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq61_HTML.gif ) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq62_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq63_HTML.gif . (see sketch of the proof in the appendix).

Remark 1.5.

In Theorem 1.3, it is not necessary to assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq65_HTML.gif are continuous up to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq66_HTML.gif . It is sufficient to know that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq67_HTML.gif are continuous. This includes terms http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq68_HTML.gif singular in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq69_HTML.gif .

The next result improves one result of Goncalves and Santos [21] because it guarantees the existence of radially symmetric solutions in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq70_HTML.gif for the problem

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ10_HTML.gif
(1.10)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq71_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq72_HTML.gif are continuous and suitable functions and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq73_HTML.gif is the ball in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq74_HTML.gif centered in the origin with radius http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq75_HTML.gif .

Theorem 1.6.

Assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq76_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq77_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq78_HTML.gif , is continuous. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq79_HTML.gif satisfies ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq80_HTML.gif and additionally
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ11_HTML.gif
(1.11)
then (1.10) admits at least one radially symmetric solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq81_HTML.gif . Besides this, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq82_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq83_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ12_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq84_HTML.gif and a particular upper solution of (1.1) for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq85_HTML.gif .

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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq86_HTML.gif is defined on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq87_HTML.gif and is locally Hölder continuous (with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq88_HTML.gif ) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq89_HTML.gif . Assume also that there exist functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq90_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ13_HTML.gif
(1.13)
and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq91_HTML.gif is locally Lipschitz continuous in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq92_HTML.gif on the set
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ14_HTML.gif
(1.14)
Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq93_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq94_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ15_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq95_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq96_HTML.gif by

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ16_HTML.gif
(2.1)

So, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq97_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq98_HTML.gif given by

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ17_HTML.gif
(2.2)

where

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ18_HTML.gif
(2.3)

It is easy to check that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ19_HTML.gif
(2.4)

and, as a consequence,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ20_HTML.gif
(2.5)

Moreover, it is also easy to verify.

Lemma 2.1.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq99_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq100_HTML.gif hold. Then, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq101_HTML.gif ,

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq102_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq103_HTML.gif ,

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq104_HTML.gif

(iv) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq105_HTML.gif ,

(v) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq106_HTML.gif ,

(vi) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq107_HTML.gif

By Lemma 2.1(iii), (iv), and (2.2), the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq108_HTML.gif , given by

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ21_HTML.gif
(2.6)

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

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ22_HTML.gif
(2.7)

Besides this, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq109_HTML.gif , for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq110_HTML.gif , and using Lemma 2.1, it follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq111_HTML.gif satisfies, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq112_HTML.gif , the following.

Lemma 2.2.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq113_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq114_HTML.gif hold. Then, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq115_HTML.gif ,

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq116_HTML.gif ,

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq117_HTML.gif

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq118_HTML.gif

(iv) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq119_HTML.gif

And, in relation to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq120_HTML.gif , we have the folowing.

Lemma 2.3.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq121_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq122_HTML.gif hold. Then, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq123_HTML.gif ,

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq124_HTML.gif ,

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq125_HTML.gif

Finally, we will define, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq127_HTML.gif , by

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ23_HTML.gif
(2.8)

So, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq128_HTML.gif is a continuous function and we have (see proof in the appendix).

Lemma 2.4.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq129_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq130_HTML.gif hold. Then,

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq131_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq132_HTML.gif

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq133_HTML.gif

(iv) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq134_HTML.gif

(v) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq135_HTML.gif

By Lemma 2.4(ii), there exists a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq136_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq137_HTML.gif , where by either ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq138_HTML.gif ) or ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq139_HTML.gif ) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq140_HTML.gif , we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ24_HTML.gif
(2.9)

So, by Lemma 2.4(v), there exists a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq141_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq142_HTML.gif . That is,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ25_HTML.gif
(2.10)

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq143_HTML.gif by

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ26_HTML.gif
(2.11)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq144_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq145_HTML.gif is given by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq146_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq147_HTML.gif is the unique positive and radially symmetric solution of problem

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ27_HTML.gif
(2.12)

More specifically, by DiBenedetto [23], http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq148_HTML.gif , for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq149_HTML.gif . In fact, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq150_HTML.gif satisfies

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ28_HTML.gif
(2.13)

So, by (2.10), (2.11), and (2.13), we have for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq151_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ29_HTML.gif
(2.14)

Hence, after some pattern calculations, we show that there is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq152_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq153_HTML.gif and

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ30_HTML.gif
(2.15)

As consequences of (2.9), (2.13) and (2.15), we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq154_HTML.gif and

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ31_HTML.gif
(2.16)

and hence, by Lemma 2.2 (i), (2.7) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq155_HTML.gif , we obtain

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ32_HTML.gif
(2.17)

that is, by using (2.2), we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ33_HTML.gif
(2.18)

In particular, making http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq156_HTML.gif , we get from (2.15), Lemma 2.2(i) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq157_HTML.gif that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq158_HTML.gif and satisfies (1.8), for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq159_HTML.gif . That is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq160_HTML.gif is an upper solution to (1.1).

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

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ34_HTML.gif
(2.19)

So, by definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq161_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq162_HTML.gif and hypothesis (1.5), we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ35_HTML.gif
(2.20)

Thus,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ36_HTML.gif
(2.21)

Recalling that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq163_HTML.gif and using (1.5) again, we obtain

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ37_HTML.gif
(2.22)

Thus by (2.9), (2.13), and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq164_HTML.gif , there is one positive constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq165_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq166_HTML.gif , for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq167_HTML.gif for the auxiliary problem

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ38_HTML.gif
(3.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq168_HTML.gif In next, to get a solution for problem (1.10), we will use a limit process in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq169_HTML.gif .

For this purpose, we observe that

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq170_HTML.gif ,

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq171_HTML.gif , by ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq172_HTML.gif ) and by (1.11), it follows that

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq173_HTML.gif is non-increasing, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq174_HTML.gif

By items (i)–(iii) above, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq175_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq176_HTML.gif fulfill the assumptions of Theorem  1.3 in [21]. Thus (3.1) admits one solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq177_HTML.gif , for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq178_HTML.gif Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq179_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq180_HTML.gif satisfying

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ39_HTML.gif
(3.2)

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

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ40_HTML.gif
(3.3)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq181_HTML.gif is the positive first eigenfunction of problem

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ41_HTML.gif
(3.4)

and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq182_HTML.gif , independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq183_HTML.gif , is chosen (using ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq184_HTML.gif )) such that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ42_HTML.gif
(3.5)

with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq185_HTML.gif denoting the first eigenvalue of problem (3.4) associated to the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq186_HTML.gif .

Hence, by (3.3),

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ43_HTML.gif
(3.6)

Using ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq187_HTML.gif ), (3.3), the above convergence and Lebesgue's theorem, we have, making http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq188_HTML.gif in (3.2), that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ44_HTML.gif
(3.7)

So, making http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq189_HTML.gif , after some calculations, we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq190_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq191_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq192_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq193_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq194_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq195_HTML.gif is a classical solution of problem (1.10) (given by Theorem 1.6) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq196_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq197_HTML.gif is a suitable function and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq198_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq199_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq200_HTML.gif is such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq201_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq202_HTML.gif .

Let

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ45_HTML.gif
(4.1)

Thus, it is easy to check the following lemma.

Lemma 4.1.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq203_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq204_HTML.gif hold. Then,

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq205_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq206_HTML.gif is non-increasing,

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq207_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq208_HTML.gif

Hence, Lemma 4.1 shows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq209_HTML.gif fulfills all assumptions of Theorem 1.6. Thus, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq210_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq211_HTML.gif there exists one http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq212_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq213_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq214_HTML.gif satisfying

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ46_HTML.gif
(4.2)

equivalently,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ47_HTML.gif
(4.3)

Consider http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq215_HTML.gif extended on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq216_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq217_HTML.gif . We claim that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ48_HTML.gif
(4.4)

Indeed, first we observe that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq218_HTML.gif satisfies Lemma 4.1(ii). So, with similar arguments to those of [21], we show http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq219_HTML.gif .

To prove http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq220_HTML.gif , first we will prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq221_HTML.gif . In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq222_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq223_HTML.gif , then there is one http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq224_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ49_HTML.gif
(4.5)

because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq225_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq226_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq227_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq228_HTML.gif .

So, using Lemma A.1 (see the appendix) with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq229_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq230_HTML.gif , we obtain

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ50_HTML.gif
(4.6)

and from Lemma 4.1(i),

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ51_HTML.gif
(4.7)

As a consequence of the contradiction hypothesis and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq231_HTML.gif , we get

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ52_HTML.gif
(4.8)

Recalling that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq232_HTML.gif , it follows that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ53_HTML.gif
(4.9)

So,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ54_HTML.gif
(4.10)

However, this is impossible. To end the proof of claim (4.4), we will suppose that there exist an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq233_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq234_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq235_HTML.gif . Hence, there are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq236_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq237_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq238_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq239_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq240_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq241_HTML.gif .

Following the same above arguments, we obtain

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ55_HTML.gif
(4.11)

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

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ56_HTML.gif
(4.12)

it follows by claim (4.4) that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ57_HTML.gif
(4.13)

Moreover, making http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq242_HTML.gif in (4.3), we use Lebesgue's theorem that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ58_HTML.gif
(4.14)

Hence, after some calculations, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq243_HTML.gif and setting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq244_HTML.gif it follows, by DiBenedetto [23], that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq245_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq246_HTML.gif . Recalling that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq247_HTML.gif and using Lemma 4.1(i), it follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq248_HTML.gif is a lower solution of (1.1) with

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ59_HTML.gif
(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),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ60_HTML.gif
(A.1)
So, using (2.2), (2.5), and Lemma 2.1(i) and (ii), we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ61_HTML.gif
(A.2)
Since, by Lemma 2.1(iv),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ62_HTML.gif
(A.3)

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

On the other hand, for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq252_HTML.gif , it follows from Lemma 2.1(vi) that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ63_HTML.gif
(A.4)

where the last equality is obtained by using ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq253_HTML.gif )-(ii). Hence, using (A.2), the proof of Lemma 2.4(iii) is concluded.

Proof.

In this case ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq254_HTML.gif ),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ64_HTML.gif
(A.5)
That is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq255_HTML.gif does not depend on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq256_HTML.gif . So, by L'Hopital and Lemma 2.2(iv),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ65_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq257_HTML.gif , for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq258_HTML.gif , satisfying

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ66_HTML.gif
(A.7)

and we define the continuous function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq259_HTML.gif by

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ67_HTML.gif
(A.8)

So, we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq260_HTML.gif and

Lemma A.1.

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq261_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ68_HTML.gif
(A.9)

Finally, we will sketch the proof of claim ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq262_HTML.gif ), implies ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq263_HTML.gif ) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq264_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq265_HTML.gif .

Below, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq266_HTML.gif will denote several positive constants and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq267_HTML.gif , the function

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ69_HTML.gif
(A.10)

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq268_HTML.gif , by estimating the integral in (A.10), we obtain

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ70_HTML.gif
(A.11)

Using the assumption http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq269_HTML.gif in the computation of the first integral above and Jensen's inequality to estimate the last one, we have

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ71_HTML.gif
(A.12)

Computing the above integral, we obtain

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ72_HTML.gif
(A.13)

Similar calculations show that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ73_HTML.gif
(A.14)

So, by ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq270_HTML.gif ),

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ74_HTML.gif
(A.15)

On the other hand, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq271_HTML.gif , set

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ75_HTML.gif
(A.16)

and note that either http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq272_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq273_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq274_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq275_HTML.gif . In the first case, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq276_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq277_HTML.gif . Hence

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ76_HTML.gif
(A.17)

So http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq278_HTML.gif has a finite limit as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq279_HTML.gif , because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq280_HTML.gif . In the second case, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq281_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq282_HTML.gif and hence,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ77_HTML.gif
(A.18)

Integrating by parts and estimating using http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq283_HTML.gif , we obtain

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ78_HTML.gif
(A.19)

Again by ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq284_HTML.gif ), we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq285_HTML.gif 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.MathSciNet
  2. Reshetnyak YuG: Index boundedness condition for mappings with bounded distortion. Siberian Mathematical Journal 1968, 9(2):281-285. 10.1007/BF02204791MATHMathSciNetView Article
  3. Uhlenbeck K: Regularity for a class of non-linear elliptic systems. Acta Mathematica 1977, 138(3-4):219-240.MATHMathSciNetView Article
  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 Article
  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 Article
  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 Article
  7. Yang Z, Xu B: Entire bounded solutions for a class of quasilinear elliptic equations. Boundary Value Problems 2007, 2007:-8.
  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 Article
  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 Article
  10. Brezis H, Kamin S:Sublinear elliptic equations in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq249_HTML.gif . Manuscripta Mathematica 1992, 74(1):87-106. 10.1007/BF02567660MATHMathSciNetView Article
  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 Article
  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 Article
  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 Article
  14. Goncalves JV, Melo AL, Santos CA:On existence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq250_HTML.gif -ground states for singular elliptic equations in the presence of a strongly nonlinear term. Advanced Nonlinear Studies 2007, 7(3):475-490.MATHMathSciNet
  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 Article
  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 press
  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 Article
  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.MATHMathSciNet
  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 Article
  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 Article
  21. Goncalves JV, Santos CAP: Positive solutions for a class of quasilinear singular equations. Electronic Journal of Differential Equations 2004, 56: 1-15.MathSciNet
  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 Article
  23. DiBenedetto E: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq251_HTML.gif 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 Article

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.