Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq1_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq2_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq3_HTML.gif are suitable functions and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq5_HTML.gif for some https://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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq7_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq8_HTML.gif , denotes the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq9_HTML.gif -Laplacian operator; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq10_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq13_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq14_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq15_HTML.gif and

https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq17_HTML.gif , the fluids are called pseudoplastics; if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq18_HTML.gif Newtonian and if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq20_HTML.gif of parameter https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq23_HTML.gif and https://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), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq25_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq26_HTML.gif with https://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 https://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

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

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

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

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

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

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

(ii) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq40_HTML.gif .

Theorem 1.1.

Consider https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq41_HTML.gif , then there exists one https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq42_HTML.gif such that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq43_HTML.gif there exists at least one https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq44_HTML.gif solution of problem (1.1). Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ4_HTML.gif
(1.4)
for some constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq45_HTML.gif . If additionally
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq46_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq47_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq48_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq49_HTML.gif , then (1.6) becomes
https://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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq50_HTML.gif , then there exists one https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq52_HTML.gif , at least one radially symmetric solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq53_HTML.gif , for some https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq55_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq56_HTML.gif if ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq57_HTML.gif ) hypothesis is replaced by

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

In fact, ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq60_HTML.gif ) implies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq61_HTML.gif ) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq62_HTML.gif , if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq65_HTML.gif are continuous up to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq66_HTML.gif . It is sufficient to know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq67_HTML.gif are continuous. This includes terms https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq68_HTML.gif singular in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq70_HTML.gif for the problem

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

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

Theorem 1.6.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq76_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq77_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq78_HTML.gif , is continuous. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq79_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq80_HTML.gif and additionally
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq81_HTML.gif . Besides this, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq83_HTML.gif satisfies
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq86_HTML.gif is defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq87_HTML.gif and is locally Hölder continuous (with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq88_HTML.gif ) in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq89_HTML.gif . Assume also that there exist functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq90_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ13_HTML.gif
(1.13)
and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq91_HTML.gif is locally Lipschitz continuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq92_HTML.gif on the set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ14_HTML.gif
(1.14)
Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq93_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq94_HTML.gif satisfying
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq96_HTML.gif by

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

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

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

where

https://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

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

and, as a consequence,

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

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

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

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

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

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

(vi) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq108_HTML.gif , given by

https://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),

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

Besides this, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq109_HTML.gif , for each https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq111_HTML.gif satisfies, for each https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq113_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq114_HTML.gif hold. Then, for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq115_HTML.gif ,

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

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

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

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

And, in relation to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq121_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq122_HTML.gif hold. Then, for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq123_HTML.gif ,

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

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

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

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

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

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

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

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

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

(v) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq136_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq137_HTML.gif , where by either ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq138_HTML.gif ) or ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq139_HTML.gif ) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq140_HTML.gif , we have

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq141_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq142_HTML.gif . That is,

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

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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq144_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq145_HTML.gif is given by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq146_HTML.gif where https://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

https://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], https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq148_HTML.gif , for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq149_HTML.gif . In fact, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq150_HTML.gif satisfies

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq151_HTML.gif ,

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq152_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq153_HTML.gif and

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq154_HTML.gif and

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq155_HTML.gif , we obtain

https://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

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

In particular, making https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq157_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq158_HTML.gif and satisfies (1.8), for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq159_HTML.gif . That is, https://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

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

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

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

Thus,

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

Recalling that https://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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq164_HTML.gif , there is one positive constant https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq166_HTML.gif , for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq167_HTML.gif for the auxiliary problem

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

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

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq171_HTML.gif , by ( https://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) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq173_HTML.gif is non-increasing, for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq174_HTML.gif

By items (i)–(iii) above, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq175_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq177_HTML.gif , for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq178_HTML.gif Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq179_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq180_HTML.gif satisfying

https://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

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

where https://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

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

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

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

with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq186_HTML.gif .

Hence, by (3.3),

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

Using ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq188_HTML.gif in (3.2), that

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

So, making https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq189_HTML.gif , after some calculations, we obtain that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq191_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq192_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq194_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq196_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq197_HTML.gif is a suitable function and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq198_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq199_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq200_HTML.gif is such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq201_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq202_HTML.gif .

Let

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

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

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

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

Hence, Lemma 4.1 shows that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq210_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq211_HTML.gif there exists one https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq212_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq213_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq214_HTML.gif satisfying

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

equivalently,

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

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

https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq219_HTML.gif .

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

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

because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq225_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq226_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq227_HTML.gif as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq229_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq230_HTML.gif , we obtain

https://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),

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq231_HTML.gif , we get

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

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

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

So,

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq233_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq234_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq235_HTML.gif . Hence, there are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq236_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq237_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq238_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq239_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq240_HTML.gif for all https://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

https://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

https://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

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

Moreover, making https://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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq243_HTML.gif and setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq244_HTML.gif it follows, by DiBenedetto [23], that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq245_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq246_HTML.gif . Recalling that https://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 https://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

https://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),
https://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
https://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),
https://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 https://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

https://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 ( https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq254_HTML.gif ),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_Equ64_HTML.gif
(A.5)
That is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq255_HTML.gif does not depend on https://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),
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq257_HTML.gif , for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq258_HTML.gif , satisfying

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq259_HTML.gif by

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

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

Lemma A.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq261_HTML.gif , then
https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq262_HTML.gif ), implies ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq263_HTML.gif ) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq264_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq265_HTML.gif .

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

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

If https://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

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

Using the assumption https://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

https://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

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

Similar calculations show that

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

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq271_HTML.gif , set

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

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

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

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq283_HTML.gif , we obtain

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

Again by ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F845946/MediaObjects/13661_2009_Article_883_IEq284_HTML.gif ), we obtain that https://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

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Copyright

© C. A. Santos 2009

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