Open Access

On a Perturbed Dirichlet Problem for a Nonlocal Differential Equation of Kirchhoff Type

Boundary Value Problems20102011:891430

DOI: 10.1155/2011/891430

Received: 24 May 2010

Accepted: 26 July 2010

Published: 9 August 2010

Abstract

We study the existence of positive solutions to the following nonlocal boundary value problem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq1_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq3_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq6_HTML.gif is a Carathéodory function, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq7_HTML.gif is a positive continuous function, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq8_HTML.gif is a real parameter. Direct variational methods are used. In particular, the proof of the main result is based on a property of the infimum on certain spheres of the energy functional associated to problem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq9_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq11_HTML.gif .

1. Introduction

This paper aims to establish the existence of positive solutions in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq12_HTML.gif to the following problem involving a nonlocal equation of Kirchhoff type:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ1_HTML.gif
(Px3bb)

Here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq13_HTML.gif is an open bounded set in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq14_HTML.gif with smooth boundary https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq17_HTML.gif is a Carathéodory function, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq18_HTML.gif is a positive continuous function, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq19_HTML.gif is a real parameter, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq20_HTML.gif is the standard norm in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq21_HTML.gif . In what follows, for every real number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq22_HTML.gif , we put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq23_HTML.gif .

By a positive solution of ( ), we mean a positive function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq25_HTML.gif which is a solution of ( ) in the weak sense, that is such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ2_HTML.gif
(1.1)
for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq27_HTML.gif . Thus, the weak solutions of ( ) are exactly the positive critical points of the associated energy functional
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ3_HTML.gif
(1.2)

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq29_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq30_HTML.gif ), the equation involved in problem ( ) is the stationary analogue of the well-known equation proposed by Kirchhoff in [1]. This is one of the motivations why problems like ( ) were studied by several authors beginning from the seminal paper of Lions [2]. In particular, among the most recent papers, we cite [37] and refer the reader to the references therein for a more complete overview on this topic.

The case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq33_HTML.gif was considered in [3] and [4], where the existence of at least one positive solution is established under various hypotheses on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq34_HTML.gif . In particular, in [3] the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq35_HTML.gif is supposed to satisfy the well-known Ambrosetti-Rabinowitz growth condition; in [4] https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq36_HTML.gif satisfies certain growth conditions at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq38_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq39_HTML.gif is nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq40_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq41_HTML.gif . Critical point theory and minimax methods are used in [3] and [4]. For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq43_HTML.gif , the existence of a nontrivial solution as well as multiple solutions for problem ( ) is established in [5] and [7] by using critical point theory and invariant sets of descent flow. In these papers, the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq45_HTML.gif is again satisfying suitable growth conditions at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq47_HTML.gif . Finally, in [6], where the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq48_HTML.gif is replaced by a more general https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq49_HTML.gif and the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq50_HTML.gif is multiplied by a positive parameter https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq51_HTML.gif , the existence of at least three solutions for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq52_HTML.gif belonging to a suitable interval depending on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq54_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq55_HTML.gif small enough (with upper bound depending on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq56_HTML.gif ) is established (see [6, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq57_HTML.gif ]). However, we note that the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq58_HTML.gif does not meet the conditions required in [6]. In particular, condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq59_HTML.gif of [6, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq60_HTML.gif ] is not satisfied by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq61_HTML.gif . Moreover, in [6] the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq62_HTML.gif is required to satisfy a subcritical growth at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq63_HTML.gif (and no other condition).

Our aim is to study the existence of positive solution to problem ( ), where, unlike previous existence results (and, in particular, those of the aforementioned papers), no growth condition is required on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq65_HTML.gif . Indeed, we only require that on a certain interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq66_HTML.gif the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq67_HTML.gif is bounded from above by a suitable constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq68_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq69_HTML.gif . Moreover, we also provide a localization of the solution by showing that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq70_HTML.gif we can choose the constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq71_HTML.gif in such way that there exists a solution to ( ) whose distance in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq73_HTML.gif from the unique solution of the unperturbed problem (that is problem ( ) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq75_HTML.gif ) is less than https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq76_HTML.gif .

2. Results

Our first main result gives some conditions in order that the energy functional associated to the unperturbed problem ( ) has a unique global minimum.

Theorem 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq79_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq80_HTML.gif be a continuous function satisfying the following conditions:

( ) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq82_HTML.gif ;

( )the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq84_HTML.gif is strictly monotone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq85_HTML.gif ;

( ) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq87_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq88_HTML.gif .

Then, the functional
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ4_HTML.gif
(2.1)

admits a unique global minimum on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq89_HTML.gif .

Proof.

From condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq90_HTML.gif we find positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq91_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ5_HTML.gif
(2.2)
Therefore, by Sobolev embedding theorems, there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq92_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ6_HTML.gif
(2.3)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq93_HTML.gif , from the previous inequality we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ7_HTML.gif
(2.4)
By standard results, the functional
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ8_HTML.gif
(2.5)
is of class https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq94_HTML.gif and sequentially weakly continuous, and the functional
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ9_HTML.gif
(2.6)

is of class https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq95_HTML.gif and sequentially weakly lower semicontinuous. Then, in view of the coercivity condition (2.4), the functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq96_HTML.gif attains its global minimum on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq97_HTML.gif at some point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq98_HTML.gif .

Now, let us to show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ10_HTML.gif
(2.7)
Indeed, fix a nonzero and nonnegative function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq99_HTML.gif , and put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq100_HTML.gif . We have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ11_HTML.gif
(2.8)

Hence, taking into account that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq101_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq102_HTML.gif small enough, one has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq103_HTML.gif . Thus, inequality (2.7) holds.

At this point, we show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq104_HTML.gif is unique. To this end, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq105_HTML.gif be another global minimum for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq106_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq107_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq108_HTML.gif functional with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ12_HTML.gif
(2.9)
for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq109_HTML.gif , we have that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq110_HTML.gif . Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq111_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq112_HTML.gif are weak solutions of the following nonlocal problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ13_HTML.gif
(2.10)
Moreover, in view of (2.7), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq113_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq114_HTML.gif are nonzero. Therefore, from the Strong Maximum Principle, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq115_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq116_HTML.gif are positive in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq117_HTML.gif as well. Now, it is well known that, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq118_HTML.gif , the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ14_HTML.gif
(2.11)
admits a unique positive solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq119_HTML.gif (see, e.g., [8, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq120_HTML.gif ]). Denote it by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq121_HTML.gif . Then, it is easy to realize that for every couple of positive parameters https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq122_HTML.gif , the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq123_HTML.gif are related by the following identity:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ15_HTML.gif
(2.12)
From (2.12) and condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq124_HTML.gif , we infer that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq126_HTML.gif are related by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ16_HTML.gif
(2.13)
Now, note that the identities
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ17_HTML.gif
(2.14)
lead to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ18_HTML.gif
(2.15)
which, in turn, imply that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ19_HTML.gif
(2.16)
Now, since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq127_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq128_HTML.gif are both global minima for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq129_HTML.gif , one has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq130_HTML.gif . It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ20_HTML.gif
(2.17)
At this point, from condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq131_HTML.gif and (2.17), we infer that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ21_HTML.gif
(2.18)

which, in view of (2.13), clearly implies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq132_HTML.gif . This concludes the proof.

Remark 2.2.

Note that condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq133_HTML.gif is satisfied if, for instance, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq134_HTML.gif is nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq135_HTML.gif and so, in particular, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq136_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq137_HTML.gif .

From now on, whenever the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq138_HTML.gif satisfies the assumption of Theorem 2.1, we denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq139_HTML.gif the unique global minimum of the functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq140_HTML.gif defined in (2.1). Moreover, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq142_HTML.gif , we denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq143_HTML.gif the closed ball in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq144_HTML.gif centered at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq145_HTML.gif with radius https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq146_HTML.gif . The next result shows that the global minimum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq147_HTML.gif is strict in the sense that the infimum of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq148_HTML.gif on every sphere centered in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq149_HTML.gif is strictly greater than https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq150_HTML.gif .

Theorem 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq151_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq152_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq153_HTML.gif be as Theorem 2.1. Then, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq154_HTML.gif one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ22_HTML.gif
(2.19)

Proof.

Put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq155_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq156_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq157_HTML.gif . Assume, by contradiction, that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ23_HTML.gif
(2.20)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ24_HTML.gif
(2.21)
Now, it is easy to check that the functional
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ25_HTML.gif
(2.22)
is sequentially weakly continuous in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq158_HTML.gif . Moreover, by the Eberlein-Smulian Theorem, every closed ball in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq159_HTML.gif is sequentially weakly compact. Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq160_HTML.gif attains its global minimum in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq161_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ26_HTML.gif
(2.23)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq162_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq163_HTML.gif . From assumption https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq164_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq165_HTML.gif turns out to be a strictly increasing function. Therefore, in view of (2.21), one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ27_HTML.gif
(2.24)
This inequality entails that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq166_HTML.gif is a global minimum for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq167_HTML.gif . Thus, thanks to Theorem 2.1, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq168_HTML.gif must be identically https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq169_HTML.gif . Using again the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq170_HTML.gif is strictly increasing, from inequality (2.24), we would get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ28_HTML.gif
(2.25)

which is impossible.

Whenever the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq171_HTML.gif is as in Theorem 2.1, we put
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ29_HTML.gif
(2.26)

for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq172_HTML.gif . Theorem 2.3 says that every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq173_HTML.gif is a positive number.

Before stating our existence result for problem ( ), we have to recall the following well-known Lemma which comes from [9, Theorems https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq175_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq176_HTML.gif ] and the regularity results of [10].

Lemma 2.4.

For every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq177_HTML.gif , denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq178_HTML.gif the (unique) solution of the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ30_HTML.gif
(2.27)
Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq179_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ31_HTML.gif
(2.28)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq180_HTML.gif depends only on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq181_HTML.gif .

Theorem 2.5 below guarantees, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq182_HTML.gif , the existence of at least one positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq183_HTML.gif for problem ( ) whose distance from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq185_HTML.gif is less than https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq186_HTML.gif provided that the perturbation term https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq187_HTML.gif is sufficiently small in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq188_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ32_HTML.gif
(2.29)

Here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq189_HTML.gif is the constant defined in Lemma 2.4 and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq190_HTML.gif . Note that no growth condition is required on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq191_HTML.gif .

Theorem 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq192_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq193_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq194_HTML.gif be as in Theorem 2.3. Moreover, fix any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq195_HTML.gif . Then, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq196_HTML.gif , there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq197_HTML.gif such that for every Carathéodory function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq198_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ33_HTML.gif
(2.30)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq199_HTML.gif is the constant defined in (2.26) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq200_HTML.gif is the embedding constant of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq201_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq202_HTML.gif , problem ( ) admits at least a positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq204_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq205_HTML.gif .

Proof.

Fix https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq206_HTML.gif . For every fixed https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq207_HTML.gif which, without loss of generality, we can suppose such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq208_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq209_HTML.gif be the number defined in (2.30). Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq210_HTML.gif be a Carathéodory function satisfying condition (2.30), and put
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ34_HTML.gif
(2.31)
as well as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ35_HTML.gif
(2.32)
Moreover, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq211_HTML.gif , put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq212_HTML.gif . By standard results, the functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq213_HTML.gif is of class https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq214_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq215_HTML.gif and sequentially weakly continuous. Now, observe that thanks to (2.30), one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ36_HTML.gif
(2.33)
Then, we can fix a number
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ37_HTML.gif
(2.34)
in such way that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ38_HTML.gif
(2.35)
Applying [11, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq216_HTML.gif ] to the restriction of the functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq217_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq218_HTML.gif to the ball https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq219_HTML.gif , it follows that the functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq220_HTML.gif admits a global minimum on the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq221_HTML.gif . Let us denote this latter by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq222_HTML.gif . Note that the particular choice of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq223_HTML.gif forces https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq224_HTML.gif to be in the interior of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq225_HTML.gif . This means that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq226_HTML.gif is actually a local minimum for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq227_HTML.gif , and so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq228_HTML.gif . In other words, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq229_HTML.gif is a weak solution of problem ( ) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq231_HTML.gif in place of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq232_HTML.gif . Moreover, since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq233_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq234_HTML.gif , it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq235_HTML.gif is nonzero. Then, by the Strong Maximum Principle, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq236_HTML.gif is positive in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq237_HTML.gif , and, by [10], https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq238_HTML.gif as well. To finish the proof is now suffice to show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ39_HTML.gif
(2.36)
Arguing by contradiction, assume that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ40_HTML.gif
(2.37)
From Lemma 2.4 and condition (2.30) we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ41_HTML.gif
(2.38)
Therefore, using (2.30) (and recalling the notation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq239_HTML.gif ), one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ42_HTML.gif
(2.39)

that is absurd. The proof is now complete.

Remarks 2.6.

To satisfy assumption (2.30) of Theorem 2.5, it is clearly useful to know some lower estimation of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq240_HTML.gif . First of all, we observe that by standard comparison results, it is easily seen that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ43_HTML.gif
(2.40)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq241_HTML.gif is the unique positive solution of the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ44_HTML.gif
(2.41)
When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq242_HTML.gif is a ball of radius https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq243_HTML.gif centered at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq244_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq245_HTML.gif , and so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq246_HTML.gif . More difficult is obtaining an estimate from below of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq247_HTML.gif : if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq248_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ45_HTML.gif
(2.42)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq249_HTML.gif is the embedding constant of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq250_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq251_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq252_HTML.gif grows as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq253_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq254_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq255_HTML.gif , it seems somewhat hard to find a lower bound for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq256_HTML.gif . However, with regard to this question, it could be interesting to study the behavior of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq257_HTML.gif on varying of the parameter https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq258_HTML.gif for every fixed https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq259_HTML.gif . For instance, how does https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq260_HTML.gif behave as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq261_HTML.gif ? Another question that could be interesting to investigate is finding the exact value of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq262_HTML.gif at least for some particular value of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq263_HTML.gif (for instance https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq264_HTML.gif ) even in the case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_IEq265_HTML.gif .

Authors’ Affiliations

(1)
Department of Mathematics, University of Messina

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© Giovanni Anello. 2011

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.