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

- Giovanni Anello
^{1}Email author

**2011**:891430

**DOI: **10.1155/2011/891430

© Giovanni Anello. 2011

**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 in , on , where , is a Carathéodory function, is a positive continuous function, and 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 in , .

## 1. Introduction

Here is an open bounded set in with smooth boundary , , is a Carathéodory function, is a positive continuous function, is a real parameter, and is the standard norm in . In what follows, for every real number , we put .

*weak sense*, that is such that

When ( ), 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 [3–7] and refer the reader to the references therein for a more complete overview on this topic.

The case was considered in [3] and [4], where the existence of at least one positive solution is established under various hypotheses on . In particular, in [3] the nonlinearity is supposed to satisfy the well-known Ambrosetti-Rabinowitz growth condition; in [4] satisfies certain growth conditions at and , and is nondecreasing on for all . Critical point theory and minimax methods are used in [3] and [4]. For and , 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 is again satisfying suitable growth conditions at and . Finally, in [6], where the nonlinearity is replaced by a more general and the nonlinearity is multiplied by a positive parameter , the existence of at least three solutions for all belonging to a suitable interval depending on and and for all small enough (with upper bound depending on ) is established (see [6, Theorem ]). However, we note that the nonlinearity does not meet the conditions required in [6]. In particular, condition of [6, Theorem ] is not satisfied by . Moreover, in [6] the nonlinearity is required to satisfy a subcritical growth at (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 . Indeed, we only require that on a certain interval the function is bounded from above by a suitable constant , uniformly in . Moreover, we also provide a localization of the solution by showing that for all we can choose the constant in such way that there exists a solution to ( ) whose distance in from the unique solution of the unperturbed problem (that is problem ( ) with ) is less than .

## 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 and . Let be a continuous function satisfying the following conditions:

( )the function is strictly monotone in ;

admits a unique global minimum on .

Proof.

is of class and sequentially weakly lower semicontinuous. Then, in view of the coercivity condition (2.4), the functional attains its global minimum on at some point .

Hence, taking into account that , for small enough, one has . Thus, inequality (2.7) holds.

which, in view of (2.13), clearly implies . This concludes the proof.

Remark 2.2.

Note that condition is satisfied if, for instance, is nondecreasing in and so, in particular, if with .

From now on, whenever the function
satisfies the assumption of Theorem 2.1, we denote by
the unique global minimum of the functional
defined in (2.1). Moreover, for every
and
, we denote by
the closed ball in
centered at
with radius
. The next result shows that the global minimum
is *strict* in the sense that the infimum of
on every sphere centered in
is strictly greater than
.

Theorem 2.3.

Proof.

which is impossible.

for every . Theorem 2.3 says that every 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 and ] and the regularity results of [10].

Lemma 2.4.

where the constant depends only on .

Here is the constant defined in Lemma 2.4 and . Note that no growth condition is required on .

Theorem 2.5.

where is the constant defined in (2.26) and is the embedding constant of in , problem ( ) admits at least a positive solution such that .

Proof.

that is absurd. The proof is now complete.

Remarks 2.6.

where is the embedding constant of in . Therefore, grows as at . If , it seems somewhat hard to find a lower bound for . However, with regard to this question, it could be interesting to study the behavior of on varying of the parameter for every fixed . For instance, how does behave as ? Another question that could be interesting to investigate is finding the exact value of at least for some particular value of (for instance ) even in the case of .

## Authors’ Affiliations

## References

- Kirchhoff G:
*Mechanik*. Teubner, Leipzig, Germany; 1883.Google Scholar - Lions J-L: On some questions in boundary value problems of mathematical physics. In
*Contemporary Developments in Continuum Mechanics and Partial Differential Equations, North-Holland Mathematics Studies*.*Volume 30*. Edited by: de la Penha GM, Medeiros LAJ. North-Holland, Amsterdam, The Netherlands; 1978:284–346.View ArticleGoogle Scholar - Alves CO, Corrêa FJSA, Ma TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type.
*Computers & Mathematics with Applications*2005, 49(1):85–93. 10.1016/j.camwa.2005.01.008View ArticleMathSciNetGoogle Scholar - Bensedik A, Bouchekif M: On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity.
*Mathematical and Computer Modelling*2009, 49(5–6):1089–1096. 10.1016/j.mcm.2008.07.032View ArticleMathSciNetGoogle Scholar - Mao A, Zhang Z: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(3):1275–1287. 10.1016/j.na.2008.02.011View ArticleMathSciNetGoogle Scholar - Ricceri B: On an elliptic Kirchhoff-type problem depending on two parameters.
*Journal of Global Optimization*2010, 46(4):543–549. 10.1007/s10898-009-9438-7View ArticleMathSciNetGoogle Scholar - Yang Y, Zhang J: Positive and negative solutions of a class of nonlocal problems.
*Nonlinear Analysis: Theory, Methods & Applications*2010, 122(1):25–30.View ArticleGoogle Scholar - Ambrosetti A, Brezis H, Cerami G: Combined effects of concave and convex nonlinearities in some elliptic problems.
*Journal of Functional Analysis*1994, 122(2):519–543. 10.1006/jfan.1994.1078View ArticleMathSciNetGoogle Scholar - Gilbarg D, Trudinger NS:
*Elliptic Partial Differential Equations of Second Order*.*Volume 22*. Springer, Berlin, Germany; 1977:x+401.View ArticleGoogle Scholar - DiBenedetto E:
local regularity of weak solutions of degenerate elliptic equations.
*Nonlinear Analysis: Theory, Methods & Applications*1983, 7(8):827–850. 10.1016/0362-546X(83)90061-5View ArticleMathSciNetGoogle Scholar - Ricceri B: A general variational principle and some of its applications.
*Journal of Computational and Applied Mathematics*2000, 113(1–2):401–410. 10.1016/S0377-0427(99)00269-1View ArticleMathSciNetGoogle Scholar

## Copyright

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.