- Research Article
- Open Access

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

- Giovanni Anello
^{1}Email author

**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 , .

## Keywords

- Weak Solution
- Global Minimum
- Dirichlet Problem
- Standard Result
- Regularity Result

## 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

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