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On a Perturbed Dirichlet Problem for a Nonlocal Differential Equation of Kirchhoff Type
Boundary Value Problems volume 2011, Article number: 891430 (2011)
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
This paper aims to establish the existence of positive solutions in to the following problem involving a nonlocal equation of Kirchhoff type:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ1_HTML.gif)
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
.
By a positive solution of ( ), we mean a positive function which is a solution of ( ) in the weak sense, that is such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ2_HTML.gif)
for all . Thus, the weak solutions of ( ) are exactly the positive critical points of the associated energy functional
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ3_HTML.gif)
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
;
( ) for some
.
Then, the functional
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ4_HTML.gif)
admits a unique global minimum on .
Proof.
From condition we find positive constants
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ5_HTML.gif)
Therefore, by Sobolev embedding theorems, there exists a positive constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ6_HTML.gif)
Since , from the previous inequality we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ7_HTML.gif)
By standard results, the functional
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ8_HTML.gif)
is of class and sequentially weakly continuous, and the functional
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ9_HTML.gif)
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
.
Now, let us to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ10_HTML.gif)
Indeed, fix a nonzero and nonnegative function , and put
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ11_HTML.gif)
Hence, taking into account that , for
small enough, one has
. Thus, inequality (2.7) holds.
At this point, we show that is unique. To this end, let
be another global minimum for
. Since
is a
functional with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ12_HTML.gif)
for all , we have that
. Thus,
and
are weak solutions of the following nonlocal problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ13_HTML.gif)
Moreover, in view of (2.7), and
are nonzero. Therefore, from the Strong Maximum Principle,
and
are positive in
as well. Now, it is well known that, for every
, the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ14_HTML.gif)
admits a unique positive solution in (see, e.g., [8, Lemma
]). Denote it by
. Then, it is easy to realize that for every couple of positive parameters
, the functions
are related by the following identity:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ15_HTML.gif)
From (2.12) and condition , we infer that
and
are related by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ16_HTML.gif)
Now, note that the identities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ17_HTML.gif)
lead to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ18_HTML.gif)
which, in turn, imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ19_HTML.gif)
Now, since and
are both global minima for
, one has
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ20_HTML.gif)
At this point, from condition and (2.17), we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ21_HTML.gif)
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.
Let ,
, and
be as Theorem 2.1. Then, for every
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ22_HTML.gif)
Proof.
Put for every
, and let
. Assume, by contradiction, that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ23_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ24_HTML.gif)
Now, it is easy to check that the functional
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ25_HTML.gif)
is sequentially weakly continuous in . Moreover, by the Eberlein-Smulian Theorem, every closed ball in
is sequentially weakly compact. Consequently,
attains its global minimum in
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ26_HTML.gif)
Let be such that
. From assumption
,
turns out to be a strictly increasing function. Therefore, in view of (2.21), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ27_HTML.gif)
This inequality entails that is a global minimum for
. Thus, thanks to Theorem 2.1,
must be identically
. Using again the fact that
is strictly increasing, from inequality (2.24), we would get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ28_HTML.gif)
which is impossible.
Whenever the function is as in Theorem 2.1, we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ29_HTML.gif)
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.
For every , denote by
the (unique) solution of the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ30_HTML.gif)
Then, , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ31_HTML.gif)
where the constant depends only on
.
Theorem 2.5 below guarantees, for every , the existence of at least one positive solution
for problem ( ) whose distance from
is less than
provided that the perturbation term
is sufficiently small in
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ32_HTML.gif)
Here is the constant defined in Lemma 2.4 and
. Note that no growth condition is required on
.
Theorem 2.5.
Let ,
, and
be as in Theorem 2.3. Moreover, fix any
. Then, for every
, there exists a positive constant
such that for every Carathéodory function
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ33_HTML.gif)
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.
Fix . For every fixed
which, without loss of generality, we can suppose such that
, let
be the number defined in (2.30). Let
be a Carathéodory function satisfying condition (2.30), and put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ34_HTML.gif)
as well as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ35_HTML.gif)
Moreover, for every , put
. By standard results, the functional
is of class
in
and sequentially weakly continuous. Now, observe that thanks to (2.30), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ36_HTML.gif)
Then, we can fix a number
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ37_HTML.gif)
in such way that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ38_HTML.gif)
Applying [11, Theorem ] to the restriction of the functionals
and
to the ball
, it follows that the functional
admits a global minimum on the set
. Let us denote this latter by
. Note that the particular choice of
forces
to be in the interior of
. This means that
is actually a local minimum for
, and so
. In other words,
is a weak solution of problem ( ) with
in place of
. Moreover, since
and
, it follows that
is nonzero. Then, by the Strong Maximum Principle,
is positive in
, and, by [10],
as well. To finish the proof is now suffice to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ39_HTML.gif)
Arguing by contradiction, assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ40_HTML.gif)
From Lemma 2.4 and condition (2.30) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ41_HTML.gif)
Therefore, using (2.30) (and recalling the notation ), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ42_HTML.gif)
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 . First of all, we observe that by standard comparison results, it is easily seen that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ43_HTML.gif)
where is the unique positive solution of the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ44_HTML.gif)
When is a ball of radius
centered at
, then
, and so
. More difficult is obtaining an estimate from below of
: if
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F891430/MediaObjects/13661_2010_Article_64_Equ45_HTML.gif)
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
.
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Anello, G. On a Perturbed Dirichlet Problem for a Nonlocal Differential Equation of Kirchhoff Type. Bound Value Probl 2011, 891430 (2011). https://doi.org/10.1155/2011/891430
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DOI: https://doi.org/10.1155/2011/891430