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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:

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

for all . Thus, the weak solutions of ( ) are exactly the positive critical points of the associated energy functional

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

admits a unique global minimum on .
Proof.
From condition we find positive constants
such that

Therefore, by Sobolev embedding theorems, there exists a positive constant such that

Since , from the previous inequality we obtain

By standard results, the functional

is of class and sequentially weakly continuous, and the functional

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

Indeed, fix a nonzero and nonnegative function , and put
. We have

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

for all , we have that
. Thus,
and
are weak solutions of the following nonlocal problem:

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

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:

From (2.12) and condition , we infer that
and
are related by

Now, note that the identities

lead to

which, in turn, imply that

Now, since and
are both global minima for
, one has
. It follows that

At this point, from condition and (2.17), we infer that

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

Proof.
Put for every
, and let
. Assume, by contradiction, that

Then,

Now, it is easy to check that the functional

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

Let be such that
. From assumption
,
turns out to be a strictly increasing function. Therefore, in view of (2.21), one has

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

which is impossible.
Whenever the function is as in Theorem 2.1, we put

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

Then, , and

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

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

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

as well as

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

Then, we can fix a number

in such way that

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

Arguing by contradiction, assume that

From Lemma 2.4 and condition (2.30) we have

Therefore, using (2.30) (and recalling the notation ), one has

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

where is the unique positive solution of the problem

When is a ball of radius
centered at
, then
, and so
. More difficult is obtaining an estimate from below of
: if
, one has

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