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

This paper aims to establish the existence of positive solutions in to the following problem involving a nonlocal equation of Kirchhoff type:

(Px3bb)

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

(1.1)

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

(1.2)

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 .

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

(2.1)

admits a unique global minimum on .

Proof.

From condition we find positive constants such that

(2.2)

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

(2.3)

Since , from the previous inequality we obtain

(2.4)

By standard results, the functional

(2.5)

is of class and sequentially weakly continuous, and the functional

(2.6)

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

(2.7)

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

(2.8)

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

(2.9)

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

(2.10)

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

(2.11)

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:

(2.12)

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

(2.13)

Now, note that the identities

(2.14)

lead to

(2.15)

which, in turn, imply that

(2.16)

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

(2.17)

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

(2.18)

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

(2.19)

Proof.

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

(2.20)

Then,

(2.21)

Now, it is easy to check that the functional

(2.22)

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

(2.23)

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

(2.24)

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

(2.25)

which is impossible.

Whenever the function is as in Theorem 2.1, we put

(2.26)

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

(2.27)

Then, , and

(2.28)

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

(2.29)

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

(2.30)

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

(2.31)

as well as

(2.32)

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

(2.33)

Then, we can fix a number

(2.34)

in such way that

(2.35)

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

(2.36)

Arguing by contradiction, assume that

(2.37)

From Lemma 2.4 and condition (2.30) we have

(2.38)

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

(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 . First of all, we observe that by standard comparison results, it is easily seen that

(2.40)

where is the unique positive solution of the problem

(2.41)

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

(2.42)

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 .