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 , .
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 . This is one of the motivations why problems like () were studied by several authors beginning from the seminal paper of Lions . 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  and , where the existence of at least one positive solution is established under various hypotheses on . In particular, in  the nonlinearity is supposed to satisfy the well-known Ambrosetti-Rabinowitz growth condition; in  satisfies certain growth conditions at and , and is nondecreasing on for all . Critical point theory and minimax methods are used in  and . For and , the existence of a nontrivial solution as well as multiple solutions for problem () is established in  and  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 , 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 . In particular, condition of [6, Theorem ] is not satisfied by . Moreover, in  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.
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 .
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
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.
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 .
Let , , and be as Theorem 2.1. Then, for every one has
Put for every , and let . Assume, by contradiction, that
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.
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 .
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 .
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 , 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.
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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