On a Perturbed Dirichlet Problem for a Nonlocal Differential Equation of Kirchhoff Type
© Giovanni Anello. 2011
Received: 24 May 2010
Accepted: 26 July 2010
Published: 9 August 2010
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 , .
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 .
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 .
admits a unique global minimum on .
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.
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 .
which is impossible.
for every . Theorem 2.3 says that every is a positive number.
where the constant depends only on .
Here is the constant defined in Lemma 2.4 and . Note that no growth condition is required on .
where is the constant defined in (2.26) and is the embedding constant of in , problem ( ) admits at least a positive solution such that .
that is absurd. The proof is now complete.
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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