© P. Candito and G. D'Aguì. 2010
Received: 26 October 2010
Accepted: 20 December 2010
Published: 29 December 2010
In these last years, the study of discrete problems subject to various boundary value conditions has been widely approached by using different abstract methods as fixed point theorems, lower and upper solutions, and Brower degree (see, e.g., [1–3] and the reference given therein). Recently, also the critical point theory has aroused the attention of many authors in the study of these problems [4–12].
In particular, for every lying in a suitable interval of parameters, at least three solutions are obtained under mutually independent conditions. First, we require that the primitive of is sublinear at infinity and satisfies appropriate local growth condition (Theorem 3.1). Next, we obtain at least three positive solutions uniformly bounded with respect to , under a suitable sign hypothesis on , an appropriate growth conditions on in a bounded interval, and without assuming asymptotic condition at infinity on (Theorem 3.4, Corollary 3.6). Moreover, the existence of at least two nontrivial solutions for problem () is obtained assuming that is sublinear at zero and superlinear at infinity (Theorem 3.5).
It is worth noticing that it is the first time that this type of results are obtained for discrete problem with Neumann boundary conditions; instead of Dirichlet problem, similar results have been already given in [6, 9, 13]. Moreover, in , the existence of multiple solutions to problem () is obtained assuming different hypotheses with respect to our assumptions (see Remark 3.7).
Investigation on the relation between continuous and discrete problems are available in the papers [15, 16]. General references on difference equations and their applications in different fields of research are given in [17, 18]. While for an overview on variational methods, we refer the reader to the comprehensive monograph .
2. Critical Point Theorems and Variational Framework
Let be a real Banach space, let be two functions of class on , and let be a positive real parameter. In order to study problem (), our main tools are critical points theorems for functional of type which insure the existence at least three critical points for every belonging to well-defined open intervals. These theorems have been obtained, respectively, in [6, 20, 21].
Theorem 2.1 (see [11, Theorem 2.6]).
Theorem 2.2 (see [7, Corollary 3.1]).
Theorem 2.3 (see [8, Theorem 2.3]).
It is worth noticing that whenever is a finite dimensional Banach space, a careful reading of the proofs of Theorems 2.1 and 2.2 shows that regarding to the regularity of the derivative of and , it is enough to require only that and are two continuous functionals on .
Next lemma describes the variational structure of problem (), for the reader convenience we give a sketch of the proof, see also ,
Finally, we point out the following strong maximum principle for problem ().
3. Main Results
Now, we give the main results.
problem (Pλf) admits at least three solutions.
It is worth noticing that a careful reading of the proof of Theorem 3.1 shows that, provided that and under the only condition ( ), problem () admits at least one solution for every and at least three solutions for every , whenever there exists for which .
and the proof is completed.
problem () admits at least three nontrivial solutions.
Therefore, since and , condition ( ) is verified. Hence, from Theorem 2.3, the functional admits three critical points, which are three solutions for (). Since for some , they are nontrivial solutions, and the conclusion is proved.
problem () admits at least three solutions.
In , by Mountain Pass Theorem, the authors established the existence of at least one solution for problem () requiring the following conditions:
Moreover, they remember that the above conditions imply, respectively, the following:
that is, 0 is a local minimum. Moreover, by ( ), by now, it is evident that the functional is anticoercive in . Hence, by the regularity of , there exists which is a global maximum for the functional. Therefore, since it is not restrictive to suppose that (otherwise, there are infinitely many critical points), our conclusion follows: if , from Corollary 2.11 of  which ensures a third critical point different from 0 and and by standards arguments if .
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