- Research Article
- Open Access
Three Solutions for a Discrete Nonlinear Neumann Problem Involving the -Laplacian
© P. Candito and G. D'Aguì. 2010
- Received: 26 October 2010
- Accepted: 20 December 2010
- Published: 29 December 2010
We investigate the existence of at least three solutions for a discrete nonlinear Neumann boundary value problem involving the -Laplacian. Our approach is based on three critical points theorems.
- Dirichlet Problem
- Fixed Point Theorem
- Nontrivial Solution
- Neumann Boundary
- Neumann Boundary Condition
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].
where is a fixed positive integer, is the discrete interval , for all , is a positive real parameter, , , is the forward difference operator, , , and is a continuous function.
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 .
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]).
Assume that there exist and , with such that
() for each the functional is coercive.
Then, for each , the functional has at least three distinct critical points in .
Theorem 2.2 (see [7, Corollary 3.1]).
Assume that there exist two positive constants , and , with such that
(b3) for each , and for every , which are local minima for the functional such that and , and one has .
Then, for each , the functional admits at least three critical points which lie in .
where we read if this case occurs.
Theorem 2.3 (see [8, Theorem 2.3]).
Let be a finite dimensional real Banach space. Assume that for each one has
Then, for each , the functional admits at least three distinct critical points.
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 .
where for every . It is easy to show that and are two functionals on .
Next lemma describes the variational structure of problem (), for the reader convenience we give a sketch of the proof, see also ,
is a Banach space. Let , be a solution of problem () if and only if is a critical point of the functional .
for every and , standard variational arguments complete the proof.
Finally, we point out the following strong maximum principle for problem ().
Then, either in , or .
so . Thus, repeating these arguments, the conclusion follows at once.
Now, we give the main results.
Assume that there exist three positive constants , , and with , and such that
problem (Pλf) admits at least three solutions.
At this point, since , it is clear that the functional turns out to be coercive.
We note that hypothesis ( ) can be replaced with the following:
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 .
Let be a continuous function in such that for some . Assume that there exist three positive constants , , and with such that
() for each ,
for all , .
for all such that .
Now, let and be two local minima for such that and . Owing to Lemmas 2.5 and 2.6, they are two positive solutions for () so , for all and for all . Hence, since one has for all , is verified.
and the proof is completed.
Let be a continuous function such that for some . Assume that there exist four constants , , , and , with , and such that
() , for all .
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.
Let be a continuous function such that for some . Assume that there exist four constants , , , and with and such that
(l2) , for all .
problem () admits at least three solutions.
where , which implies condition ( ).
In , by Mountain Pass Theorem, the authors established the existence of at least one solution for problem () requiring the following conditions:
(θ1) for uniformly in ,
for every and .
Moreover, they remember that the above conditions imply, respectively, the following:
(θ3) for uniformly in ,
(θ4) there exist two positive constants and such that
Next result shows that under more general conditions than ( ) and ( ), problem ( ) has at least two nontrivial solutions.
Assume that ( ) holds and in addition
Then, problem ( ) has at least two nontrivial solutions.
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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