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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 564363, 10 pages
http://dx.doi.org/10.1155/2010/564363
Research Article

Multiplicity Results for a Perturbed Elliptic Neumann Problem

1Department of Science for Engineering and Architecture (Mathematics Section), Engineering Faculty, University of Messina, 98166 Messina, Italy
2Department of Mathematics of Messina and DIMET, University of Reggio Calabria, 89060 Reggio Calabria, Italy

Received 21 April 2010; Accepted 9 July 2010

Academic Editor: Pavel Drábek

Copyright © 2010 Gabriele Bonanno and Giuseppina D'Aguì. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The existence of three solutions for elliptic Neumann problems with a perturbed nonlinear term depending on two real parameters is investigated. Our approach is based on variational methods.

1. Introduction

Here and in the sequel, is a bounded open set, with a boundary of class , with , ; and are -Carathéodory functions.

The aim of this paper is to study the following perturbed boundary value problem with Neumann conditions: where is the -Laplacian, is the outer unit normal to , and are positive real parameters.

Nonlinear boundary value problems involving the -Laplacian operator (with ) arise from a variety of physical problems. They are used in non-Newtonian fluids, reaction-diffusion problems, flow through porous media, and petroleum extraction (see, e.g., [1, 2]).

In the last years, several researchers have studied nonlinear problems of this type through different approaches. In [1], the authors have obtained results on the existence of a solution for the problem by using the perturbation result on sums of ranges of nonlinear accretive operators. Subsequently, Wei and Agarwal, in [2], have studied the same problem by developing some new techniques in the wake of [1]. Problem (), when , and does not depend on , has been studied in [3]. In this paper, the authors have obtained the existence of at least three solutions for small , by using Implicit Function Theorem and Morse Theory. By using variational methods and in particular critical point results given by Ricceri in [4], Faraci, in her nice paper [5], has dealt with a Neumann Problem involving the -Laplacian (for any ) of type In particular, [5, Theorems , ] assure the existence of three solutions for the problem given above.

In the present paper, we establish some results (Theorems 3.1, 3.2), which assure the existence of at least three weak solutions for the problem (). In particular the following result is a consequence of Theorem 3.2.

Theorem 1.1. Let be a nonnegative continuous function such that Then, for every and for every positive continuous function there exists such that, for each , the problem has at least three nonzero classical solutions.

With respect to [3, 5], we stress that our results hold under different assumptions (see Remarks 3.4 and 3.5). In particular, in Theorem 1.1, no asymptotic condition at infinity is required on the nonlinear term. We also point out that in Theorems 3.1 and 3.2, precise estimates of parameters and are given.

2. Preliminaries and Basic Notations

Our main tools are three critical point theorems that we recall here in a convenient form. The first has been obtained in [6], and it is a more precise version of Theorem of [7]. The second has been established in [7].

Theorem 2.1 (see [6, Theorem ]). Let be a reflexive real Banach space, a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on , a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that 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 ]). Let be a reflexive real Banach space, a convex, coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on , a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Assume that there exist two positive constants , and , with , such that ; ; for each , , and for every , which are local minima for the functional , and such that and one has . Then, for each the functional admits three critical points which lie in .

Now we recall some basic definitions and notations.

A function is called an -Carathéodory function if is measurable for all , is continuous for almost every , for all one has . Clearly, if is continuous in , then it is -Carathéodory.

We also recall that a weak solution of the problem () is any , such that

Put

If is convex, an explicit upper bound for the constant is (see, e.g., [8, Remark ]).

Put for all and for all .

Moreover, set for all and for all . Clearly, and .

3. Main Results

In this section, we present our main results on the existence of at least three weak solutions for the problem ().

In order to introduce our first result, fixing such that and picking , put where we read so that, for instance, when () and .

Theorem 3.1. Let be an -Carathéodory function, and put for all . Assume that there exist two positive constants , with , such that (i) where is given by (2.4); (ii). Then, for every and for every -Carathédory function such that (iii), where for all , there exists given by (3.2) such that, for each , Problem () has at least three weak solutions.

Proof. Fix , , and as in the conclusion. Take endowed with the norm On the space , we consider the norm . Since , is compactly embedded in , we have Put, for each , Since the critical points of the functional on are weak solutions of problem (), our aim is to apply Theorem 2.1 to and . To this end, taking into account that the regularity assumptions of Theorem 2.1 on and are satisfied, we will verify and .
Put taking into account (3.4), one has Now, fix . Clearly and moreover . One has Hence, Since , one has Furthermore, ; this means that
Then, Hence, from (3.8) and (3.10), condition of Theorem 2.1 is verified.
Finally, since , we can fix such that and . Therefore, there exists a function such that for each .
Now, fix ; from (ii) there is a function such that for each . It follows that, for each , This leads to the coercivity of , and condition of Theorem 2.1 is verified. Since, from (3.8) and (3.10), Theorem 2.1 assures the existence of three critical points for the functional , and the proof is complete.

Now, we state a variant of Theorem 3.1. Here no asymptotic condition on is requested; on the other hand, the functions , are supposed to be nonnegative.

Fix such that ,,  , and picking put

Theorem 3.2. Assume that there exist three positive constants , with , such that (j) for each ; (jj)(jjj) Then, for every and for every nonnegative -Carathédory function , there exists given by (3.16) such that, for each , the problem () has at least three weak solutions , , such that

Proof. Without loss of generality, we can assume for all . Fix , , and as in the conclusion and take , and as in the proof of Theorem 3.1.
We observe that the regularity assumptions of Theorem 2.2 on and are satisfied.
Then, our aim is to verify and . To this end, put as in Theorem 3.1, , and . Therefore, one has and, since and , one has Therefore, and of Theorem 2.2 are verified.
Finally, we verify that satisfies assumption of Theorem 2.2.
Let and be two local minima for . Then and are critical points for , and so, they are weak solutions for the problem (). For every positive parameter , and for every one has hence, owing to the Weak Maximum Principle (see for instance [9]) we obtain . Then, it follows that , and that , and, hence,
From Theorem 2.2, the functional has at least three distinct critical points which are weak solutions of () and the conclusion is achieved.

Example 3.3. Consider the following problem where Then, owing to Theorem 3.2, for each , the problem given above has at least three weak solutions. It is enough to choose, for instance, and .

Remark 3.4. We observe that [5, Theorem ] cannot be applied to the problem of Example 3.3 since the assumptionwithsuch that does not hold. Moreover, also [5, Theorem ] cannot be applied since condition is not verified.
Finally, we observe that [5, Theorem ] and [5, Theorem ] ensure only two solutions and two nonzero solutions, respectively.
Furthermore, contrary to theorems in [5], owing to our results, we have precise values of for which the problem admits solutions.
On the other hand, we observe that in [5], the case is investigated too.

Remark 3.5. It is easy to verify that our results and theorems in [3] are mutually independent. In particular, we remark that none of theorems in [3] can be applied to the problem in Example 3.3. We also observe that in [3] no estimate of small is given.

Proof of Theorem 1.1. Our aim is to apply Theorem 3.2 by choosing Put and Therefore, taking into account that , one has Moreover, since , there is a positive constant such that and Hence, a simple computation shows that all assumptions of Theorem 3.2 are satisfied, and the conclusion follows.

Remark 3.6. We explicitly observe that in Theorem 3.2, as well as in Theorem 1.1, no asymptotic condition at infinity on the nonlinear term is requested.

Example 3.7. The function satisfies all assumptions of Theorem 1.1.

Remark 3.8. We explicitly observe that the very nice Theorem of [10] cannot be applied to the function of Example 3.7 since

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