Open Access

Multiplicity of Positive and Nodal Solutions for Nonhomogeneous Elliptic Problems in Unbounded Cylinder Domains

Boundary Value Problems20092009:687385

DOI: 10.1155/2009/687385

Received: 13 March 2009

Accepted: 7 May 2009

Published: 9 June 2009

Abstract

We show that if and satisfy some suitable conditions, then the Dirichlet problem in has a solution that changes sign in , in addition to two positive solutions where is an unbounded cylinder domain in .

1. Introduction

Throughout this paper, let be the generic point of with , , where
(11)
In this paper, we study the multiplicity results of both positive and nodal solutions for the nonhomogeneous elliptic problems
(12)

where is a bounded smooth domain, is a smooth unbounded cylinder domain in .

It is assumed that and satisfy the following assumptions:

(a 1) is continuous and on and
(13)

(f 1) ;

(f 2) in which we defined
(14)
(f 3)there exist positive constants such that
(15)

where is the first positive eigenvalue of the Dirichlet problem in .

For the homogeneous case, that is, , Zhu [1] has established the existence of a positive solution and a nodal solution of problem (1.2) in provided satisfies in and as for some positive constants and . More recently, Hsu [2] extended the results of Zhu [1] with to an unbounded cylinder . Let us recall that, by a nodal solution we mean the solution of problem (1.2) with change of sign.

For the nonhomogeneous case ( ),Adachi and Tanaka [3] have showed that problem (1.2) has at least four positive solutions in for and satisfy some suitable conditions, but we place particular emphasis on the existence of nodal solutions. More recently, Chen [4] considered the multiplicity results of both positive and nodal solutions of problem (1.2) in . She has showed that problem (1.2) has at least two positive solutions and one nodal solution in when and satisfy some suitable assumptions.

In the present paper, motivated by [4] we extend and improve the paper by Chen [4]. We will deal with unbounded cylinder domains instead of the entire space and also obtain the same results as in [4]. Our arguments are similar to those in [5, 6], which are based on Ekeland's variational principle [7].

Now, we state our main results.

Theorem 1.1.

Assume hold and satisfies assumption .

(a2)there exist positive constants such that
(16)

Then problem (1.2) has at least two positive solutions and in . Furthermore, and satisfy , and is a local minimizer of where is the energy functional of problem (1.2).

Theorem 1.2.

Assume hold and satisfies assumption .

(a3)there exist positive constants , and such that
(17)

Then problem (1.2) has a nodal solution in in addition to two positive solutions and .

For the case , we also have obtained the same results as in Theorems 1.1 and 1.2.

Theorem 1.3.

Assume hold and satisfies assumption .

(a2)there exist positive constants such that
(18)

Then problem (1.2) has at least two positive solutions and in . Furthermore, and satisfy , and is a local minimizer of where is the energy functional of problem (1.2).

Theorem 1.4.

Assume hold and satisfies assumption below.

(a3)there exist positive constants and such that
(19)

Then problem (1.2) has a nodal solution in in addition to two positive solutions and .

Among the other interesting problems which are similar to problem (1.2), Bahri and Berestycki [8] and Struwe [9] have investigated the following equation:
(110)

where , , and is a bounded domain in . They found that (1.10) possesses infinitely many solutions. More recently, Tarantello [5] proved that if is the critical Sobolev exponent and satisfying suitable conditions, then (1.10) admits two solutions. For the case when is an unbounded domain, Cao and Zhou [10], Cîrstea and Rădulescu [11], and Ghergu and Rădulescu [12] have been investigated the analogue equation (1.10) involving a subcritical exponent in . Furthermore, Rădulescu and Smets [13] proved existence results for nonautonomous perturbations of critical singular elliptic boundary value problems on infinite cones.

This paper is organized as follows. In Section 2, we give some notations and preliminary results. In Section 3, we will prove Theorem 1.1. In Section 4, we establish the existence of nodal solutions.

2. Preliminaries

In this paper, we always assume that is an unbounded cylinder domain or . Let for , and let be the first positive eigenfunction of the Dirichlet problem in with eigenvalue unless otherwise specified. We denote by and ( ) universal constants, maybe the constants here should be allowed to depend on and , unless some statement is given. Now we begin our discussion by giving some definitions and some known results.

We define
(21)
Let be the Sobolev space of the completion of under the norm with the dual space , and denote the usual scalar product in . The energy functional of problem (1.2) is given by
(22)

here and from now on, we omit " " and " " in all the integration if there is no other indication. It is well known that is of in and the solutions of problem (1.2) are the critical points of the energy functional (see Rabinowitz [14]).

As the energy functional is not bounded on , it is useful to consider the functional on the Nehari manifold
(23)
Thus, if and only if
(24)

Easy computation shows that is bounded from below in the set . Note that contains every nonzero solution of (1.2).

Similarly to the method used in Tarantello [5], we split into three parts:
(25)

Let us introduce the problem at infinity associated with problem (1.2) as

(26)
We state here some known results for problem (2.6). First of all, we recall that by Esteban [15] and Lien et al. [16], problem (2.6) has a ground state solution such that
(27)
where , and
(28)

Furthermore, from Hsu [2] we can deduce that for any there exist positive constants such that, for all ,

(29)

We also quote the following lemma (see Hsu [17] or K. -J. Chen et al. [18] for the proof) about the decay of positive solution of problem (1.2) which we will use later.

Lemma 2.1.

Assume and hold. If is a positive solution of problem (1.2), then

(i) for all ;

(ii) as uniformly for and for any ;

(iii)for any there exist positive constants such that, for all ,
(210)

We end this preliminaries by the following definition.

Definition 2.2.

Let , be a Banach space and .

(i) is a -sequence in for if and strongly in as
  1. (ii)

    We say that satisfies the condition if any -sequence in for has a convergent subsequence.

     

3. Proof of Theorem 1.1

In this section, we will establish the existence of two positive solutions of problem (1.2).

First, we quote some lemmas for later use (see the proof of Tarantello [5] or Chen [4, Lemmas 2.2, 2.3, and 2.4]).

Lemma 3.1.

Assume and hold, then for every , there exists a unique such that . In particular, we have
(31)
and . Moreover, if , then there exists a unique such that . In particular,
(32)

and .

Lemma 3.2.

Assume and hold, then for every , we have
(33)

Lemma 3.3.

Assume and hold, then for every , there exist a and a -map satisfying that
(34)

Apply Lemmas 3.1, 3.2, 3.3, and Ekeland variational principle [7], and we can establish the existence of the first positive solution.

Proposition 3.4.

Assume and hold, then the minimization problem is achieved at a point which is a critical point for . Moreover, if and , then is a positive solution of problem (1.2) and is a local minimizer of .

Proof.

Modifying the proof of Chen [4, Proposition 2.5]. Here we omit it.

Since and , thus, in the search of our second positive solution, it is natural to consider the second minimization problem:
(35)

We will establish the existence of the second positive solution of problem (1.2) by proving that satisfies the -condition.

Proposition 3.5.

Assume and hold, then satisfies the -condition with .

Proof.

Let be a -sequence for with . It is easy to see that is bounded in , so we can find a such that weakly in up to a subsequence and is a critical point of . Furthermore, we may assume a.e. in , strongly in for all . Hence we have that and
(36)
Set . Then by (3.6) and Brézis and Lieb lemma (see [19]), we obtain
(37)
Moreover, by Vitali's lemma and ,
(38)
In view of assumptions and (3.7), (3.8), and by Lemma 3.2, we obtain
(39)
(310)
Hence, we may assume that
(311)
By the definition of , we have , combining with (3.11) and , and we get that . Either or . If , the proof is complete. Assume that , from (2.7), (3.9), and (3.11), we get
(312)

which is a contradiction. Therefore, and we conclude that strongly in .

Let , let , and let be a constant, we denote and for where is the ground state solution of problem (2.6) and is the first positive solution of problem (1.2).

Proposition 3.6.

Assume and hold, then there exists such that
(313)

The following estimates are important to find a path which lies below the first level of the break down of the condition. Here we use an interaction phenomenon between and .

To give a proof of Proposition 3.6, we need to establish some lemmas.

Lemma 3.7.

Let , and is a domain in . Then for any , there exists a positive constant such that
(314)

Proof.

From (2.10), we have for ,
(315)

Lemma 3.8.

Let be a domain in , and let be a vector in . If satisfies
(316)
then
(317)
or
(318)

Proof.

We know then
(319)

Since as the lemma follows from the Lebesgue's dominated convergence theorem.

Now, we give the proof of Proposition 3.6.

The Proof of Proposition 3.6

Recall , where is a domain in . For , let
(320)
We also remark that for all
(321)
and for any and there exists such that for all
(322)
Since is continuous in , there exists such that for all
(323)
and by the fact that as uniformly in , then there exists such that
(324)
Thus, we only need to show that there exists a constant such that
(325)
Straightforward computation gives us
(326)
where
(327)
Thus, we only need to prove that there exists a constant such that
(328)
Now we estimate and . Without loss of generality, we may assume that . Thus, we can choose small enough such that
(329)
By (3.21),
(330)
Let , and by applying (3.22), we obtain
(331)
Let . Then applying (3.14), we have for
(332)
Next from , (2.9), (3.29), and Lemma 3.8, there exists a such that for any ,
(333)
From (3.29), we have for ,
(334)
Finally, we can choose large enough such that
(335)

Thus from (3.26) and (3.32)–(3.35), we obtain (3.13). This completes the proof of Proposition 3.6.

Proposition 3.9.

For , there exists a -sequence for . In particular, we have .

Proof.

Set and define the map given by . Since the continuity of follows immediately from its uniqueness and extremal property, thus is continuous with continuous inverse given by . Clearly disconnecting is exactly two components:
(336)

and .

We will prove that there exists such that . Denote . Since , we have

(337)
Thus
(338)
Therefore, there exists such that for . Since , then
(339)

hence .

disconnects in exactly two components, so we can find an such that . Therefore , which follows from Proposition 3.6.

Analogously to the proof of Proposition 3.4, by the Ekeland variational principle we can show that there exists a -sequence for .

Proposition 3.10.

Assume and hold, then the functional has a minimizer which is also a critical point of and for .

Proof.

From Propsitions 3.5 and 3.9, we can deduce that strongly in . Consequently, is a critical point of , (since is closed) and .

By Lemma 3.1, we can choose a number such that . Since . Applying Lemma 3.1 again, we conclude that

(340)

Hence . So we can always take . By the maximum principle for weak solutions (see Gilbarg and Trudinger [20]) we can show that if , then in .

The proof of Theorem 1.1

By Propositions 3.4 and 3.10, we obtain the conclusion of Theorem 1.1.

4. Existence of Nodal Solution

In this section, we will study the existence of nodal solutions for problem (1.2). To this end, we need to compare some different minimization problems. Define
(41)
Here, we use notation . Set
(42)
(43)

Then we have

Proposition 4.1.
  1. (a)

    If , then the minimization problem (4.2) attains its infimum at a point which defines a sign changing critical point of .

     
  2. (b)

    Analogously, if the same conclusion holds for the minimization problem (4.3).

     

Proof.

The proof is almost the same as that in Tarantello [6, Proposition 3.1] .

The above proposition would yield the conclusion for the main theorem only if the given relations between , and could be established. While it is not clear whether or not such inequalities should hold, we will use these values to compare with another minimization problem. Namely, set
(44)
and define
(45)

It is clear that . Since satisfies condition only locally, we need the following upper bound for . Recall that , and where and is the ground state solution of problem (2.6).

Lemma 4.2.

Assume , and hold. For any fixed , there exist such that
(46)
and for large,
(47)

Proof.

To prove (4.6), it suffices to show that there exist and such that
(48)
To this purpose, let
(49)
For , denote by and the positive values given by Lemma 3.1 according to which we have
(410)
Note that and are continuous with respect to satifying
(411)

Therefore, by the continuity of , we can find such that . This gives (4.8) with and .

To prove (4.7), we only need to estimate for and . First, it is obvious that the structure of guarantees the existence of (independent of large) such that , for all . On the other hand, for , since is continuous in , there exists small enough such that

(412)
At this point, we find large , such that holds for all and :
(413)
By (4.13) and the following elementary inequality:
(414)
where is some positive constant, we have
(415)

Without loss of generality, we may assume , and where , and are given in and , respectively.

(i)First, by the Hölder inequality and (2.9),
(416)
From (2.9), (2.10), and applying Lemma 3.8, there exists a such that for
(417)
Similarly, we also obtain
(418)
and there exists a such that for
(419)
(ii)Since satisfies assumption and by Lemma 3.8, there exists a such that for ,
(420)
By , (2.9), and Lemma 3.8, there exists a such that for ,
(421)
(iii)Note that the constants ( ) in (i), (ii) are independent of . Thus, by (i), (ii), and let , we can find a such that for ,
(422)
Combining (4.15) and (4.22), we obtain that there exists a such that for ,
(423)

This completes the proof of Lemma 4.2.

Proposition 4.3.

Assume and hold. If and , then the minimization problem attains its infimum at which defines a changing sign critical point of .

Proof.

It is obvious that is closed. Exactly as in the proof of [6, Proposition 3.2], by means of Ekeland's principle, we derive a -sequence for . In particular, we have , for some constants and . Thus, we can take a subsequence, also denoted by , such that weakly in . We start by showing that .

Indeed, if by contradiction we assume, for instant, that , then we can deduce that

(424)
On the other hand,
(425)
By (4.24) and , we may assume that
(426)
Using the argument in the proof of Proposition 3.5, by (2.7), (4.24), and (4.25), we can deduce that and
(427)

However, by Lemma 4.2, ; that is, which contradicts (4.27). A similar argument applies to . Therefore, is a weak solution of problem (1.2) changing sign and .

Set and with weakly in . Note that

(428)
In view of Proposition 3.9 and Lemma 4.2, we also have
(429)
Therefore, we must have
(430)
Without loss of generality, we suppose
(431)
By (4.24), we have
(432)

We claim that . Indeed, we assume is bounded below, as above, (4.28) and (4.32) imply , contradicting (4.31). In the same way, if , we can also prove . Hence we have or ; that is, or . By assumptions and , we conclude that .

If we write with weakly in , we have

(433)
Furthermore, by Lemma 4.2, we have
(434)

We claim that . Indeed, we assume is bounded below, as above, (4.33) imply , contradicting (4.34). Consequently, strongly in and .

The Proof of Theorems 1.2–1.4

The conclusion of Theorem 1.2 follows immediately from Theorem 1.2 and Propositions 4.1 and 4.3. With the same argument, we also have that Theorems 1.3 and 1.4 hold for .

Authors’ Affiliations

(1)
Center for General Education, Chang Gung University

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© Tsing-San Hsu 2009

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