Abstract and Applied Analysis

Volume 2012 (2012), Article ID 568120, 31 pages

doi:10.1155/2012/568120

Research Article

Existence Results for Quasilinear Elliptic Equations with Indefinite Weight

Mieko Tanaka

Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjuku-ku 162-8601, Japan

Received 19 January 2012; Accepted 7 March 2012

Academic Editor: Juan J. Nieto

Copyright © 2012 Mieko Tanaka. 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.

We provide the existence of a solution for quasilinear elliptic equation in under the Neumann boundary condition. Here, we consider the condition that as and as . As a special case, our result implies that the following -Laplace equation has at least one solution: in on for every , , and with . Moreover, in the nonresonant case, that is, is not an eigenvalue of the -Laplacian with weight , we present the existence of a solution of the above -Laplace equation for every , and .

1. Introduction

In this paper, we consider the existence of a solution for the following quasilinear elliptic equation:

where is a bounded domain with boundary , denotes the outward unit normal vector on , , and . We assume that is a Carathéodory function on satisfying

and that is bounded on a bounded set (admitting in the nonresonant case). Here, is a map which is strictly monotone in the second variable and satisfies certain regularity conditions (see the following assumption ). The equation contains the corresponding -Laplacian problem as a special case. Although the operator is nonhomogeneous in the second variable in general, we assume that is asymptotically -homogeneous at infinity in the following sense .

Throughout this paper, we assume that the map satisfies the following assumptions and :

there exist a positive function and a continuous function on such that

, where for all and

(i); (ii) there exists such that

(iii) there exists such that

(iv) there exists such that

A similar hypothesis to () is considered in the study of quasilinear elliptic problems (cf. [1, Example  2.2], [2–6]). It is easily seen that many examples as in the above references satisfy the condition (). Also, the following example satisfies our hypotheses:

In particular, for , that is, stands for the usual -Laplacian , we can take in (). Conversely, in the case where holds in , by the inequalities in Remark 1.4 (ii) and (iii), we see whence .

Concerning the weight , throughout this paper, we assume that

holds, where denotes the Lebesgue measure of a measurable set .

Because is asymptotically -homogeneous at infinity, the solvability of our equation is related to the following homogeneous equation (see Theorem 1.1):

where is the positive function as in (). We say that is an eigenvalue of if the equation has a nontrivial solution.

There are few existence results of a solution to our equation (and also the -Laplace equation). For example, if and hold, then the standard argument guarantees the existence of a solution. For the -Laplacian as a special case of our problem, it is shown in [7] that the equation

has a unique positive solution provided , and , where is the principal eigenvalue defined in Section 2.1 with . In [8], although the resonant case where or is considered under the assumptions to , its result does not cover the case of with , where (, ) is th eigenvalue of the -Laplacian with weight . For the Laplace problem under the Neumann boundary condition, we can refer to [9, 10]. Under the Dirichlet boundary condition, the existence results for the Laplace problem are well known when and is not an eigenvalue of the Laplacian (cf. [11]). Moreover, under the Dirichlet (or blow-up) boundary condition, many authors study various equations involving the -Laplace (Laplace) operator with (indefinite) weight. For example, we refer to [12] for boundary blow-up problems with Laplacian, [13] for periodic reaction-diffusion problems and [14, 15] for singular quasilinear elliptic problems.

Recently, the present author shows the existence of a solution for our problem in the case where is between the principal eigenvalue and the second eigenvalue in [6] (for ). In addition, a similar situation is treated in [5]. However, existence results are not seen in the case when is greater than the second eigenvalue for our problem. Therefore, the first purpose of this paper is to present an existence result of a solution in the nonresonant case where is not an eigenvalue of . Then, it studied the existence of at least one solution in the resonant case under assumptions that cover the case with and .

For the proof of our result, it is necessary to study the weighted eigenvalue problem . Thus, in Section 2, we introduce two sequences and of an eigenvalue of defined by Ljusternik-Schnirelman theory or Drábek-Robinson's method (cf. [16]), respectively. Then, we show several properties of above eigenvalues. In Section 3, we give the proof in the nonresonant case by using . In Sections 4 and 5, we handle the resonant case.

1.1. Statements of Our Existence Results

First, we state the existence result of a solution in the nonresonant case.

Theorem 1.1. Assume that is not an eigenvalue of . Then, has at least one solution.

To state our existence result in the resonant case, we introduce some conditions. Set

where is the function as in ().

there exist and such that

there exist and such that

there exist and such that

there exist and such that

Theorem 1.2. Assume one of the following conditions:

(i) and () or () hold; (ii), and one of (), (), () and () hold; (iii), and () or () hold;

Then, has at least one solution.

In the special case where and for , we easily see that () or () holds with provided or , respectively. Therefore, the following result is proved according to Theorem 1.2.

Corollary 1.3. Let , and . Then, the following equation has at least one solution:

1.2. Properties of the Map

In what follows, the norm on is given by , where denotes the norm of for (). Setting , then we can easily see that

for every .

Remark 1.4. It is easily seen that the following assertions hold under condition ():

(i)for all , is maximal monotone and strictly monotone in ; (ii) for every ; (iii) for every ; (iv) is convex in for all and satisfies the following inequalities:

for every , where and are the positive constants in ().

The following result is proved in [3]. It plays an important role for our poof.

Proposition 1.5 (see [3, Proposition 1]). Let be the map defined by

for , . Then, has the property, that is, any sequence weakly convergent to with strongly converges to .

2. The Weighted Eigenvalue Problems

2.1. Preliminaries

The following lemmas can be easily shown by way of contradiction because is equivalent to (note that is positive). Here, we omit the proofs (refer to [7]).

Lemma 2.1. Assume . Then, there exists a constant such that for every with .

Lemma 2.2. Assume that and . Then, there exists a constant such that

for every .

Lemma 2.3. Assume that in . Then, for every there existed such that

for every .

First, we recall the following principle eigenvalue :

Because of , we have the following result as the same argument as in the case of the -Laplacian.

Proposition 2.4 (see [7, Proposition  2.2]). The following assertions hold:

(i)If holds, then ; (ii)If holds, then is a simple eigenvalue and it admits a positive eigenfunction. In addition, the open interval contains no eigenvalues of .

Lemma 2.5. Assume . Then, one has for every and with .

Proof. We choose a minimizer for because Proposition 2.4 guarantees the existence of it. Then, for every , we have

by the definition of . By applying the same argument to a minimizer for , we obtain for with .

2.2. Other Eigenvalues

Here, we introduce two unbounded sequences and as follows:

where denotes the Krasnoselskii genus of (see [17, Definition 5.1] for the definition) and denotes the usual unit sphere in . We see that is defined by Ljusternik-Schnirelman theory and it is known that the definition of is introduced by Drábek and Robinson ([16]) under the -Laplace Dirichlet problem with .

Remark 2.6. The following assertions can be shown easily:

(i); (ii) and for every ; (iii) for every ; (iv) for every ; (v) and for every ,

see [18] for the proof of (ii).

Define a functional on by for . Because is a regular value of , it is well known that the norm of the derivative at of the restriction of to is defined as follows:

where denotes the tangent space of at , that is, . Here, we recall the definition of the Palais-Smale condition for .

Definition 2.7. is said to satisfy the bounded Palais-Smale condition if any bounded sequence such that has a convergent subsequence. Moreover, we say that satisfies the Palais-Smale condition at level if any sequence such that and as has a convergent subsequence. In addition, we say that satisfies the Palais-Smale condition if satisfies the Palais-Smale condition for every .

The following result can be proved by the same argument as in [19, Proposition 3.3] (which treats the case of the -Laplacian, i.e., ) because of . Here, we omit the proof.

Lemma 2.8. The following assertions hold:

(i) satisfies the bounded Palais-Smale condition; (ii) satisfies the Palais-Smale condition provided .

Proposition 2.9. and are eigenvalues of such that

Proof. In the case of , since satisfies the Palais-Smale condition, we can apply the first deformation lemma on manifold (refer to [20]). Thus, by the standard argument, we can prove that and are critical values of . This means that and are eigenvalues of by the Lagrange multiplier rule. In addition, we can easily show by the standard argument via the first deformation lemma on manifold (refer to [21, Proposition  3.14.7], [22] or [17] in the case of a Banach space). Hence, holds because of for every .

In the case of , by the same argument as in [18], our conclusion can be proved. For readers' convenience, we give a sketch of the proof. For , we define and . Moreover, we set minimax values and of by

Because any Palais-Smale sequence of is bounded, it is easily shown that satisfies the Palais-Smale condition (refer to [19, Proposition  3.3]) Hence, it can be proved that and are critical values of . Furthermore, it follows from the argument as in [18, Lemma  3.5] that and as . Therefore, by noting that is -homogeneous, we can obtain a solution with for in , on , where or . Because of , it follows from the standard argument that has a subsequence strongly convergent to a solution for

where . Thus, and are eigenvalues of . To prove , by considering a function for , we have (refer to Proposition 2.10). Because we can apply our fist assertion to (note ), we obtain .

Proposition 2.10. Let if and if . Then, the following assertions hold:

(i)if in , then ; (ii)if in , then ; (iii)if and in , then .

Moreover, the same conclusion holds for .

Proof. We only treat because we can give the proof for similarly.

(i) Let in . Fix an arbitrary . Then, by the definition of , there exists a such that . Set for (note ), then holds. Therefore, by the definition of , we have

 because of for every . Since is arbitrary, we obtain .(ii) Let in and fix an arbitrary . By the definition of , there exists a such that . Since is compact and , we set . Then, due to Hölder's inequality and in , there exists an such that

 for every and . Therefore, by a similar argument to (i), we obtain

 for sufficiently large . Hence, follows. Since is arbitrary, our conclusion is proved.(iii) Let in and . We fix an arbitrary . Due to our assertion (ii), there exists an such that . For every , by the definition of , we can take satisfying .

Here, we will prove

If satisfies , then we obtain

by Lemma 2.2 and Hölder's inequality (note and ), where is a constant (independent of and ) obtained by the continuity of into . Therefore, if we take an satisfying for every , then we obtain

for every provided and . Similarly, in the case where changes sign, for every satisfying , we have

Hence, by taking a sufficiently large , we get the inequality

for every with and . In the case of in , by using Lemma 2.3 instead of Lemma 2.2, we have a similar inequality

for every provided (some sufficiently large ). Consequently, our claim follows from (2.15), (2.17), and (2.18).

Let us return to the proof of (iii). Because

holds by (2.13), and the continuity of into , we see the inequality

for every and (some sufficiently large ). By considering , we obtain

Because of , we get for sufficiently large , and hence our conclusion holds.

Finally, we recall the second eigenvalue of obtained by the mountain pass theorem.

where .

Since holds, the following result can be shown by the same argument as in [19] (although they handle the asymmetry case, it is sufficient to consider the case of in this paper). See [19, Theorem 3.2] for the proof.

Theorem 2.11. is an eigenvalue of which satisfies . Moreover, there is no eigenvalues of between and .

Now, we have the following result.

Proposition 2.12.

holds, where is a minimax value defined by (2.22).

Proof. First, we prove the inequality . Because is an eigenvalue (note that the following equation is homogeneous), we can choose a solution with for

Note that is a sign-changing function because any eigenfunction associated with any eigenvalue greater than the principal eigenvalue changes sign (refer to [18, Proposition  4.3]). Thus, we have

by taking as test function (recall that ). Hence, we may assume that by the normalization. Set . Then, because is homeomorphic to , there exists such that . Since the value of is equal to on , we obtain

by the definition of and .

Next, we will prove the inequality by dividing into two cases: and .

Case of : by way of contradiction, we assume that . Then, follows from Theorem 2.11. Note that satisfies the Palais-Smale condition in this case (see Lemma 2.8), and hence we can apply the first deformation lemma to . Therefore, by the standard argument (cf. [22], [17, Lemma 5.6]), we see that , where . This means that is an infinite set, that is, the following equation has infinite many solutions:

due to the Lagrange multiplier's rule. This contradicts to the fact described as in Proposition 2.4 that is simple. As a result, we have shown that holds in the case of (note ).

Case of : According to Proposition 2.10 (i) for , we have for every since we can apply the first result to . Because we prove by the same argument as in [6, Lemma 2.9] (for the case ), our conclusion is proved by taking in the inequality .

3. Proof of Theorem 1.1

We define a functional on as follows:

for ((1.15) or (1.9) for the definition of , , and ). It is easily seen that is well defined and class of on by (1.1), (1.16) and the continuity of .

Remark 3.1. Let be a critical point of , namely, satisfies the equality

for every . Then, by the Moser iteration process (refer to Theorem C in [4]). Therefore, () follows from the regularity result in [23]. Furthermore, due to [24, Theorem 3], satisfies in the distribution sense and the boundary condition

for every (see [24] for the definition of ). Since and for every , satisfies the Neumann boundary condition, that is, for every .

3.1. The Palais-Smale Condition in the Nonresonant Case

First, we recall the definition of the Palais-Smale condition.

Definition 3.2. A functional on a Banach space is said to satisfy the Palais-Smale condition at if a Palais-Smale sequence at level , namely,

has a convergent subsequence. We say that satisfies the Palais-Smale condition if satisfies the Palais-Smale condition at any . Moreover, we say that satisfies the bounded Palais-Smale condition if any bounded sequence such that is bounded and as has a convergent subsequence.

Concerning the Palais-Smale condition, we state the following result developed from [6, Proposition 7].

Proposition 3.3. If is not an eigenvalue of , then satisfies the Palais-Smale condition.

Proof. Let be a Palais-Smale sequence of , namely,

for some . It is sufficient to prove only the boundedness of because the operator described in Proposition 1.5 has the property.

To prove the boundedness of , it suffices to show that is bounded because of the inequality (obtained by (1.1)) and the following inequality:

where we use Remark 1.4 (iii) in the last inequality. By way of contradiction, we may assume that as by choosing a subsequence if necessary. Set . Then, since the inequality (3.6) guarantees that is bounded in , we may suppose, by choosing a subsequence, that in and in for some .

Here, we will prove that

where . Fix an arbitrary . It follows from (1.1) that there exists a such that

Then, we obtain

Since we are assuming that as , there exists such that for every

holds. This shows that because is arbitrary.

Here, we recall the following result proved in [6]:

for every . Thus, by considering

we see that strongly converges to in (note that -Laplacian has the property). Therefore, by taking a limit in for any and by noting (3.7) and (3.11), we know that is a nontrivial solution (note ) of

This means that is an eigenvalue of . This is a contradiction. Hence, is bounded.

3.2. Key Lemmas

To show the linking lemma, we define

.

Lemma 3.4. Let be odd and . Then, for every with , where is the set introduced in (3.14) and is the upper hemisphere in with boundary .

Proof. Fix any such that . If satisfies , then holds. So, we may assume that for every . Define as follows:

By the definition of , there exists such that . Since is odd and is even, we may suppose . So, this yields the inequality , whence holds.

Lemma 3.5. Let . Then, there exists such that

where is defined by (2.5).

Proof. Choose such that . By the definition of , there exists such that

Due to the compactness of , we put . By the property of the function as in () and Young's inequality, for every there exist constants and such that

for every and . Moreover, the hypothesis (1.1) ensures that for every there exist constants satisfying

for every and a.e. . Hence, we have

for every , , and since , (3.17), (3.18) and (3.19), where and . By taking and satisfying , we show that as . Thus, our conclusion follows because is symmetric.

3.3. The Case

Lemma 3.6. Let and . Then, is bounded from below, coercive and weakly lower semicontinuous (w.l.s.c.) on .

Proof. is w.l.s.c. on because is convex and continuous on (cf. [25, Theorem 1.2]). Thus, is also w.l.s.c. on since the inclusion from to is compact.

Choose such that , where . By an easy estimation, (3.18) and (3.19) as in Lemma 3.5, we have

for every and .

Let satisfy . Then, the following inequality follows from Lemma 2.2:

where is a positive constant independent of with and .

For every such that , we obtain

by the definition of , Lemma 2.1 and , where is a constant obtained by Lemma 2.1.

Consequently, if we choose a satisfying , then we obtain positive constants and (independent of ) such that

for every by (3.21), (3.22), and (3.23). Because of , our conclusion is shown.

Lemma 3.7. Let in and . If holds, then is bounded from below, coercive and w.l.s.c. on .

Proof. First, as the same reason in Lemma 3.6, it follows that is w.l.s.c. on . By a similar argument to Lemma 3.6, for every and where , we obtain

for every (note ). Here, from Lemma 2.3,

for every follows, where , and is a constant obtained in Lemma 2.3. Therefore, by choosing a such that , we can prove our conclusion.

Lemma 3.8. Let and . Then, is bounded from below on , where is the set introduced in (3.14).

Proof. Due to the same inequalities concerning and as in Lemma 3.5, for every and , there exists such that

for every , where . Choose positive constants and such that (note ).

First, we consider the case of in . For every , we obtain

by Lemma 2.3 with (note and ).

Next, we handle with the case where changes sign. Let satisfy . Then, we have for such

by Lemma 2.2, where and .

On the other hand, for with , the following inequality follows from Lemma 2.2:

Consequently, by (3.27), (3.29), (3.28), and (3.30), there exists independent of such that

for every . Hence, our conclusion is shown by taking satisfying .

Proof of Theorem 1.1 in the Case . First, if either on and or (i.e., ) holds, then Lemma 3.7 or Lemma 3.6 guarantees the existence of a global minimizer of , respectively (cf. [25, Theorem 1.1]). Hence, has a solution.

Since is an eigenvalue of if and only if is one of , it suffices to consider the case of . Furthermore, by Proposition 2.9, Remark 2.6 (i), and our hypothesis that is not an eigenvalue of , we may assume that there exists a such that . By Lemmas 3.5 and 3.8, we can choose and satisfying

Put

Then, it follows from Lemma 3.4 and (3.32) that holds. Since satisfies the Palais-Smale condition by Proposition 3.3, the minimax theorem guarantees (cf. [25, Theorem 4.6]) that is a critical value of . Hence, has at least one solution.

3.4. The Case

First, we introduce an approximate functional as follows:

Lemma 3.9. Let . Then, there exists an such that for each , is bounded from below on , where is the set introduced in (3.14).

Proof. Choose such that . Then, for every , Lemma 3.8 guarantees that bounded from below on because of and .

Proof of Theorem 1.1 in the Case . By noting that and , we may assume that for some . Let be a natural number obtained by Lemma 3.9. Due to Proposition 2.10 (i) and (ii), there exists an such that

for every . Thus, for every , we can take and satisfying

by applying Lemmas 3.5 and 3.9 to (note (3.35)). Set

for each . Then, for each , we can obtain satisfying

by applying Ekeland's variational principle to each (refer to [25, Theorem 4.3]). In addition, we can see that is bounded in . Indeed, if there exists a subsequence satisfying as , then we can show that is an eigenvalue of by the same argument as in Proposition 3.3. This contradicts to our assumption that is not an eigenvalue of . Moreover, the boundedness of follows from a similar inequality to (3.6) as in Proposition 3.3 under the boundedness of .

Therefore, we may assume, by choosing a subsequence that is a Palais-Smale sequence of since is bounded on a bounded set and according to the following inequality:

Therefore, because satisfies the Palais-Smale condition by Proposition 3.3, has a critical point, whence has at least one solution.

4. Proof of Theorem 1.2

First, we will prove the following result concerning the Palais-Smale condition under the additional hypothesis () or ().

Proposition 4.1. Assume that one of the following conditions hold:

(i) and () or (); (ii) and one of (), (), () and ().

Then, satisfies the Palais-Smale condition.

Proof. As the same reason in Proposition 3.3, it suffices to prove the boundedness of a Palais-Smale sequence such that (for some ) and as . By way of contradiction, we may assume that as by choosing a subsequence. Set . Then, by the same argument as in Proposition 3.3, has a subsequence strongly convergent to being a nontrivial solution of

To simplify the notation, we denote the above subsequence strongly convergent to by , again. Thus, as for a.e. (note ).

Assume () or (). Then, we can obtain

Indeed, it follows from () that there exist and independent of such that if and a.e. , and for every and a.e. . Therefore, since a.e. and (note ), we have (4.2) if () holds, by applying Fatou's lemma to the following inequality:

In the case of (), by considering instead of as in the above argument, we can show our claim (4.2).

Furthermore, by Hölder's inequality, we have

in the case of () because in , where and are constants as in (). Similarly, we obtain

in the case of ().

Hence, we have a contradiction because of (4.2), (4.4), or (4.5) by taking a limit inferior or superior in the following equality:

where we use the fact that is bounded because of .

Assume and () or (): because is a nontrivial solution of (4.1) with , is not a constant function, that is, . Therefore, we have as for a.e. . Because of , we can show

by a similar argument to one for in the above. In addition, we can easily obtain the following inequality:

in the case of (), respectively. Hence, we have a contradiction by considering .

By a similar way to the case , we introduce the following approximate functionals on :

Note on provided .

Proposition 4.2. If either and () or () (resp., either and () or ()) and satisfies

then is bounded in .

Proof. First, we note that the boundedness of guarantees that is bounded by (refer to (3.6) as in the proof of Proposition 3.3). Moreover, because of the following equality:

we can prove the boundedness of by the same argument as in Proposition 4.1.

Proof of Theorem 1.2. Because of , we may assume . In the case where and for some , the proof of Theorem 1.1 implies the existence of a critical point of because satisfies the Palais-Smale condition by Proposition 4.1. Concerning other cases, in the next section, we will prove the existence of a bounded sequence satisfying or in as . Because is bounded on a bounded set, we may assume that converges to some by choosing a subsequence. In addition, by noting the inequality , we easily see that is a bounded Palais-Smale sequence of . Therefore, has a critical point since satisfies the Palais-Smale condition by Proposition 4.1.

5. Construction of a Bounded Palais-Smale Sequence

In this section, due to the reason stated in the proof of Theorem 1.2, we will construct a bounded sequence satisfying or in as . It implies the existence of a bounded Palais-Smale sequence of .

5.1. The Case

Assume ()

In this c ase, we can show that for each , has a global minimizer . Indeed, for , there exists such that for every by (1.1), (1.16) and (refer to the inequality as in the proof of Lemma 3.5). This means that is coercive and bounded from below on . Therefore, has a global minimizer since is w.l.s.c. on as the same reason in Lemma 3.6.

Furthermore, because of in and , it follows from Proposition 4.2 that is bounded.

Assume ()

Choose such that , where is the second eigenvalue of (so the weight function and see (2.22) for the definition). Then, by noting that , we have

by Lemma 3.8, where is a subset defined by (3.14) with the weight , that is,

Moreover, for every holds because for every . Since as by (1.1), there exists such that .

Define

for . By the definition of , we easily see that for every (refer to [6] or Lemma 3.4). Hence,

holds, whence is bounded from below. Moreover, by applying Ekeland's variational principle to each , we can obtain a sequence satisfying and . Since is bounded from below, it follows from Proposition 4.2 that is bounded. As a result, we can construct a bounded sequence satisfying as in .

5.2. The Case with

Assume () or ()

Since we see that and (according to Lemma 2.5), is coercive, bounded from below and w.l.s.c. on by Lemma 3.6. Thus, we obtain a global minimizer of for sufficiently large such that . Because of for every , Proposition 4.2 guarantees that is bounded.

Assume () or ()

First, we note that and by Lemma 2.5 for sufficiently large such that . Moreover, it follows from Proposition 2.10 and that there exists an satisfying and

for every . By applying Theorem 1.1 to each case of a weight (note that is not an eigenvalue of by (5.5), there exists satisfying (note ) and

where the last inequality follows from Lemma 3.4 with . On the other hand, because for every and , we have

for every , where the last inequality follows from Lemma 3.8. Thus, is bounded from below. Hence, Proposition 4.2 guarantees the boundedness of .

5.3. The Case with

Assume () or ()

We may assume by taking anew if necessary (note that we have already proved the case of in Section 4). Here, we can choose an such that , and

for every by and Proposition 2.10 (i), (iii). Note the following inequality:

for every and , where the last inequality is obtained by . Let . It follows from Lemma 3.8 and (5.8) that is bounded from below on . Hence, (5.9) yields that is also bounded from below on , namely,

On the other hand, because of (see (5.8)), Lemma 3.5 guarantees the existence of satisfying

Thus, for each , we can take such that

(note (5.9) for the first inequality). Set

for . Since for every by Lemma 3.4 and , we have . Therefore, Ekeland's variational principle (refer to [25, Theorem  4.3]) guarantees the existence of satisfying and .

Finally, to show the boundedness of due to Proposition 4.2, we will prove that is bounded from above. For each , we define a continuous map from to by

Then, holds. This leads to

because of (5.9), (5.11), and the compactness of .

Assume () or ()

Because the case of is already shown (see Sections 5.1 and 5.2), We may assume for some by taking anew if necessary. Here, we can choose an such that and

for every by and Proposition 2.10 (i), (iii). Moreover, we note the following inequality:

for every and . It follows from Lemma 3.8 and (5.16) (note (5.17) also) that is bounded from below on with . Hence, (5.17) implies

for every . Because of , there exist and such that

by Lemma 3.5. Define

for . Then, occurs (see (5.18)) since for every by Lemma 3.4. This means that is bounded from below. Consequently, we can obtain a desired bounded sequence by the same argument as in Sections 5.1 and 5.2.

5.4. The Case (iii) as in Theorem 1.2

First, note that we are assuming the hypothesis () or () in this case (iii). In addition, as the reason in the proof of Theorem 1.2, it suffices to handle with .

Let satisfy . According to Proposition 2.10 (i) and (ii), we can take an such that and

for every . The following inequality follows from the easy estimates:

for every and . Let and set . Because of (5.21), Lemma 3.8 implies that is bounded from below on with (note ). Hence, (5.22) yields that

for each . On the other hand, because of (see (5.21)), Lemma 3.5 guarantees the existence of satisfying

Therefore, for each , we can choose such that

(note (5.22) for the first inequality). Set

for . Since for every by Lemma 3.4, we have . Moreover, by the same argument as in Section 5.3 (note (5.24)), we have

and hence our conclusion is shown.

Remark 5.1. If holds, then we can not show the continuity of with respect to (refer to Proposition 2.10). Hence, we are not able to construct a bounded Palais-Smale sequence under () or (). However, if we have the additional information about the existence of a suitable such that in , and when occurs, then we can still easily prove that equation has a solution in the case also where , and () or (). In fact, let for some . Note the following inequality:

for every and . Fix such that and . Set . Because of (the last inequality follows from Proposition 2.10 (i)), Lemma 3.8 implies that is bounded from below on (note ). By combining this fact and (5.28), we have

Because of , for each , we can take a satisfying

as by Lemma 3.5.

Since any extension of () links by Lemma 3.4, we can construct a desired sequence by the same argument as in Section 5.3 under () or ().

Acknowledgments

The author would like to express her sincere thanks to Professor Shizuo Miyajima for helpful comments and encouragement. The author thanks the referees for his helpful comments and suggestions.

References

D. Motreanu and N. S. Papageorgiou, “Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator,” Proceedings of the American Mathematical Society, vol. 139, no. 10, pp. 3527–3535, 2011.

L. Damascelli, “Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results,” Annales de l'Institut Henri Poincaré, vol. 15, no. 4, pp. 493–516, 1998.

D. Motreanu, V. V. Motreanu, and N. S. Papageorgiou, “Papageorgiou, multiple constant sign and nodal solutions for nonlinear neumann eigenvalue problems,” Annali della Scuola Normale Superiore di Pisa, vol. 10, no. 5, pp. 729–755, 2011.

S. Miyajima, D. Motreanu, and M. Tanaka, “Multiple existence results of solutions for the Neumann problems via super- and sub-solutions,” Journal of Functional Analysis, vol. 262, pp. 1921–1953, 2012.

S. T. Robinson, “On the second eigenvalue for nonhomogeneous quasi-linear operators,” SIAM Journal on Mathematical Analysis, vol. 35, no. 5, pp. 1241–1249, 2004.

M. Tanaka, “The antimaximum principle and the existence of a solution for the generalized -Laplace equations with indefinite weight,” Differential Equations & Applications, vol. 4, no. 4, 2012.

T. Godoy, J.-P. Gossez, and S. Paczka, “On the antimaximum principle for the -Laplacian with indefinite weight,” Nonlinear Analysis, vol. 51, no. 3, pp. 449–467, 2002.

A. Anane and A. Dakkak, “Nonresonance conditions on the potential for a Neumann problem,” in Partial Differential Equations, vol. 229 of Lecture Notes in Pure and Applied Mathematics, pp. 85–102, Dekker, New York, NY, USA, 2002.

J.-P. Gossez and P. Omari, “A necessary and sufficient condition of nonresonance for a semilinear Neumann problem,” Proceedings of the American Mathematical Society, vol. 114, no. 2, pp. 433–442, 1992.

P. Omari and F. Zanolin, “Nonresonance conditions on the potential for a second-order periodic boundary value problem,” Proceedings of the American Mathematical Society, vol. 117, no. 1, pp. 125–135, 1993.

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, vol. 34 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1995.

Y. Chen and M. Wang, “Large solutions for quasilinear elliptic equation with nonlinear gradient term,” Nonlinear Analysis, vol. 12, no. 1, pp. 455–463, 2011.

P. Y. H. Pang, Y. Wang, and J. Yin, “Periodic solutions for a class of reaction-diffusion equations with -Laplacian,” Nonlinear Analysis, vol. 11, no. 1, pp. 323–331, 2010.

G. Jia, Q. Zhao, and C.-Y. Dai, “Singular quasilinear elliptic problems with indefinite weights and critical potential,” Acta Mathematicae Applicatae Sinica, vol. 28, pp. 157–164, 2012.

G. Zhang, S. Man, and W. Zhang, “On a class of critical singular quasilinear elliptic problem with indefinite weights,” Nonlinear Analysis, vol. 74, no. 14, pp. 4771–4784, 2011.

P. Drábek and S. B. Robinson, “Resonance problems for the p-Laplacian,” Journal of Functional Analysis, vol. 169, no. 1, pp. 189–200, 1999.

M. Struwe, Variational Methods, Springer, New York, NY, USA, 1999.

S. E. Habib and N. Tsouli, “On the spectrum of the -Laplacian operator for Neumann eigenvalue problems with weights,” Electronic Journal of Differential Equations, vol. 2005, pp. 181–190, 2005.

M. Arias, J. Campos, M. Cuesta, and J.-P. Gossez, “An asymmetric Neumann problem with weights,” Annales de l'Institut Henri Poincaré, vol. 25, no. 2, pp. 267–280, 2008.

J.-N. Corvellec, “A general approach to the min-max principle,” Zeitschrift für Analysis und ihre Anwendungen, vol. 16, no. 2, pp. 405–433, 1997.

K. Perera, R. P. Agarwal, and D. O'Regan, Morse Theoretic Aspects of p-Laplacian Type Operators, vol. 161 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2010.

A. Szulkin, “Ljusternik-Schnirelmann theory on -manifolds,” Annales de l'Institut Henri Poincaré, vol. 5, no. 2, pp. 119–139, 1988.

G. M. Lieberman, “Boundary regularity for solutions of degenerate elliptic equations,” Nonlinear Analysis, vol. 12, no. 11, pp. 1203–1219, 1988.

E. Casas and L. A. Fernandez, “A Green's formula for quasilinear elliptic operators,” Journal of Mathematical Analysis and Applications, vol. 142, no. 1, pp. 62–73, 1989.

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, NY, USA, 1989.