#### Abstract

Assume that is a positive continuous function in and satisfies some suitable conditions. We prove that the quasilinear elliptic equation in admits at least two solutions in (one is a positive ground-state solution and the other is a sign-changing solution).

#### 1. Introduction

For , and we consider the quasilinear elliptic equations

where is the -Laplacian operator, that is,

Let be a positive continuous function in and satisfy

Associated with (1.1) and (1.2), we define the functionals , and for

It is easy to verify that the functionals , and are

For the case Lions [1, 2] proved that if and then (1.1) has a positive ground-state solution in Benci and Cerami [3] proved that (1.2) does not have any ground-state solution in an exterior domain. Bahri and Li [4] proved that there is at least one positive solution of (1.1) in (or an exterior domain) when and for Cao [5] has studied the multiplicity of solutions (one is a positive ground-state solution and the other is a nodal solution) of (1.1) with Neumann condition in an exterior domain as follows. Assume that and for , , then (1.1) has at least two nontrivial solutions (one is a positive ground-state solution and the other is a nodal solution) in an exterior domain.

This article is motivated by the above papers. If is a positive continuous function in and satisfies (Q1), then we prove that (1.1) admits a positive ground-state solution in Combine it with some ideas of Cerami et al. [6] to show that if also satisfies for then a nodal solution of (1.1) exists.

#### 2. Preliminaries

We define the Palais-Smale (denoted by (PS)) sequences and (PS)-conditions in for as follows

Definition 2.1. (i) For a sequence is a -sequence in for if and strongly in as where is the dual space of and
(ii) satisfies the -condition in if every -sequence in for contains a convergent subsequence.

Lemma 2.2. Let and let be a -sequence in for then is a bounded sequence in Moreover, as and

Proof. Since we have that if and if For sufficiently large we get It follows that is bounded in Then as Thus, that is, as and

Define

where , and

where

Lemma 2.3. Let be a sign-changing solution of (1.1) Then

Proof. Define and Since is a sign-changing solution of (1.1), then is nonnegative and nonzero. Multiply (1.1) by and integrate it to obtain that is, and Similarly, Hence,

Lemma 2.4. (i) For each there exists a positive number such that and
(ii) Let and let be a sequence in for such that and Then there is a sequence in such that , and as

Proof. (i) For each and let Thus, Define then that is,
(ii) By (i), there exists a sequence in such that that is, for each Since and we have that Hence, as

Lemma 2.5. There exists such that for each where is independent of

Proof. For each by the Sobolev inequality, we obtain that This implies that for each

By Lemma 2.5,

Lemma 2.6. Let such that then is a nonzero solution of (1.1) in

Proof. Suppose that then Since by the Lagrange multiplier theorem, there is a such that in Then we have Thus, and in Therefore, is a nonzero solution of (1.1) in with

Lemma 2.7. There is a -sequence in for

Proof. Let be a minimizing sequence of Applying the Ekeland principle, there exists a sequence such that , and strongly in as Let for each then Thus, there exists a sequence such that where as Since we have that Hence, as This implies that strongly in as that is, is a -sequence in for

Remark 2.8. The above definitions and lemmas also hold for , and

#### 3. Existence of a Ground-State Solution

Using the arguments by Lions [1, 2], Benci and Cerami [3], Struwe [7], and Alves [8], we have the following decomposition lemma.

Lemma 3.1 (Palais-Smale Decomposition Lemma for ). Assume that is a positive continuous function in and Let be a -sequence in for Then there are a subsequence a positive integer sequences in functions in and in for such that In addition, if then and for

Lemma 3.2. Let be a -sequence in for with Then there exist a subsequence and a nonzero such that strongly in and that is, satisfies the -condition in

Proof. Since is a -sequence in for with by Lemma 2.2, is bounded in Thus, there exist a subsequence and such that weakly in It is easy to check that is a solution of (1.1) in Applying Palais-Smale Decomposition Lemma 3.1, we get Then and Hence, strongly in and

Let be the positive ground-state solution of (1.2) in Using the same arguments by Li and Yan [9] and Marcos do Ó [10, Lemma 3.8], or see Serrin and Tang [11, page 899] and Li and Zhao [12, Theorem 1.1], we obtain the following results:

(i) for some and (ii)for any there exist positive numbers and such that where

Remark 3.3. Similarly, we also show that all positive solutions of (1.1) in have exponential decay.

By Lemma 3.2, we can prove the following theorem.

Theorem 3.4. Assume that is a positive continuous function in and satisfies (Q1). Then there exists a positive ground-state solution of (1.1) in

Proof. Let be the positive ground-state solution of (1.2) in then is a minimizer of and By Lemma 2.4(i), there exists a positive number such that that is, Since on a set of positive measure, we can deduce that Therefore, Applying Lemma 3.2, there exists such that From the results of Lemmas 2.6 and 2.3, by Maximum Principle, is a positive ground-state solution of (1.1) in

#### 4. Existence of a Nodal Solution

In this section, assume that is a positive continuous function in and satisfies (Q1). In order to prove Lemma 4.8, also satisfies the following condition (Q2): there exist some constants and such that

Let be a functional in defined by

We define

where and

Lemma 4.1. (i) If changes sign, then there exist positive numbers such that and
(ii) There exists such that for each

Proof. (i) Since and are nonzero and nonnegative, by Lemma 2.4(i) it is easy to obtain the result.
(ii) For each by Lemma 2.4(i), there exists such that Then By Lemma 2.5, we have Hence, for each

Consider these minimization problem

By Lemma 4.1,

Lemma 4.2. There exists a sequence such that and strongly in as

Proof. It is similar to the proof of Zhu [13].

Lemma 4.3. Let and be real-valued functions in If in then one has the following inequalities:(i)(ii)(iii)(iv)

Lemma 4.4. Let be a -sequence in for satisfying Then there exists such that converges to strongly in and is a higher-energy solution of (1.1) such that

Proof. By the definition of the -sequence in for it is easy to see that is a bounded sequence in and satisfies By Lemma 4.1(ii) there exists such that Using the Palais-Smale Decomposition Lemma 3.1, then we have where is a solution of (1.1) in and is a solution of (1.2) in Since for each and we have Now we want to show that On the contrary, suppose that (i) is a sign-changing solution of (1.2) by Lemma 2.3 and Remark 2.8, we have which is a contradiction.(ii) is a constant-sign solution of (1.2) we may assume that Applying the Decomposition Lemma 3.1, there exists a sequence in such that and By the Sobolev continuous embedding inequality, we obtain Since by Lemma 4.3, then (a)Suppose that we obtain as Then which is a contradiction.(b)Suppose that We have which is a contradiction.
By (i) and (ii) then Thus, as and Finally, we claim that is a sign-changing solution of (1.1) in If by Lemma 4.3, then Similarly, we have the inequality (4.12) which is a contradiction. Moreover, by Lemma 2.3,

Recall that is the positive ground-state solution of (1.2) in For any there exist positive numbers and such that

where Define

Clearly,

Lemma 4.5. There are and real numbers and such that for where , and is the positive ground-state solution of (1.1) in

Proof. Applying the mean value theorem by Miranda [14], the proof is similar to that of Zhu [13] (or see Hsu [15, page 728]).

We need the following lemmas to prove that for sufficiently large

Lemma 4.6. Let be a domain in If satisfies then

Proof. Since we have Since as then the lemma follows from the Lebesgue-dominated convergence theorem.

Lemma 4.7. For all one has the following inequality:

Proof. See Yang [16, Lemma 4.2.].

Lemma 4.8. There exists an such that for where is a positive ground-state solution of (1.1) in

Proof. By Lemma 4.7, then Since and using the inequality for any , and some positive constant then (i)Since for some constants and by Lemma 4.6, we have that there exists an such that for (ii)Applying Lemma 4.6, there exists an such that for Similarly, we also obtain that there exists an such that for By (i) and (ii) choosing we can find an such that for

Theorem 4.9. Assume that is a positive continuous function in and satisfies (Q1) and (Q2), then (1.1) has a positive solution and a nodal solution in

Proof. By Lemmas 4.2, 4.4, 4.5, and 4.8 and Theorem 3.4, we obtain the proof.