MULTIPLICITY OF SOLUTIONS FOR A CLASS OF QUASILINEAR PROBLEM IN EXTERIOR DOMAINS WITH NEUMANN CONDITIONS

We study the existence and multiplicity of solutions for a class 
of quasilinear elliptic problem in exterior domain with Neumann 
boundary conditions.


Introduction
In this paper, we are concerned with the existence and multiplicity of solutions for the following class of quasilinear elliptic problem with Neumann conditions: where Ω ⊂ R N is a bounded domain with smooth boundary, 1 < p < N, and ∆ p u is the p-Laplacian operator, that is, Q is a continuous function satisfying (1.3) and the nonlinearity f : R → R is an odd function of C 1 class satisfying the following hypotheses.
In [6], Cao also studied the problem (1.1) for p = 2, f (u) = |u| η−1 u, and Q satisfying the condition (1.3). The author showed that the problem has at least two solutions, where the first solution is related to the minimization problem (1.6) and the second solution is nodal, that is, a solution of (1.1) with change of sign. In that paper, one of the main points is a compactness global result proved in [5].
In this work, motivated by [6], we prove the existence of ground-state and nodal solutions to (1.1). We used variational methods such as mountain pass theorem without Palais-Smale condition (see [14]) to obtain a positive ground-state solution. In relation to nodal solutions, we apply the implicit function theorem. Here, we adapt to p-Laplacian operator and to a general nonlinearity f some ideas found in [5,6,13]. However, the arguments explored in the above articles cannot be carried out straightforwardly in our case because some estimates become more subtle to be established. A main point in this paper is a version of a compactness global lemma (CGL) to study the behavior of Palais-Smale sequences, which is a version for p-Laplacian from a result shown by Benci and Cerami in [5].
To state our main results, we need some definitions and notations.
If h is a Lebesgue integrable function and B is a measurable set, we write B h for B hdx. Moreover, if h ∈ W 1,p (R N \ Ω), we denote by h its usual norm. We denote by I : W 1,p (R N \ Ω) → R the functional related to (1.1) given by where F(u) = u 0 f (t)dt. We have the following problem: and by I ∞ : W 1,p (R N ) → R the functional related to (1.8) given by where C is a positive constant and m > p(q + 1)/((q + 1) − p). Then, (1.1) has a positive ground-state solution.
Using the ground-state obtained in the above theorem together with some estimates given in Sections 4 and 5, we establish a second theorem which shows the existence of a nodal solution. For this result, we will need the following hypothesis: (1. where C is a positive constant and γ < q/(q + 1). Then, (1.1) has a nodal solution.
Remark 1.3. In the proof of Theorems 1.1 and 1.2, we used variational methods and adapted some arguments explored by Cao in [6]. These results complete the study made in [6] in the sense that we consider the p-Laplacian operator and a general class of nonlinearity.

Technical lemmas
In this section, we state some results necessary for the proof of Theorems 1.1 and 1.2. It is known that, under assumptions (f 1 ), (f 2 ), and (f 3 ), the arguments used in [3] show that (1.8) possesses a ground-state solution. About the behavior of the solutions at infinity, we have the following result.
Proof. The proof follows by similar arguments found in [11, Theorem 3.1].

Remark 2.2.
With the same arguments used in the proof of the above lemma, we can show that all positive weak solutions of (1.1) have exponential decaying.
254 Multiplicity of solutions for a quasilinear problem The next lemma shows an important inequality related to the vectors of R N , and its proof can be found in [15,Lemma 4.2].
Lemma 2.4. Let F ∈ C 2 (R,R + ) be a convex and even function such that F(0) = 0 and f (s) = F (s) ≥ 0 for all s ∈ [0,∞). Then, for all u,v ≥ 0, Proof. Indeed, we have two cases to consider. If v ≤ u, by convexity, we have If u ≤ v, we repeat the above argument to find From (2.6) and (2.7) the lemma follows.

Behavior of the Palais-Smale sequence
In this section, we prove some important lemmas to establish the CGL. The CGL is a key result for the understanding of the behavior of Palais-Smale sequence. We recall that a sequence ( be an open set and g n : Proof. We will show only (I) because the same arguments can be used in the proof of (II). We begin remarking that hence, by (f 1 ), that is, For each > 0, we obtain using Young's inequality that We consider the function G ,n given by Therefore, by Lebesgue's theorem, we have From the definition of G ,n , it follows that Thus, we obtain the following inequality 14) The next result can be found in [2].
..,k, and nontrivial solutions u 1 ,...,u k of the problem (1.8), such that Proof. The arguments used in this proof follow the same ideas found in [2,5]. The sequence (u n ) is bounded, thus there exists u 0 ∈ W 1,p (R N \ Ω) such that Adapting arguments found in [1,9,10,15], it follows that I (u 0 ) = 0. Define the function Then (3.20) It follows, using Lemmas 2.4 and 3.2, that Suppose that Consequently, by (f 1 ), (f 2 ), and (f 3 ), there exists α > 0 such that Now, we decompose R N into N-dimensional unit hypercubes Q i with vertex having integer coordinates and put It follows from Sobolev imbeddings and the last equality that {y 1 m } is unbounded, that is, Using (3.22) and the fact that D 1 m → R N , we conclude that u 1 is a nontrivial solution of (1.8). Define If Ψ 2 m (· − y 1 m ) → 0, the theorem is finished, otherwise for the contrary case, we repeat the arguments and we will find u 1 ,u 2 ,...,u k nontrivial solutions for (1.8) and sequences (y j m) with |y j m| → ∞ such that Notice that there exists ξ > 0 verifying where Inequality (3.33) along with (3.32) tell us that the iteration must finish at some index k ∈ N. This completes the proof of this lemma. where c ∞ is the mountain pass level of the energy functional associated to (1.8).

Existence of ground-state solution
In this section, we will prove the existence of a positive ground-state solution for the functional I. To this end, we suppose that f (t) = 0 as t ≤ 0. The first lemma is related to the mountain pass geometry, and its proof uses well-known arguments.  It is not difficult to see that (γ n ) is bounded and therefore γ n → γ o for some subsequence still denoted by (γ n ). We claim that γ o = 1. In fact, since |x n | → ∞, it follows from (4.8) that Sinceū is a ground-state, we get (4.10) Therefore, by (f 3 ), we have that γ o = 1. From (f 1 ), we obtain Fix r n ∈ (0,n) and observe that (4.17) Consequently, using the estimates obtained, s n t n ≤ C 8 e pan e bn(η+1) + e pna e mrn + e pan n N e (n−rn)(q+1)b . (4.18) Since a/b → 1 as δ → 0 (see Lemma 2.1), there exists > 0 such that . (4.19) Choosing r n = n(1 − p(a + )/b(q + 1)), we obtain s n /t n → 0 and hence c 1 < c ∞ .

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Proof of Theorem 1.1. It follows from Corollary 3.4 and mountain pass theorem (see Ambrosetti and Rabinowitz [4]) that I has a critical point u 1 in the level c 1 . We claim that u 1 is nonnegative. Indeed, we know that I (u 1 )u 1 − = 0, thus Hence u − 1 = 0. Using the strong maximum principle, we have u 1 > 0 in R N \ Ω. Thus, we conclude that u 1 is a ground-state solution.

Existence of nodal solution
In this section, we will show that there is a solution of (1.1) that changes sign. Here, we adapt for our case some arguments explored by Cerami et al. [7] (see also Cao [6] and Noussair and Wei [13]). We start with some notations. Consider the closed set Using well-known arguments, we can show that there exists a constant µ 1 > 0 verifying Proof. It is easy to verify that I is bounded from below on ᏹ. Hence we may apply the Ekeland variational principle to obtain a minimizing sequence {u n } ⊂ ᏹ for c satisfying c ≤ I u n ≤ c + 1 n , (5.5) Using standard arguments, we have that u n is bounded. We claim that To this end, for each ϕ ∈ W 1,p (R N \ Ω) and n ∈ N, we introduce the functions h i n : R 3 → R, i = 1,2, given by This shows that for t ∈ (−δ n ,δ n ), v n = u n + tϕ + s n (t)u + n + l n (t)u − n ∈ ᏹ.
From the boundedness of u n in W 1,p (R N \ Ω) and (5.11), it follows that {s n (0)} is bounded. A similar argument can be applied for the sequence {l n (0)} to conclude that it is also bounded. From (5.6), we have which implies that Then, for all ϕ ∈ W 1,p (R N \ Ω) with ϕ ≤ 1, we get Proof. Letū be a ground-state of (1.8). Defineū n (x) =ū(x − x n ) and u n = αu 1 − βū n , where u 1 is a positive ground-state of (1.1), x n = (0,...0,n), α,β > 0. Consider the functions Thus, for n large enough, we get By the mean value theorem (see [12]), we have α * , β * such that 1/ p ≤ α * , β * ≤ p, where Since u 1 is a solution of (1.1) andū n is related with a ground-state of (  As an immediate consequence of Lemma 3.3 and the last proposition, we get the following lemma. Lemma 5.3. Let (u n ) ⊂ ᏹ be the sequence obtained in Lemma 5.1. Then (u n ) has a subsequence converging strongly in W 1,p (R N \ Ω).
Proof. It is easy to see that (u n ) is bounded in W 1,p (R N \ Ω). Denote by u the weak limit of (u n ) in W 1,p (R N \ Ω). Thus, either u n → u in W 1,p (R N \ Ω) or there exist k functions u j with 1 ≤ j ≤ k satisfying Lemma 3.3. It is clear that k ≤ 1. Suppose that u ≡ 0. Since c 2 > 0, we have k = 1 and hence, u is a nodal solution of (1.1).