Existence of Multiple Solutions for a p-Kirchhoff Problem with Nonlinear Boundary Conditions

The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem −M(∫Ω | x|−ap | ∇u|p)div(|x|−ap | ∇u|p−2∇u) = λh(x) | u|r−2 u, x ∈ Ω, M(∫Ω | x|−ap | ∇u|p) | x|−ap | ∇u|p−2 (∂u/∂ν) = g(x) | u|q−2 u, on ∂Ω, where 1 < (N + 1)/2 < p < N, a < (N − p)/p. By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.


Introduction and Main Result
In this paper, we consider the existence of multiple solutions for the singular elliptic problem: where 1 < ( + 1)/2 < < , < ( − )/ , > 0, Ω is an exterior domain of R : that is, and Ω = R \ , where is a bounded domain in R with the smooth boundary ( = Ω), and 0 ∈ Ω. ( ) and ℎ( ) are continuous functions, ( ) = + with the parameters , > 0. Problem like (1) is usually called nonlocal problem because of the presence of the integral over the entire domain, and this implies that (1) is no longer a pointwise identity. In fact, such kind of problem can be traced back to the work of Kirchhoff. In [1], Kirchhoff investigated the model of the form where , 0 , ℎ, , and are all positive constants. This equation extends the classical d' Alembert's wave equation by considering the effects of changes in the length of the strings during the vibrations. Problem (1) is related to the stationary analogue of problem (2). After Kirchhoff 's work, various models of Kirchhoff-type have been studied by many authors: we refer the readers to [2][3][4][5][6][7][8][9]. In [4], by the variational methods, Bensedik and Bouchekif considered the problem where Ω is a bounded domain in R . One of the assumptions made on ( , ) in (3) is that (f 1 ) ( , ) is continuous function on Ω × R such that The authors proved that problem (3) has a positive solution or has no solution when some other assumptions are fulfilled. In our paper, however, the weight functions ℎ( ) and ( ) are permitted to change sign. Thus, the methods in [4] cannot be directly applied on (1).

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The Scientific World Journal In recent years, some other authors considered the Kirchhoff-type equations with -Laplacian [10][11][12][13]. In fact, motivated by [4,5] and our previous work [14], we consider the existence of multiple solutions for problem (1) on the Nehari manifold by variational methods. We prove that problem (1) has at least two positive solutions. Since Ω ⊂ R is an unbounded domain and the problem is singular, the loss of compactness of the Sobolev embedding renders variational technique more delicate.
In order to state our result, we introduce a weighted Sobolev space = 1, (Ω), which is the completion of the space ∞ 0 (Ω) with the norm of For ≥ 1 and = ( ) > 0 in Ω, we define the space (Ω, ) as being the set of Lebesgue measurable functions : Ω → R 1 , which satisfies The following weighted Sobolev-Hardy inequality is due to Caffarelli et al. [15], which is called the Caffarelli-Kohn-Nirenberg inequality. There is a constant 1 > 0 such that where −∞ < < ( − )/ , ≤ < + 1, = + 1 − , and * = /( − ). Throughout this paper, we make the following assumptions: Now, we give the definition of weak solution for problem (1).
Our main result is in the following.
This paper is organized as follows. In Section 2, we give some properties of the Nehari manifold and set up the variational framework for problem (1). In Section 3, we consider the multiplicity results and prove Theorem 2.

Preliminary Results
It is clear that problem (1) has a variational structure. Let ( ) : → R 1 be the corresponding Euler functional of problem (1), which is defined by wherê( ) = ∫ 0 ( ) . Then, we see that the functional ( ) ∈ 1 ( , R 1 ), and for ∀ ∈ , there holds Particularly, we have It is well known that the weak solution of problem (1) is the critical point of ( ). Thus, to prove the existence of weak solutions for problem (1), it is sufficient to show that ( ) admits a sequence of critical points. Since ( ) is not bounded below on , it is useful to consider the functional ( ) on the Nehari manifold [17,18]: where ⟨, ⟩ denotes the usual duality. Then, it follows from (12) that ∈ if and only if The Scientific World Journal 3 Then, we get from (10)- (14) that We define Then, (14) implies that It is natural to split into three parts: Now, we give some important properties of + , − , and 0 .
Proof. Suppose that there exists ∈ 0 . If > 2 , then it follows from (19) and (21) that which implies that .
The Scientific World Journal On the other hand, we can similarly get from (20) and (21) that which yields that .

Lemma 5.
If 0 is a local minimizer of ( ) on and 0 ∉ 0 , then 0 is a critical point of ( ).
Thus, (71) implies that that is, Since Ω is compact and ∈ ( Ω) ∩ ∞ ( Ω), we obtain by the trace embedding theorem in [20] that This concludes the proof.

Existence of Solutions
In this part, we will give the proof of the existence of nonnegative and nontrivial solutions. Before this, we need to prove the following two important lemmas.
Similar to the proof of Lemma 10, we may assume that ⇀ − 0 weakly in . For ∈ − , we deduce by Lemma 6 and (15) that ∫ Ω ( )| | > 0; furthermore, ∫ Ω ( )| − 0 | > 0. We also want to prove that → − 0 strongly in . In fact, if not, we have By virtue of Lemma 7, there exists − 0 > 0 such that and a simple transformation shows that Therefore, Lemma 9 together with (84)-(87) gives that which is a contradiction, and we complete the proof.

Conclusions
The object of this paper is to prove the existence of multiple solutions for the nonlinear Kirchhoff-type problem (1). By the variational methods, we discuss the problem on the Nehari manifold and give the sufficient conditions for the existence of solutions. We overcome the difficulty due to the loss of compactness on the unbounded domain. In the future work, we are interested to consider similar problems, but the term on the right will be replaced by abstract functions.