KIRCHHOFF TYPE PROBLEMS WITH POTENTIAL WELL AND INDEFINITE POTENTIAL. 1. Introduction In this article, we will study the Kirchhoff type problem

Electronic Journal of Differential Equations, Vol. 216 (216), No. 178, pp. 1 13. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu KIRCHHOFF TYPE PROBLEMS WITH POTENTIAL WELL AND INDEFINITE POTENTIAL YUANZE WU, YISHENG HUANG, ZENG LIU Abstract. In this article, we study the Kirchhoff type problem Z α u 2 dx + 1 u + (λa(x) + a )u = u p 2 u in, u H 1 ( ), where 4 < p < 6, α and λ are two positive parameters, a R is a (possibly negative) constant and a(x) is the potential well. Using the variational method, we show the existence of nontrivial solutions. We also obtain the concentration behavior of the solutions as λ +. 1. Introduction In this article, we will study the Kirchhoff type problem ( ) α u 2 dx + 1 u + (λa(x) + a )u = u p 2 u in, u H 1 ( ), (1.1) where 4 < p < 6, α and λ are two positive parameters, a R is a constant and a(x) is a potential satisfying some conditions to be specified later. The Kirchhoff type problems in bounded domains is one of most popular nonlocal problems in the study areas of elliptic equations (cf. [5, 6, 16, 18, 22, 23, 28] and the references therein). One motivation comes from the very important application to such problems in physics. Indeed, The Kirchhoff type problem in bounded domains is related to the stationary analogue of the model u tt ( α u 2 dx + β ) u = h(x, u) in (, T ), u = on (, T ), u(x, ) = u (x), u t (x, ) = u (x), (1.2) where T > is a constant, u, u are continuous functions. Such model was first proposed by Kirchhoff in 1883 as an extension of the classical D Alembert s wave equations for free vibration of elastic strings, Kirchhoff s model takes into account the changes in length of the string produced by transverse vibrations. In (1.2), u denotes the displacement, h(x, u) the external force and β the initial tension while 21 Mathematics Subject Classification. 35B38, 35B4, 35J1, 35J2. Key words and phrases. Kirchhoff type problem; indefinite potential; potential well; variational method. c 216 Texas State University. Submitted February 19, 216. Published July 6, 216. 1

2 Y. WU, Y. HUANG, Z. LIU EJDE-216/178 α is related to the intrinsic properties of the string (such as Youngs modulus). For more details on the physical background of Kirchhoff type problems, we refer the readers to [1, 13]. The Kirchhoff type nonlocal term was introduced to the elliptic equations in by He and Zou in [11], where, by using the variational method, some existence results of the nontrivial solutions were obtained. Since then, many papers have been devoted to such topic, see for example [2, 1, 12, 15, 17, 24, 26] and the references therein. In particular, in a recent article [24], Sun and Wu have studied the Kirchhoff type problem ( ) µ u 2 dx + ν u + λa(x)u = f(x, u) in, u H 1 (R N ), where µ, ν, λ > are parameters and a(x) satisfies the following conditions: (A1) a(x) C( ) and a(x) on. (A2) There exists a > such that A < +, where A = {x : a(x) < a } and A is the Lebesgue measure of the set A. (A3) = inta 1 () is a bounded domain and has smooth boundaries with = a 1 (). Using the variational method, they obtain some existence and non-existence results of the nontrivial solutions when f(x, u) is 1-asymptotically linear, 3-asymptotically linear or 4-asymptotically linear at infinity. Under the conditions (A1) (A3), λa(x) is called as the steep potential well for λ sufficiently large and the depth of the well is controlled by the parameter λ. Such potentials were first introduced by Bartsch and Wang in [3] for the scalar Schrödinger equations. An interesting phenomenon for this kind of Schrödinger equations is that, one can expect to find the solutions which are concentrated at the bottom of the wells as the depth goes to infinity. Because this interesting property, such topic for the scalar Schrödinger equations was studied extensively in the past decade. We refer the readers to [4, 7, 14, 21, 25] and the references therein. Recently, the steep potential well was also considered for some other elliptic equations and systems, see for example [8, 9, 19, 27, 29] and the references therein. To our best knowledge, most of the literatures on this topic are devoted to the definite case while the indefinite case was only considered in [4, 7] for the the scalar Schrödinger equations and in [29] for the Schrödinger-Poisson systems. Inspired by the above facts, we wonder what will happen for the Kirchhoff type problem with steep potential wells in the indefinite case of a <? To our best knowledge, this kind of problems has not been studied yet in the literatures. Thus, the purpose of this paper is to explore the preceding problems. Before stating our results, we shall introduce some notation. By condition (A3), it is well known that in the case of a, all the eigenvalues {γ i } of the problem u = γ a u u H 1 () (1.3) satisfy γ 1 < γ 2 < γ 3 < < γ i <... with γ i + as i and the multiplicity of γ i is finite for every i N. In particular, γ 1 is simple. For each i N, denote the corresponding eigenfunctions and the eigenspace of γ i by {ϕ i,j } j=1,2,...,ki and N i =span{ϕ i,j } j=1,2,...,ki respectively, where k i are the multiplicity of γ i, then ϕ i,j can be chosen so that ϕ i,j L2 () = 1 a and {ϕ 2 i,j } can form a basis of H 1 ().

EJDE-216/178 KIRCHHOFF TYPE PROBLEMS 3 Let k = inf{k : γ k > 1}, (1.4) then our main result in this paper can be stated as follows. Theorem 1.1. Suppose that (A1) (A3) hold. If either a or a < with γ k 1 < 1 then there exist positive constants α and Λ such that (P α,λ ) has a nontrivial solution u α,λ for all λ > Λ and α (, α ). Moreover, u α,λ u α strongly in H 1 ( ) as λ + up to a subsequence and u α is a nontrivial solution of the following Kirchhoff type problem: ( ) α u 2 dx + 1 u + a u = u p 2 u in, (1.5) u = on. Remark 1.2. (a) If a < with a large enough then it is easy to see that k > 1. It follows that (1.1) is indefinite in a suitable Hilbert space (see Lemma 2.5 for more details). To out best knowledge, Theorem 1.1 is the first result for the Kirchhoff type problem in for the indefinite case. (b) Theorem 1.1 also gives the existence of nontrivial solutions to (1.5). Note that (1.5) is also indefinite if a < with a large enough. Thus, to our best knowledge, it is also the first result for the Kirchhoff type problem on bounded domains in the indefinite case. Through this paper, C and C i (i = 1, 2,... ) will be indiscriminately used to denote various positive constants. o n (1) and o λ (1) will always denote the quantities tending towards zero as n and λ + respectively. 2. Variational setting By condition (A1), we see that for every a R and λ > max{, a a }, E = {u D 1,2 ( ) : a(x)u 2 dx < + } equipped with the inner product u, v λ = ( u v + (λa(x) + a ) + uv)dx is a Hilbert space, which we will denote by E λ, where (λa(x)+a ) + = max{λa(x)+ a, }. The corresponding norm on E λ is ( ) 1/2. u λ = ( u 2 + (λa(x) + a ) + u 2 )dx It follows from the Hölder inequality, the Sobolev inequality and the conditions (A1) (A2) that for every u E λ with λ > max{, a /a }, u 2 dx = u 2 dx + u 2 dx A \A A 2/3 S R 1 u 2 1 dx + (λa(x) + a ) + u 2 dx a 3 + a λ max { A 2/3 S 1 1, a + a λ} ( u 2 + (λa(x) + a ) + u 2 )dx

4 Y. WU, Y. HUANG, Z. LIU EJDE-216/178 and ( ) 1/p ( ) 1/2 u p dx S 1/2 p ( u 2 + u 2 )dx R 3 Sp 1/2 1 + max{ A 2/3 S 1 1, a + a λ } ( ) 1/2, ( u 2 + (λa(x) + a ) + u 2 )dx where S and S p are the best Sobolev embedding constant from D 1,2 ( ) to L 6 ( ) and H 1 ( ) to L p ( ) respectively; that is, and S = inf{ u 2 L 2 ( ) : u D1,2 ( ), u 2 L 6 ( ) = 1} S p = inf{ u 2 L 2 ( ) + u 2 L 2 ( ) : u H1 ( ), u 2 L p ( ) = 1}, where Lp ( ) is the usual norm in L p ( ) for all p 1. Let d λ = max{ A 2/3 S 1 1, a +a λ }. Then we have u L2 (R N ) d λ u λ and u Lp ( ) Sp 1/2 1 + d 2 λ u λ, (2.1) which yields that E λ is embedded continuously into H 1 ( ) for λ > max{, a a }. Moreover, by using (2.1), the conditions (A1) (A2) and by following a standard argument, we can show that corresponding energy functional J α,λ (u) to the Problem (1.1), given by J α,λ (u) = α 4 u 4 L 2 ( ) + 1 ( u 2 + (λa(x) + a )u 2 )dx 1 2 R p u p L p ( ), 3 is C 2 in E λ for λ > max{, a a }. For the sake of convenience, we re-write the energy functional J λ (u) by J α,λ (u) = α 4 u 4 L 2 ( ) + 1 2 u 2 λ 1 2 D λ(u, u) 1 p u p L p ( ), where D λ (u, v) = (λa(x)+a ) uvdx and (λa(x)+a ) = max{ (λa(x)+a ), }. In what follows, inspired by [7, 29], we shall make some further observations on the functional D λ (u, u). By condition (A1), (λa(x) + a )u 2 dx for all u E λ with λ > in the case of a. It follows that D λ (u, u) is definite on E λ with λ > in the case of a. Let us consider the case of a < in what follows. Let A λ := {x : λa(x) + a < }, then by the condition (A3), we have A λ, which means that A λ for every λ >, and moreover, by the conditions (A1) (A2), the real number Λ := inf{λ > : A λ < + }. satisfies < Λ a a. For λ > Λ, we define F λ := {u E λ : supp u \A λ }. It follows from the conditions (A1) (A3) that F λ is nonempty, closed and convex with F λ E λ. Hence, E λ = F λ F λ and F λ for λ > Λ in the case of a <, where F λ is the orthogonal complement of F λ in E λ.

EJDE-216/178 KIRCHHOFF TYPE PROBLEMS 5 Lemma 2.1. Let β(λ) := inf u 2 λ, u Fλ D λ where D λ := {u E λ : D λ (u, u) = 1}. If the conditions (A1) (A3) hold, then β(λ) is nondecreasing as the function of λ on (Λ, + ) and β(λ) can be attained by some e(λ) Fλ. Furthermore, (e(λ), β(λ)) (ϕ 1, γ 1 ) strongly in H 1 ( ) R as λ + up to a subsequence. Proof. First, thanks to the definition of Λ, we see that D λ (u, u) and u 2 λ are weakly continuous and weakly low semi-continuous on Fλ respectively. Thus, we can use a standard argument to show that β(λ) can be attained by some e(λ) Fλ D λ for all λ > Λ. Next, we show that β(λ) is nondecreasing as the function of λ on (Λ, + ). Indeed, let λ 1 λ 2, then by the definition of E λ, we have E λ1 = E λ2 in the sense of sets. It follows that F λ2 F λ1, which implies Fλ 1 Fλ 2. Note that u 2 λ 1 u 2 λ 2 and D λ1 (u, u) D λ2 (u, u) for all u E λ1 by the condition (A1). Thus, due to the definition of β(λ 1 ) and β(λ 2 ), we can see that β(λ 2 ) β(λ 1 ); that is, β(λ) is nondecreasing as a functional of λ on (Λ, + ). Finally, we shall prove that (e(λ), β(λ)) (ϕ 1, γ 1 ) strongly in H 1 ( ) R as λ + up to a subsequence. In fact, since (λa(x) + a ) [e(λ)] 2 dx = 1, it implies that lim (a(x) + a λ + R λ )+ [e(λ)] 2 dx =. (2.2) 3 Note that H 1 () Fλ for all λ > Λ due to the condition (A3), we can easily show that < β(λ) γ 1 for all λ > Λ. It follows from (2.1) that {e(λ)} is bounded in H 1 ( ) for λ. Without loss of generality, we assume that e(λ) e weakly in H 1 ( ) and β(λ) β as λ +. By the Sobolev embedding theorem, the condition (A2) and (2.2), we must have (e, β ) H 1 () R + satisfying e outside and a 2 e 2 dx = 1 and (e(λ), β(λ)) (e, β ) strongly in L 2 ( ) R as λ + up to a subsequence, which gives γ 1 ( e(λ) 2 + (λa(x) + a ) + [e(λ)] 2 )dx R 3 e 2 (2.3) dx + o λ (1) γ 1 + o λ (1). Hence, (e(λ), β(λ)) (e, γ 1 ) strongly in H 1 ( ) R as λ + up to a subsequence and (e, γ 1 ) satisfies γ 1 = e 2 dx = inf u 2 dx u H 1()\{} a 2 u 2 dx. Thus, e α = ϕ 1 and the proof is complete. We re-denote the above (e(λ), β(λ)) by (e 1 (λ), β 1 (λ)) and define F λ,1 := { u F λ : u 2 λ D λ (u, u) = β 1(λ) }. Since γ 2 > γ 1 β 1 (λ) for λ > Λ, it is easy to see that Fλ,1 F λ. Thus, we have Fλ = F λ,1 F,, λ,1, where Fλ,1 is the orthogonal complement of F λ,1 in F λ.

6 Y. WU, Y. HUANG, Z. LIU EJDE-216/178 Lemma 2.2. Suppose that (A1) (A3) hold. Then there exists Λ 1 Λ such that Fλ,1 = span{e 1(λ)} and β 2 (λ) can be attained by some e 2 (λ) F, λ,1 for λ > Λ 1, where β 2 (λ) := inf u 2 λ. F, λ,1 D λ Furthermore, (e 2 (λ), β 2 (λ)) (ϕ 2,j, γ 2 ) strongly in H 1 ( ) R as λ + up to a subsequence for some j N with 1 j k 2. Proof. Since D λ (u, u) and u 2 λ are weakly continuous and weakly low semi-continuous on F,, λ,1 respectively, by the fact that Fλ,1 is weakly closed for λ > Λ, we can also use a standard argument to show that β 2 (λ) can be attained by some e 2 (λ) F, λ,1 for λ > Λ. For the sake of clarity, the remaining proof will be performed through the following steps. Step 1. We prove that there exists Λ 1 Λ such that F λ,1 = span{e 1(λ)} for λ > Λ 1. Indeed, suppose on the contrary that there exist e 1(λ n ), e 1(λ n ) F α,λ,1 with e 1(λ n ), e 1(λ n ) Eλn,E λn =, (λ n a(x) + a ) [e 1(λ n )] 2 dx = (λ n a(x) + a ) [e 1(λ n )] 2 dx = 1 for {λ n } satisfying λ n + as n. By Lemma 2.1, we can see that e 1(λ n ) ϕ 1 and e 1(λ n ) ϕ 1 strongly in H 1 ( ) as n up to a subsequence. It follows from (2.3) and Lemma 2.1 once more that which is a contradiction. 2γ 1 = 2β 1 (λ n ) + o n (1) = e 1(λ n ) 2 λ n + e 1(λ n ) 2 λ n + o n (1) = (e 1(λ n ) e 1(λ n )) 2 L 2 ( ) + o n(1) = o n (1), (2.4) Step 2. We show that lim sup λ + β 2 (λ) γ 2. In fact, by Step 1, we have ϕ 2,1 = d λ e 1(λ) + ϕ 2,1,λ, where d λ is a constant and ϕ 2,1,λ is the projection of ϕ 2,1 in F, λ,1. Thus, e 1(λ), ϕ 2,1 Eλ,E λ = d λ e 1(λ) 2 λ. It follows from the condition (A3) and Lemma 2.1 that d λ as λ + up to a subsequence. Now, by the definition of β 2 (λ), we can see from Lemma 2.1, (2.3) and a variant of the Lebesgue dominated convergence theorem (cf. [2, Theorem 2.2]) that ϕ 2,1,λ lim sup β 2 (λ) lim sup 2 λ λ + λ + D λ (ϕ 2,1,λ, ϕ 2,1,λ ) = lim sup λ + ϕ 2,1 d λ e 1(λ) 2 λ D λ (ϕ 2,1 d λ e 1(λ), ϕ 2,1 d λ e 1(λ)) = ϕ 2,1 2 L 2 ( ) a 2 ϕ 2,1 2 = γ 2. L 2 ( ) Step 3. We prove that lim sup λ + β 2 (λ) γ 2 and (e 2 (λ), β 2 (λ)) (ϕ 2,j, γ 2 ) strongly in H 1 ( ) R as λ + up to a subsequence for some j N with 1 j k 2. Actually, by Step 2, we know that {e 2 (λ)} is bounded in D 1,2 ( ). Similarly as in the proof of Lemma 2.1, we can see that (e 2 (λ), β 2 (λ)) (e 2, β 2) strongly in

EJDE-216/178 KIRCHHOFF TYPE PROBLEMS 7 L 2 ( ) R as λ + up to a subsequence with e 2 H 1 () and e 2 outside. Since D λ (u, u) is weakly continuous on F λ, we also have a 2 e 2 2 dx = 1. Furthermore, by the theory of Lagrange multipliers, we can also see that (e 2, β 2) satisfies (1.3). It follows from a variant of the Lebesgue dominated convergence theorem (cf. [2, Theorem 2.2]) that e 2 2 L 2 ( ) = β 2 a e 2 2 L 2 ( ) = β 2 (λ)d λ (e 2 (λ), e 2(λ)) + o λ (1) = e 2 (λ) 2 dx + o λ (1) e 2 2 L 2 ( ). Thus, (e 2 (λ), β 2 (λ)) (e 2, β 2) strongly in H 1 ( ) R as λ + up to a subsequence. By Step 2, we must have β 2 = γ 1 or β 2 = γ 2. If lim sup λ + β 2 (λ) < γ 2 then there exists {λ n } such that (e 2 (λ n ), β 2 (λ n )) (ϕ 1, γ 1 ) strongly in L 2 ( ) R as n up to a subsequence. It follows from Lemma 2.1, (2.3) and Step 2 that = e 1 (λ n ), e 2 (λ n ) λn,λ n = ϕ 1 2 L 2 ( ) + o n(1), which is a contradiction. Let F λ,2 := { u F λ : u 2 λ D λ (u, u) = β 2(λ) }. Since γ 3 > γ 2, it yields from Lemma 2.2 and condition (A3) that F λ,1 F λ,2 F λ. Lemma 2.3. Suppose that (A1) (A3) hold. Then there exists Λ 2 Λ 1 such that dim(f λ,2 ) k 2 for λ > Λ 2. Proof. Let e 2 (λ), e 2(λ) F λ,2. By Lemma 2.2, e 2(λ) ϕ 2,j and e 2(λ) ϕ 2,j strongly in H 1 ( ) R as λ + up to a subsequence for some j, j N with 1 j, j k 2. Clearly, two cases may occur: (1) ϕ 2,j = ϕ 2,j ; (2) ϕ 2,j ϕ 2,j and ϕ 2,jϕ 2,j dx =. If case (1) happens then by a similar argument used in the proof of (2.4), we can get that γ 2 =, which is a contradiction. Thus, we must have case (2). It follows that there exists Λ 2 Λ 1 such that dim(fλ,2 ) k 2 for λ > Λ 2. Now, by iterating, for m = 3, 4,..., we can define β m (λ) as follows: where F, β m (λ) := inf u 2 λ, F, λ,m D λ λ,m := {u F λ : u, v λ =, for all v m 1 { Fλ,i := u Fλ u 2 } λ : D λ (u, u) = β i(λ). i=1 F λ,i}, Similarly as Lemmas 2.2 and 2.3, we can obtain the following result.

8 Y. WU, Y. HUANG, Z. LIU EJDE-216/178 Lemma 2.4. Suppose that (A1) (A3) hold. Then there exists Λ m Λ m 1 such that β m (λ) can be attained by some e m (λ) F, λ,m for λ > Λ m. Furthermore, (e m (λ), β m (λ)) (ϕ m,j, γ m ) strongly in H 1 ( ) R as λ + up to a subsequence for some j N with 1 j k m and dim(fλ,m ) k m for λ > Λ m, where { Fλ,m := u Fλ u 2 } λ : D λ (u, u) = β m(λ). Let k be given by (1.4), then by Lemmas 2.1, 2.2 and 2.4, k 1 i=1 F λ,i, and Fλ,k are well defined for λ > Λ k. Lemma 2.5. Suppose that (A1) (A3) hold. If γ k 1 < 1 then there exists Λ k such that for λ > Λ k, it holds that Λ k (1) u 2 λ D λ(u, u) 1 2 (1 1 γ ) u 2 k 1 λ in k 1 i=1 F λ,i ; (2) u 2 λ D λ(u, u) 1 2 (1 1 γ ) u 2, k λ in Fλ,k. The proof of the above lemma follows immediately from Lemmas 2.1, 2.2 and 2.4. Remark 2.6. By Lemmas 2.2 2.4, we also have k 1 i=1 F λ,i = in the case of γ 1 > 1 while k 1 i=1 F λ,i and dim( k 1 i=1 F λ,i ) k 1 i=1 k i in the case of γ 1 < 1. 3. Nontrivial solution We first consider the case of a <. By the decomposition of E λ, we will find the nontrivial solution by the linking theorem. Let us first verify that J α,λ (u) has a linking structure in E λ in the case of a <. Lemma 3.1. Suppose that (A1) (A3) hold and a <. For every α >, if β k 1 < 1 then there exists ρ > independent of λ such that F, λ,k inf J α,λ (u) d (3.1) S λ,ρ for all λ > Λ k, where S λ,ρ := {u E λ : u λ = ρ} and d is a constant independent of λ and α. Proof. By (2.1) and Lemma 2.5, for every u F, λ,k, we have J α,λ (u) = α 4 u 4 L 2 ( ) + 1 2 u 2 λ 1 2 D λ(u, u) 1 p u p L p ( ) Note that d λ = 1 4 (1 1 ) u 2 λ S p 2 p γ k ( 1 u 2 λ 4 (1 1 ) S p γ k max{ A 2/3 S 1 1, a +a λ d λ (1 + d 2 λ) p 2 u p λ p 2 max{ A 2/3 S 1, ) (1 + d 2 λ) p 2 u p 2 λ. }, so that 1 a + a Λk } (3.2)

EJDE-216/178 KIRCHHOFF TYPE PROBLEMS 9 for λ > Λ k. It follows from (3.2) that there exists ρ > independent on λ such that (3.1) holds for all λ > Λ k. Let Q λ,k := {u = v + te k (λ) : t and v k 1 i=1 F λ,i}. Lemma 3.2. Suppose that (A1) (A3) hold and a <. If γ k 1 < 1 then there exist α > and R > ρ independent of λ such that sup Q R λ,k J α,λ (u) 1 2 d for all λ > Λ k in the case of α (, α ), where d is given in lemma 3.1, Q R λ,k := Q λ,k B λ,r and B λ,r := {u E λ : u λ R }. Proof. Let u λ Q R λ,k. Then one of the following two cases must happen: (a) u λ = Rũ λ with ũ λ k 1 i=1 F λ,i and ũ λ λ 1. (b) u λ = Rũ λ with ũ λ Q 1 λ,k \ k 1 i=1 Fλ,i and ũ λ λ = 1. If the case (b) happens then by Lemma 2.5, we deduce that J α,λ (u λ ) = J α,λ (Rũ λ ) α 4 R4 + 1 2 (1 1 )R 2 1 γ k p Rũ λ p L p ( ). (3.3) Since ũ λ k i=1 F λ,i, by Lemmas 2.1, 2.2 and 2.4, ũ λ = ũ + o λ (1) strongly in H 1 ( ) for some ũ span{ϕ i,j } i=1,2,...,k j=1,2,...,k i and ũ 2 dx = 1. Thus, ũ λ p L p ( ) = ũ p L p ( ) + o λ(1) by the Sobolev embedding theorem. Note that dim span{ϕ i,j } i=1,2,...,k j=1,2,...,k i ) k 1 i=1 k i + 1 for all λ > Λ k by Remark 2.6. Therefore, there exists a constant M > such that u Lp ( ) M for all u span{ϕ i,j } i=1,2,...,k j=1,2,...,k i with u 2 dx = 1. In particular, ũ L p ( ) M. It follows from 4 < p < 6 and (3.3) that there exists a constant R (> ρ) such that J α,λ (R ũ λ ) for all λ > Λ k. Now, we consider the case of (a). By Lemma 2.5 once more, we know that J α,λ (u λ ) = J α,λ (Rũ λ ) α 4 R4. Thus, there exists α > such that J α,λ (u λ ) 1 2 d for λ > Λ k and α (, α ). From Lemmas 3.1 and 3.2, we can see that J α,λ (u) has a linking structure in E λ with λ > Λ k and α (, α ) in the case of a <. By the linking theorem, there exists {u n } E λ such that (1 + u n λ )J α,λ (u n) = o n (1) strongly in Eλ and J α,λ (u n ) = c α,λ + o n (1), where Eλ is the dual space of E λ. Furthermore, c α,λ [d, α 4 R4 + 1 2 (1 1 γ )R k ]. 2 Note that in the special case γ 1 > 1, the linking structure is actually the mountain pass geometry. Thus, the linking theorem can be replaced by the mountain pass theorem and we can also obtain a sequence {u n } E λ such that (1 + u n λ )J α,λ (u n) = o n (1) strongly in Eλ and J α,λ(u n ) = c α,λ + o n (1). In the case a, since 4 < p < 6 and the fact that D λ (u, u) = in E λ, by using a standard argument, we can verify that J α,λ (u) has a mountain pass geometry in E λ for λ > ; that is, (a) inf Sλ,ρ J α,λ (u) C for some ρ > ;

1 Y. WU, Y. HUANG, Z. LIU EJDE-216/178 (b) J α,λ (R φ) for some R > ρ and φ H 1 (). This also gives the existence of a sequence {u n } E λ such that (1 + u n λ )J α,λ(u n ) = o n (1) strongly in E λ and J α,λ(u n ) = c α,λ + o n (1) with c α,λ [C α, C α], where C α, C α are two positive constants independent of λ. In a word, in both cases of a < and a, for λ > Λ k, there exists {u n } E λ such that (1 + u n λ )J α,λ (u n) = o n (1) strongly in E λ and J α,λ(u n ) = c α,λ + o n (1) with c α,λ [C α, C α]. Lemma 3.3. Suppose that (A1) (A3) hold. For every α >, if either a or a < with β k 1 < 1 then { u n λ } is bounded. Proof. Since λ > Λ k, by the condition (A2) and the Hölder and the Sobolev inequalities, we obtain that D λ (u n, u n ) a u n 2 dx a A 2 3 S 1 u n 2 L 2 ( ). A Note that (1 + u n λ )J α,λ (u n) = o n (1) strongly in Eλ and J α,λ(u n ) = c α,λ + o n (1), by the Young inequality and the fact that 4 < p < 6, we deduce that c α,λ + o n (1) = J α,λ (u n ) 1 p J α,λ(u n ), u n E λ,e λ = α( 1 4 1 p ) u n 4 L 2 ( ) + (1 2 1 p ) u n 2 λ ( 1 2 1 p )D λ(u n, u n ) p 4 4p (α u n 4 L 2 ( ) + u n 2 λ) p 2 2p a A 2 3 S 1 u n 2 L 2 ( ) p 4 8p (α u n 4 L 2 ( ) + u n 2 2(p 2)2 λ) α(p 4)p a 2 A 4 3 S 2, where, E λ,e λ is the duality pairing of Eλ and E λ. The preceding inequality, together with c α,λ [C α, C α] and 4 < p < 6, implies { u n λ } is bounded. By Lemma 3.3, we can see that u n = u α,λ + o n (1) weakly in E λ for some u α,λ E λ up to a subsequence. Without loss of generality, we may assume that u n = u α,λ + o n (1) weakly in E λ. Lemma 3.4. Suppose that (A1) (A3) hold. For every α >, if either a or a < with β k 1 < 1 then there exists Λ k > Λ k such that u α,λ is a nontrivial solution of (1.1) for λ > Λ k. Proof. We first prove that u α,λ in E λ. Indeed, suppose on the contrary, then by the Sobolev embedding theorem, we can see that u n = o n (1) strongly in L 2 loc (R3 ), which, together with (A2), implies u n = o n (1) strongly in L 2 (A ). It follows from Lemma 3.3, conditions (A1) (A2) and the Hölder and the Sobolev inequality that ( ) 6 p ( ) p 2 u n p dx u n 2 4 dx u n 6 4 dx ( S 3(p 2) 4 u n 3(p 2) 2 L 2 ( ) S 3(p 2) 4 (C 1 + o n (1)) 5p 1 4 u n 2 dx + o n (1) \A ( 1 a + a λ ) 6 p 4 ) 6 p 4 u n 2 λ + o n (1). (3.4)

EJDE-216/178 KIRCHHOFF TYPE PROBLEMS 11 On the other hand, by conditions (A1) (A2) once more, we have D λ (u n, u n ) a u n 2 dx = o n (1). (3.5) A Therefore, we deduce from the fact that (1 + u n λ )J α,λ (u n) = o n (1) strongly in E λ that α u n 4 L 2 (R 2 ) + u n 2 λ S 3(p 2) (C 1 + o n (1)) 5p 1 4 ( 1 ) 6 p 4 u n 2 λ + o n (1), a + a λ which yields that there exists Λ k > Λ k dependent of α such that u n = o n (1) strongly in E λ with λ > Λ k. It is impossible since c α,λ C α > for all λ > Λ k. Therefore u α,λ in E λ. It remains to show that J α,β (u α,β) = in Eλ. In fact, without loss of generality, we may assume that u n 2 L 2 ( ) = A+o n(1) and consider the following energy functional I α,λ (u) = αa 2 u 2 L 2 ( ) + 1 2 u 2 λ 1 2 D λ(u, u) 1 p u p L p ( ). Clearly, by (2.1), I α,λ (u) is of C 2 in E λ for λ > Λ k. Since (1 + u n λ )J α,λ (u n) = o n (1) strongly in Eλ, it is easy to see from u n 2 λ = A+o n(1) and u n = u α,λ +o n (1) weakly in E λ that I α,λ (u n), u n u α,β E λ,e λ = o n (1) and I α,λ (u n) = o n (1) strongly in Eλ, so that I α,λ (u α,λ) = in Eλ. In particular, I α,λ (u α,λ), u n u α,β E λ,e λ =. Now, we obtain o n (1) = I α,λ(u n ) I α,λ(u α,λ ), u n u α,β E λ,e λ = αa u n u α,β 2 L 2 ( ) + u n u α,β 2 λ D λ (u n u α,β, u n u α,β ) u n u α,β p L p ( ). Since u n u α,β = o n (1) weakly in E λ, by using similar arguments in the proofs of (3.4) and (3.5), we can see that u n u α,β = o n (1) strongly in E λ for λ sufficiently large, say λ > Λ k. Thus, we must have that J α,β (u α,β) = in Eλ for λ > Λ k. The following lemma will give a description on the concentration behavior of the nontrivial solutions u α,λ as λ +. Lemma 3.5. Suppose that (A1) (A3) hold. For every α >, if either a or a < with β k 1 < 1 then we have u α,λ u α strongly in H 1 ( ) as λ + up to a subsequence. Furthermore, u α is a nontrivial solution of (1.5). Proof. Let u α,λn be the nontrivial solution obtained in Lemma 3.4 with λ n + as n. By Lemma 3.3, we can see that ( u α,λn 2 + (λ n a(x) + a ) + u α,λn 2 )dx C 1 for all n N. It follows that {u α,λn } is bounded in D 1,2 ( ) for n and (a(x) + a ) + u α,λn 2 dx = o n (1). R λ 3 n Without loss of generality, we may assume that u α,λn = u α + o n (1) weakly in D 1,2 ( ). Thanks to the Sobolev embedding theorem and conditions (A1) (A3),

12 Y. WU, Y. HUANG, Z. LIU EJDE-216/178 we can see that u α,λn = u α +o n (1) strongly in L 2 ( ) and u α H 1 () with u α on \. Therefore, by the Hölder and the Sobolev inequality, we obtain u α,λn u α L p ( ) 6 p 2p u α,λn u α L 2 ( ) ( u α,λ n L6 ( ) + u α L6 ( )) 3p 6 2p = o n (1). On the other hand, by a variant of the Lebesgue dominated convergence theorem (cf. [2, Theorem 2.2]) and the condition (A1), we also have D λn (u α,λn u α, u α,λn u α ) = o n (1). Therefore, u α p dx = u α,λn p L p ( ) + o n(1) = D λn (u α,λn, u α,λn ) + u α,λn 2 λ n + α u α,λn 4 L 2 ( ) α u α 4 + u α 2 + a u α 2 dx + o n (1). Note that u α H 1 () H 1 ( ), it is easy to see from J α,λ n (u α,λn ) = in Eλ n that u α is a solution of (1.5). In particular, α u α 4 + u α 2 + a u α 2 dx = u α p dx. Thus, u α,λn = u α + o n (1) strongly in D 1,2 ( ) and λ n a(x)u 2 α,λ n dx = o n (1). It follows that u α,λn = u α + o n (1) strongly in H 1 ( ). Thanks to c α,λ C α >, u α must be nonzero. Hence, u α is a nontrivial solution of (1.5). Proof of Theorem 1.1. The statement of the theorem follows immediately from Lemmas 3.4 and 3.5. Acknowledgements. Y. Wu is supported by the Fundamental Research Funds for the Central Universities (214QNA67). References [1] A. Azzollini; The Kirchhoff equation in perturbed by a local nonlinearity, Differential Integral Equations, 25 (212), 543 554. [2] C. Alves, G. Figueiredo; Nonlinear perturbations of a periodic Kirchhoff equation in, Nonlinear Anal. TMA, 75 (212), 275 2759. [3] T. Bartsch, Z.-Q. Wang; Existence and multiplicity results for superlinear elliptic problems on, Comm. Partial Differential Equations, 2(1995), 1725 1741. [4] T. Bartsch, Z. Tang; Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential, Discrete Contin. Dyn. Syst., 33(213), 7 26. [5] C. Chen, Y. Kuo, T. Wu; The Nehari manifold for a Kirchhoff type problem involving signchanging weight functions, J. Differential Equations, 25 (211), 1876 198. [6] B. Cheng, X. Wu, J. Liu; Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity, Nonlinear Differ. Equ. Appl., 19 (212), 521 537. [7] Y. Ding, A. Szulkin; Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (27), 397 419. [8] M. Furtado, E. Silva, M. Xavier; Multiplicity and concentration of solutions for elliptic systems with vanishing potentials, J. Differential Equations, 249(21), 2377 2396. [9] Y. Guo, Z. Tang; Multibump bound states for quasilinear Schrödinger systems with critical frequency, J. Fixed Point Theory Appl., 12(212), 135 174.

EJDE-216/178 KIRCHHOFF TYPE PROBLEMS 13 [1] Y. He, G. Li, S. Peng; Concentrating bound states for Kirchhoff type problems in involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (214), 441 468. [11] X. He, W. Zou; Existence and concentration behavior of positive solutions for a Kirchhoff equation in, J. Differential Equations, 252 (212), 1813 1834. [12] Y. Huang, Z. Liu, Y. Wu; On finding solutions of a Kirchhoff type problem, Proc. Amer. Math. Soc., 144 (216), 319 333. [13] G. Kirchhoff; Mechanik. Teubner, Leipzig (1883). [14] X. Liu, Y. Huang, J. Liu; Sign-changing solutions for an asymptotically linear Schrödinger equations with deepening potential well, Adv. Differential Equations, 16(211), 1 3. [15] Y. Li, F. Li, J. Shi; Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (212), 2285 2294. [16] Z. Liang, F. Li, J. Shi; Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31(214), 155 167. [17] G. Li, H. Ye; Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in, J. Differential Equations, 257 (214), 566 6. [18] C. Lei, J. Liao, C. Tang; Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (215), 521 538. [19] Y. Jiang, H.-S. Zhou; Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (211), 582 68. [2] B. Panda, O. Kapoor; On equidistant sets in normed linear spaces, Bull. Austral. Math. Soc., 11 (1974), 443-454. [21] Y. Sato, K. Tanaka; Sign-changing multi-bump solutions nonlinear Schrödinger equations with steep potential wells, Trans. Amer. Math. Soc., 361(29), 625 6253. [22] D. Naimen; Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, Nonlinear Differ. Equ. Appl., 21 (214), 885 914. [23] D. Naimen; The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations, 257 (214), 1168 1193. [24] J. Sun, T.-F. Wu; Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256(214), 1771 1792. [25] Z. Wang, H.-S. Zhou; Positive solutions for nonlinear Schrödinger equations with deepening potential well, J. Eur. Math. Soc., 11(29), 545 573. [26] Y. Wu, Y. Huang, Z. Liu; On a Kirchhoff type problem in R N, J. Math. Anal. Appl., 425 (215), 548 564. [27] Y. Ye, C. Tang; Existence and multiplicity of solutions for Schr ödinger-poisson equations with sign-changing potential, Calc. Var. Partial Differential Equations, 53(215), 383 411. [28] Z. Zhang, K. Perera; Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (26), 456 463. [29] L. Zhao, H. Liu, F. Zhao; Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential Equations, 255 (213), 1 23. Yuanze Wu (corresponding author) College of Sciences, China University of Mining and Technology, Xuzhou 221116, China E-mail address: wuyz8536@cumt.edu.cn Yisheng Huang Department of Mathematics, Soochow University, Suzhou 2156, China E-mail address: yishengh@suda.edu.cn Zeng Liu Department of Mathematics, Suzhou University of Science and Technology, Suzhou 2159, China E-mail address: luckliuz@163.com