Existence of sign-changing solution with least energy for a class of Kirchhoff-type equations in R N

Existence of sign-changing solution with least energy for a class of Kirchhoff-type equations in R ^N

Xianzhong Yao

and Chunlai Mu

1Faculty of Applied Mathematics, Shanxi University of Finance and Economics Taiyuan 030006, P.R. China

2College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China

Received 21 January 2017, appeared 9 May 2017 Communicated by Dimitri Mugnai

Abstract. We consider the existence of least energy sign-changing (nodal) solution of Kirchhoff-type elliptic problems with general nonlinearity. Using a truncated tech- nique and constrained minimization on the nodal Nehari manifold, we obtain that the Kirchhoff-type elliptic problem possesses one least energy sign-changing solution by applying a Pohožaev type identity. Moreover, the energy of the sign-changing solution is strictly more than the ground state energy.

Keywords: Kirchhoff-type, ground state solution, sign-changing solution, Pohožaev type identity.

2010 Mathematics Subject Classification: 35J50, 35B38.

1 Introduction

In this paper, we are concerned with the following Kirchhoff-type elliptic problem with gen- eral nonlinearity:

a+λ

Z

R^N|∇u|²dx+λb Z

R^Nu²dx

[−_∆u+bu] = f(u), x∈ _R^N, (1.1) where a,b > 0 are constants, λ > 0 is a parameter and N ≥ 3. Moreover, f ∈ C¹(_R,R⁺) satisfies the following hypotheses:

(f₁) |f(t)| ≤C(|t|+|t|^q−1)forq∈(2, 2^∗), 2^∗ = _N^2N₋₂; (f₂) f(t) =o(|t|)_ast→_0;

(f3) lim

t→_∞ f(t)

|t| = +_∞;

(f₄) ^f_|⁽_t^t_|⁾ is strictly increasing inR\{0}.

BCorresponding author. Email: yaoxz@sxufe.edu.cn

Kirchhoff-type problems are often referred to as being nonlocal because of the presence of the integral terms. It is related to the stationary analogue of the equation that arise in the study of string or membrane vibrations, namely

ρ∂²u

∂t² − P₀

h + ^E 2L

Z _L

0

|^∂u

∂x|²dx ∂²u

∂x² =0, (1.2)

which was presented by Kirchhoff [10] in 1883. This model is an extension of the classical d’Alembert wave equation by considering the effects of the changes on the length of the elastic string during the free vibrations. The parameters in the Kirchhoff’s model have the following meanings: Lis the length of the string,his the area of cross-section,Eis the Young modulus of the material,ρis the mass density andP₀is the initial tension. Some early classical studies of Kirchhoff-type equations were those of Pohožaev [22] and Bernstein [3]. However, Kirchhoff’s model received great attention only after Lions [13] proposed following abstract framework for the model (1.2),

(u_tt−(a+bR

Ω|∇u|²dx)_∆u= f(x,u), x ∈_Ω,

u=0 x ∈_∂Ω. ^(1.3)

The existence and concentration behavior of solutions to Kirchhoff-type elliptic problem have been extensively studied in the past decade. Most researchers paid their attention to fo- cus on existence of positive solutions, ground state, radial and nonradial solutions and semi- classical state under some different assumptions, see for example [1,4,6,7,11,12,17,19–21,24,26]

and references therein. While existence of sign-changing solutions has been received few at- tention, and there are very few results on existence of sign-changing solutions to Kirchhoff- type problem. Only Zhang et al. [18,28] investigated the existence of sign-changing solution of the Kirchhoff-type problem (1.4),

(−(a+bR

Ω|∇u|²dx)_∆u= f(u), x∈_Ω,

u=0 x∈_∂Ω, ^(1.4)

where a > 0, b≥ 0 and Ω ⊂ _R^N (N ≥ 1)is a bounded domain with smooth boundary. By using variational methods and invariant sets of descent flow, they demonstrated that equa- tions (1.4) possesses a sign-changing solution with nonlinearity f satisfying some suitable conditions.

In recent years, there has been increasing attention to the existence of sign-changing (nodal) solutions to Kirchhoff-type problem. In [23], Shuai considered equations (1.4) in N =1, 2, 3 with f ∈ C¹(_R,R)satisfying following conditions:

(H₁) f(t) =o(|t|)ast→0;

(H2) for some constant p ∈ (4, 2^∗), lim

t→_∞ f(t)

t^p−1 = 0, where 2^∗ = +_∞ for N = 1, 2, if N = 3, 2^∗ =6;

(H3) lim

t→_∞ F(t)

t⁴ = +_{∞, where}F(t) =Rt

0 f(s)ds;

(H₄) ^f_|^(t)

t|³ is an increasing function inR\{0}.

Employing constraint variational method and quantitative deformation lemma, author as- serted that there is one least energy sign-changing solution (nodal solution), which has pre- cisely two nodal domains. Moreover, the energy of sign-changing solution is strictly larger than the ground state energy. While Figueiredo and Nascimento in [5] discussed the following more general problem than (1.4), for N=3,

(−M R

Ω|∇u|²dx

∆u= f(u), x∈_Ω,

u=₀ x∈∂Ω, (1.5)

where M,f ∈ C¹(_R,R)fulfill some assumptions:

(M₁) functionM is increasing andM(0):=m₀>0;

(M₂) ^M_t^(t) is a decreasing function fort>0;

(He₃) there isθ ∈(4, 6)such that 0<θF(t)≤ f(t)t, fort6=0.

Under the conditions (M₁),(M₂) and (H₁), (H₂),(He₃), (H₄), they explored that there exists one least energy nodal solution to the problem (1.5). For more results, we refer to [2,16,27] for some variant version of Kirchhoff-type problem.

From the discussion above, we discover that researchers usually need suppose that f sat- isfies (H₄) and (H3) or (He3), which ensure the boundedness of a minimum sequence for the corresponding functional of the Kirchhoff-type problem. As well it also guarantees that the nodal Nehari manifold of corresponding functional of the Kirchhoff-type problem is not empty. Then their results can be derived by usual variational methods and quantitative de- formation lemma. In this paper, we replace the conditions (H₄) and (H₃) or (He₃) by the hypotheses (f₄) and (f₃), which is weaker than the conditions in foregoing literatures. A typical case is that f(u) =|u|^p−1ufor p ∈ (1, 5), however, the results in the references above is valid only for p∈ (3, 5). To the best authors’ knowledge, there is no result on the existence of least energy sign-changing (nodal) solution to Kirchhoff-type problem with nonlinearity f satisfying the hypotheses(f₃)and(f₄).

To character our results, we need first to introduce the energy functional for corresponding Kirchhoff-type problem (1.1) and nodal Nehari manifold. Let H¹(_R^N) be the usual Sobolev space equipped with the inner product and norm

(u,v) =

Z

R³∇u∇v+buvdx, kuk= (u,u)^1/2, andL^p(_R^N)is the usual Lebesgue space endowed with the norm

|u|_p= _Z

R^N|u|^pdx 1/p

, for 1≤ p< _∞, |u|_∞ = sup

x∈_R^N

|u(x)|, as well as

D^1,2(_R^N):={u∈ L^2∗(_R^N):∇u∈ L²(_R^N)}

with normkuk_D1,2(_R^N) =|∇u|₂. It is well known that the embedding of H¹(_R^N)intoL^p(_R^N) for p ∈ [2, 2^∗] is continuous but not compact. Denote the subspace H¹_r(_R^N) := {u ∈ H¹(_R^N) : uis radial symmetric function}and hereafter, for simplicity, H := H_r¹(_R^N). Then H,→ L^p(_R^N)compactly for p∈(_{2, 2}^∗), see [25, Corollary 1.26].

Define the energy functional associated with equation (1.1), J_λ : H→_Rgiven by J_λ(u) = ^a

2kuk²+^λ

4kuk⁴−

Z

R^NF(u)dx.

Obviously, J_λ belong toC¹(H,R). For anyu,v∈ H, there is hJ_λ⁰(u),vi= a(u,v) +λkuk²(u,v)−

Z

R^N f(u)vdx.

It is well-known that each weak solution of equation (1.1) corresponds a critical point of J_λ. We define the Nehari manifold for the corresponding energy functional J_λ

N_λ ={u∈ H\{0}:hJ_λ⁰(u),ui=0}, and the nodal Nehari manifold

M_λ = {u ∈H :u^±6=0,hJ_λ⁰(u),u^±i=0}, where

u⁺(x) =max{u(x), 0} and u⁻(x) =min{u(x), 0}. Moreover, denote

ec_λ :=inf{J_λ(u):u∈ N_λ} and c_λ :=inf{J_λ(u):u∈ M_λ}.

When u is a nontrivial solution to equation (1.1) and J_λ(u) ≤ J_λ(v), where v is any solution of equation (1.1), then we say that u∈ H is a ground state (least energy) solution to equation (1.1) andu is one sign-changing (nodal) solution to equation (1.1) if u^± 6= 0. By Lemma 2.3 below, we have thatN_λandM_λ are not empty andM_λ ⊂ N_λ. From the definition ofN_λand M_λ, we know that all nontrivial solutions and sign-changing solutions to equation (1.1) are included inN_λ andM_λ, respectively.

Now, we give our main results as follows.

Theorem 1.1. Assume the conditions(f₁)–(f₄)hold. Then there exists a positiveΛsuch that, for any λ∈(0,Λ), the problem (1.1) have a ground state solution u_λwhich is constant sign and a least energy sign-changing solution v_λ satisfying

c_λ = J_λ(v_λ)> J_λ(u_λ) =_ec_λ >0.

The remainder of this paper is organized as follows. In Section 2, we present the abstract framework of the problem as well as some preliminary results. Theorem1.1will be proved in Section 3.

2 Preliminaries

In this section, we show examples how theorems, definitions, lists and formulae should be formatted.

In this section, we give some notations and lemmas. According to the foregoing discussion, we know that it is very difficult to obtain bounded minimum sequences for the associated

functional J_λ. So we here use a truncated technique, following [8,9,11], to handle it. We introduce a cut-off function φ∈ C^∞(_R,R)satisfying















φ(t) =_1, t ∈[_{0, 1}]_, 0≤φ(t)≤1, t ∈(1, 2), φ(t) =0, t ∈[2,∞),

|φ⁰|_∞ ≤2,

and then consider the following truncated functional J_λ,κ :H→_Rdefined by J_λ,κ(u) = ^a

2kuk²+ ^λ

4hκ(u)kuk⁴−

Z

R^NF(u)dx, where for every κ>0,

h_κ(u) =φ kuk²

κ²

.

It is easy to know thatJ_λ,κ belong toC¹(H,R). Forκ>0 enough large, we can take advantage of J_λ,κ to obtain a critical pointw_λ of J_λ,κ, then, by the definition ofφ and J_λ,κ, we know that w_λis a critical point of J_λ if we show thatkw_λk ≤κ. We define the Nehari manifold of J_λ,κas follows

N_λ,κ ={u∈ H\{0}:hJ_λ,κ⁰ (u),ui=0} and the nodal Nehari manifold

M_λ,κ ={u∈ H:u^±6=0,hJ_λ,κ⁰ (u),u^±i=0}. Moreover, denote

ec_λ,κ :=inf{J_λ,κ(u):u∈ N_λ,κ}, c_λ,κ:=inf{J_λ,κ(u):u∈ M_λ,κ}.

Notation 2.1. Throughout this paper, we denote by “→” and “*” the strong and weak con- vergence in the related function space, respectively. B_r(x) := {y ∈_R^N : |x−y|<r}. We use o(1)to denote any quantity which tends to zero as n→ _∞. We will use the symbol C and C_i for denoting positive constants unless otherwise stated explicitly and the value of C andC_i is allowed to change from line to line and also in the same formula.

Lemma 2.2. For all u∈ N_λ,κ, the following results hold:

(i) for anyλ>0, There exists r >0such thatkuk ≥r;

(ii) J_λ,κ has a lower bound inN_λ,κ. Proof. For anyu∈ N_λ,κ, there is

akuk²+λhκ(u)kuk⁴+ ^λ 2κ²φ⁰

kuk² κ²

kuk⁶=

Z

R^N f(u)udx, (2.1) By(f₁), (f2)and Sobolev’s inequality, it is easy to obtain the result(i)ifkuk² ≥2κ², otherwise, the following inequality holds

akuk²+λh_κ(u)kuk⁴+ ^λ 2κ²φ⁰

kuk² κ²

kuk⁶≥akuk²− ^λ κ²kuk⁶,

owing to(f₁)and(f₂), we have, for smallε>0, Z

R^N f(u)udx≤ε|u|²₂+Cε|u|^q_q. (2.2) Combining the three formulas above and Sobolev inequality, we obtain that

akuk²− ^λ

κ²kuk⁶ ≤

Z

R^N f(u)udx≤ε|u|²₂+Cε|u|^q_q ≤εC₁kuk²+C₂kuk^q. It follows the assertion(i).

Next we show the item(ii)_{. If}kuk²≥2κ²for allu∈ N, by the definition ofφ, we observe J_λ,κ(u) = ^a

2kuk²−

Z

R^NF(u)dx, and by (2.1), it holds

akuk² =

Z

R^N f(u)udx.

Since(f₄)implies that 2F(t) ≤ f(t)t fort ∈ R, we deduce that J_λ,κ(u)> 0 and the result is finished. Suppose, by contradiction, that there isu∈ N such that kuk² <2κ². In which case, the result is valid byJ_λ,κ ∈ C¹(H,R). Thus the conclusion is established.

Lemma 2.3. For any u ∈ H with u^± 6= 0, then there is a pair (t_u,s_u) ∈ _R⁺×_R⁺ such that t_uu⁺+s_uu⁻∈ M_λ,κ forλsmall. In particular,M_λ,κ 6=_∅and for all(t,s)∈_R⁺×_R⁺, there is

J_λ,κ(tuu⁺+suu⁻)≥ J_λ,κ(tu⁺+su⁻).

Proof. For anyu∈ Hwithu^±6=0, define functiong :[0,∞)×[0,∞)→_Rgiven by g(t,s)_:= J_λ,κ(tu⁺+su⁻)

and its gradientΦ:[0,∞)×[0,∞)→_R×R, denoted by Φ(t,s):= _Φ1(t,s),Φ2(t,s)=^∂g

∂t(t,s),∂g

∂s(t,s)

=hJ_λ,κ⁰ (tu⁺+su⁻),u⁺i,hJ_λ,κ⁰ (tu⁺+su⁻),u⁻i. We simply compute, by(f₁)(f₂)and Sobolev inequality,

g(t,s)≥ ^at

2

2 ku⁺k²−εt²|u⁺|²₂−Ct^q|u⁺|^q_q+ ^as

2

2 ku⁻k²−εs²|u⁻|²₂−Cs^q|u⁻|^q_q

≥ ^at

2

2 ku⁺k²−εC₁t²ku⁺k²−C₂t^qku⁺k^q+ ^as

2

2 ku⁻k²−εC₃s²ku⁻k²−C₄s^qku⁻k^q, for smallε >0 and some positive constantsC_i (i= 1, 2, 3, 4). Therefore, g(t,s)is positive for (t,s)small. Since(f₃), fortlarge enough, there exists a large M >0 such that

f(t)≥ M|t|. (2.3)

Thus, for(t,s)large enough, we compute g(t,s) =J_λ,κ(tu⁺+su⁻)

= ^a

2t²ku⁺k²+ ^a

2s²ku⁻k²+ ^λ

4h_κ(tu⁺+su⁻)ktu⁺+su⁻k⁴−

Z

R^NF(tu⁺+su⁻)dx

= ^a

2t²ku⁺k²+ ^a

2s²ku⁻k²−

Z

R^NF(tu⁺) +F(su⁻)dx

≤ ^a

2t²ku⁺k²+ ^a

2s²ku⁻k²−Mt² Z

R^N|u⁺|²dx−Ms² Z

R^N|u⁻|²dx,

(2.4)

therefore, for(t,s)large enough, we haveg(t,s)→ −∞. So there is a pair of(t_u,s_u)such that g(t_u,s_u) =_max

t,s≥0g(t,s)_.

We next claim thatt_u,s_u >0. Indeed, without loss of generality, assuming the pair of(t_u, 0)is a maximum point ofg(t,s), we get that

∂

∂sg(tu,s) =asku⁻k²+ ^λ

4hκ(tuu⁺+su⁻)(4t²_usku⁺kku⁻k²+4s³ku⁻k²) + ^λs

2κ²h⁰_κ(tuu⁺+su⁻)ktuu⁺+su⁻k⁴ku⁻k²−

Z

R^N f(su⁻)u⁻dx

≥ asku⁻k²−

Z

R^N f(su⁻)u⁻dx−^λs κ²

kt_uu⁺+su⁻k⁴ku⁻k²_,

(2.5)

since condition (f₂), for λ,s enough small, we see that _∂s^∂g(t_u,s) > 0, which implies that g(t_u,s)is increasing forssmall. This contradicts that the pair of (t_u, 0)is a maximum point of g(t,s). Consequently,(tu,su)is a positive maximum point ofg(t,s).

Finally, we prove thatt_uu⁺+s_uu⁻∈ M_λ,κ. According to the definition ofΦ, we note that t_uu⁺+s_uu⁻ ∈ M_λ,κis equivalent toΦ(t,s) =_{0 for any}t,s >0. Because the pair of(t_u,s_u)_is a positive maximum point ofg(t,s), we observe that

∂

∂tg(t,s)|_(tu,su) = ^∂

∂sg(t,s)|_(tu,su)=0, is equal to

hJ_λ,κ⁰ (t_uu⁺+s_uu⁻),u⁺i= hJ_λ,κ⁰ (t_uu⁺+s_uu⁻),u⁻i=0, which is same as

Φ(tu,su) =0.

Thence, by virtue of the definition of nodal Nehari manifolds, we show that t_uu⁺+s_uu⁻ ∈ M_λ,κ, which finishes the proof.

Corollary 2.4. For any u∈ H\{0}, then there exists a tu∈_R⁺such that tuu∈ N_λ,κ forλsmall. In particular,N_λ,κ 6=_∅and for all t∈_R⁺, there is

J_λ,κ(t_uu)≥ J_λ,κ(tu)_.

Lemma 2.5(see Lions [14,15]). Let r>0and p∈ [2, 2^∗). If{un}is bounded in H and lim

n→_∞

sup

y∈_R^N Z

Br(y)

|u_n|^pdx=0, then we have un→0in L^q(_R^N)for q∈(2, 2^∗).

Lemma 2.6. Let{u_n} ⊂ N_λ,κbe a minimum sequence of J_λ,κat levelec_λ,κ, then{u_n}is bounded in H.

Proof. Arguing by contradiction, supposekunk →_∞asn →_{∞, and set}vn := _k^u_uⁿ

nk. Then there exists a v∈ Hsuch thatv_n *vin H, up to a subsequence. Moreover, for p∈[2, 2^∗), we have either{vn}is vanishing, i.e.,

lim

n→_∞

sup

y∈_R^N Z

Br(y)

|vn|^pdx=0

or non-vanishing, i.e., there existr,δ >0 and a sequence{y_n} ⊂_R^Nsuch that

nlim→_∞ Z

Br(yn)

|vn|^pdx≥ δ>_0.

We next shall prove neither vanishing nor non-vanishing occurs and this will provide the desired contradiction. If{v_n}is vanishing, by Lemma2.5, this implies v_n →0 in L^q(_R^N)for q∈ (_{2, 2}^∗). Then, for everyt>0, we have, in view of (f₁)_,(f₂)and Sobolev’s inequality,

ec_λ,κ+o(1) = J_λ,κ(u_n)≥ J_λ,κ(tv_n)

= ^at

2

2 kvnk²+ ^λ

4h(vn)kvnk⁴−

Z

R^N F(tvn)dx

≥ ^at

2

2 −εt² Z

R^Nv²_ndx−Cεt^q Z

R^N|v_n|^qdx

≥ ^at

2

2 −εC₁t²−Cεt^q Z

R^N|v_n|^qdx

→ ^at

2

2 −εC₁t²,

asn→∞. This yields a contradiction for enough larget .

Should non-vanishing occur, we then check that for enough largen, by (f₃) 0≤ ^Jλ,κ(u_n)

kunk² = ^a 2 −

Z

R^N

F(u_n)

u²_n |vn|²dx

≤ ^a 2−

Z

|un|>M

F(un)

u²_n |vn|²dx−

Z

|un|≤M

F(un)

u²_n |vn|²dx

≤ ^a 2−M

Z

|un|>M

|v_n|²dx

≤ ^a 2−M

Z

[|un|>M]∩Br(yn)

|v_n|²dx

≤ ^a 2−M

Z

Br(yn)

|v_n|²dx

≤ ^a 2−Mδ

<_0,

where Mis enough large. This is a contradiction and completes the proof.

Lemma 2.7. Let{u_n} ⊂ M_λ,κ be a minimum sequence for J_λ,κat level c_λ,κ, then{u_n}has a conver- gent subsequence in H.

Proof. Let{u_n} ⊂ M_λ,κ be such that

J_λ,κ(u_n)→c_λ,κ, asn→_∞.

Then, by Lemma2.6, we know that u_n is bounded in H and there exists a u ∈ H, up to a subsequence, such that

u_n *u in H,

u_n →u in L^p(_R^N)for p∈(2, 2^∗), u_n →u a.e. in R^N.

(2.6)

From(f₁)and(f₂), we have, forεsmall,

|f(t)| ≤ε|t|+C_ε|t|^q−1, for any t∈_R, (2.7) thus by Hölder’s inequality and Sobolev’s inequality, we get

Z

R^N f(u_n)(u_n−u)dx

≤ε|u_n|₂|u_n−u|₂+C_ε Z

R^N|u_n|^q−1|u_n−u|dx

≤ε|u_n|₂|u_n−u|₂+C_ε|u_n|^q_q⁻¹|u_n−u|_q.

(2.8)

Thus thanks to boundedness of{u_n}in Hand (2.6), we obtain that Z

R^N f(un)(un−u)dx→0 as n→_∞.

Then note that fornenough large,

o(1) =hJ_λ,κ⁰ (u_n),u_n−ui=a(u_n,u_n−u) +λh_κ(u_n)ku_nk²(u_n,u_n−u) + ^λ

2κ²h⁰_κ(u_n)ku_nk⁴(u_n,u_n−u)−

Z

R^N f(u_n)(u_n−u)dx

=

a+λhκ(un)kunk²+ ^λ

2κ²h⁰_κ(un)kunk⁴

(un,un−u) +o(1). (2.9)

It forces, as n→_∞,

a+λhκ(u)kunk²+ ^λ

2κ²h⁰_κ(un)kunk⁴

(un,un−u) =o(1).

From the definition ofh, we easily obtain(u_n,u_n−u)→ 0 andku_nk → kuk. Combining this with (2.6), we demonstrate thatu_n →uin H. This finishes the proof.

When {u_n} ⊂ N_λ,κ, using similar procedure of the proof above, we know that result of Lemma2.7also holds at levelce_λ,κ.

Lemma 2.8. The c_λ,κ is attained by some u∈ M_λ,κforλsmall, which is a critical point of J_λ,κ in H.

Proof. Let{u_n} ⊂ M_λ,κ be such that J_λ,κ(u_n) →c_λ,κ as n→ _∞. By Lemma2.7, we know that there exists au∈ Hsuch that

u_n→u, u⁺_n →v, u⁻_n →w,

(2.10)

in Hasn→_{∞. Since}u_n∈ M_λ,κ,

aku⁺_nk²+λh_κ(u_n)ku_nk²ku⁺_nk²+ ^λ

2κ²h⁰_κ(u_n)ku_nk⁴ku²_n=

Z

R^N f(u⁺_n)u⁺_ndx (2.11) Then, by (f₁), (f₂)and Sobolev’s inequality, we have

aku⁺_nk²−_4λκ⁴≤ aku⁺_nk²− ^λ κ²

ku_nk⁴ku⁺_nk²≤ ε Z

R^N|u⁺_n|²dx+C_ε Z

R^N|u⁺_n|^qdx

≤ εC₁ku⁺_nk²+C₂ku⁺_nk^q.

Soku⁺_nk ≥C₃ > 0, similarly,ku⁻_nk ≥ C₄ > 0. This implies thatv,w6= 0. Since His a Hilbert space and the project mappingu7→ u^±is continuous in H, we get u⁺= v andu⁻ =w, then u = u⁺+u⁻ is a sign-changing function. Next we prove u ∈ M_λ,κ. From u_n ∈ M_λ,κ, note that

hJ_λ,κ⁰ (_un)_,u⁺_ni=hJ_λ,κ⁰ (_un)_,u⁻_ni=_0, by (2.10) and passing to the limit, we obtain

hJ_λ,κ⁰ (u)_,u⁺i=hJ_λ,κ⁰ (u)_,u⁻i=_0,

which impliesu ∈ M_λ,κ and J_λ,κ(u) = c_λ,κ. Consequently, J_λ,κ|_Mλ,κ attains its minimum atu, thenuis a nontrivial critical point of J_λ,κ inM_λ,κ.

It remains to see thatuis a critical point of J_λ,κin H. Becauseuis a critical point ofJ_λ,κ in M_λ,κ, we have that J_λ,κ⁰ (u) = _{0 in}M_λ,κ. Moreover, there exists a Lagrange multiplierµsuch that

J_λ,κ⁰ (u)−_µΨ⁰(u) =0, (2.12) whereΨ(u) =hJ_λ,κ⁰ (u),ui. It suffices to prove that µ=0. By (2.12), we have

hJ_λ,κ⁰ (u),vi −µh_Ψ⁰(u),vi=0, for anyv∈ H. (2.13) Takingv=u, we compute that

h_Ψ⁰(u),ui=2akuk²+4λh_κ(u)kuk⁴+^5λ

κ²h⁰_κ(u)kuk⁶+ ^λ

κ⁴h_κ⁰⁰(u)kuk⁸−

Z

R^N f⁰(u)u²+ f(u)udx

= λ

2hκ(u)kuk⁴+ ⁴

κ²h⁰_κ(u)kuk⁶+ ¹

κ⁴h⁰⁰_κ(u)kuk⁸

−

Z

R^N f⁰(u)u²− f(u)udx

≤ λ

8κ⁴+64κ⁴+16κ⁴h⁰⁰_κ(u)−

Z

R^N f⁰(u)u²− f(u)udx.

In virtue of(f₄), we know that there exists a positive constant αsuch that Z

R^N f⁰(u)u²− f(u)udx≥α>0.

Therefore, h_Ψ⁰(u),ui< 0 for enough smallλ, together with (2.13), it showes thatµ= 0. The proof is completed.

Corollary 2.9. Theec_λ,κis attained by some u∈ N_λ,κ, which is a critical point of J_λ,κ in H.

The proof is similar to that of Lemma2.8, hence it is omitted here.

3 Proof of main results

According to the lemmas and corollaries in Section 2, we easily obtain the following results.

Theorem 3.1. Assume the conditions(f₁)–(f₄)hold, for λsmall, functional J_λ,κ possesses one least energy critical point u_λ which is constant sign and one least energy sign-changing critical point v_λ. Moreover, the energy of the sign-changing critical point is strictly greater than the least energy, that is,

c_λ,κ = J_λ,κ(v_λ)> J_λ,κ(u_λ) =_ec_λ,κ >0.

Proof. By the the lemmas and corollaries in Section 2, we know that J_λ,κ possesses a least energy critical point uλ and a least energy sign-changing critical pointvλ.

For v⁺_λ, in view of the foregoing discussions, there exists a t = t(v⁺_λ) > 0 such that tv⁺_λ ∈ N_λ,κ, then

0<_ec_λ,κ = J_λ,κ(u_λ)≤ J_λ,κ(tv⁺_λ) = J_λ,κ(tv⁺_λ +0v⁻_λ)< J_λ,κ(v⁺_λ +v⁻_λ) =c_λ,κ.

Finally, we will prove that u_λ is constant sign. Suppose thatu_λ is sign-changing, thenu_λ ∈ M_λ,κand

ec_λ,κ = _Jλ,κ(_uλ)≥ J_λ,κ(_vλ) =_cλ,κ >_ecλ,κ, this is absurd. We complete the proof.

Next we give an important identity to obtain thatu_λ andv_λ are bounded uniformly inH.

That is a Pohožaev type identity, which was proved in [11, Lemma 2.6], here we omit the details.

Lemma 3.2. If u∈ H is a weak solution of

a+λhκ(u)kuk²+ ^λ

2κ²h⁰_κ(u)kuk⁴

[−_∆u+bu] = f(u), x∈_R^N, (3.1) then forλsmall, the following Pohožaev type identity holds

N−2 2

Z

R^N|∇u|²dx+ ^Nb 2

Z

R^N|u|²dx a+λhκ(u)kuk²+ ^λ

2κ²h⁰_κ(u)kuk⁴

= N Z

R^N F(u)dx. (3.2) Lemma 3.3. For u_λand v_λobtained in Theorem3.1, ifκ>0is large enough andλ>0is sufficiently small, then u_λand v_λ are bounded in H, that is,ku_λk,kv_λk ≤κ.

Proof. This result was proved in [11, Lemma 2.7]. However, it plays a key role in proving Theorem 1.1 and for the sake of completeness and convenience to reader, we here give the detail. From J_λ,κ(v_λ) =c_λ,κ, we also write it as

1

2aNkv_λk²+¹

4Nh_κ(v_λ)kv_λk⁴−N Z

R^NF(v_λ)dx=c_λ,κN (3.3) By J_λ,κ⁰ (v_λ) =0, we know that (3.2) holds. Combining (3.2) and (3.3), we get that, forλsmall,

a 2

Z

R^N|∇v_λ|²dx ≤

a+λhκ(v_λ)kv_λk²+ ^λ

2κ²h⁰_κ(v_λ)kv_λk⁴ _Z

R^N|∇v_λ|²dx

=c_λ,κN+^λ

4Nh_κ(v_λ)kv_λk⁴+ ^λN

4κ²h_κ⁰(v_λ)kv_λk⁶.

(3.4)

Now we start to estimate the right hand side of (3.4). As the procedure in the proof of Lemma2.3, we have, by the definition of h,

c_λ,κ ≤ J_λ,κ(ϕ+ψ)

= ^a

2kϕk²+ ^a

2kψk²+^λ

4h_κ(ϕ+ψ)kϕ+ψk⁴−

Z

R^NF(ϕ+ψ)dx

= ^a 2+ ^a

2 −

Z

R^NF(ϕ+ψ)dx+^λ

4hκ(ϕ+ψ)kϕ+ψk⁴

≤ ^a 2+ ^a

2 −C₁ Z

B_R(0)ϕ²dx−C₁ Z

B_R(0)ψ²dx+C+λκ⁴

≤C₁+λκ⁴.

We also have that

λ

4Nhκ(v_λ)kv_λk⁴≤λNκ⁴, and

λN 4κ² h⁰_κ(v_λ)

kv_λk⁶≤4λNκ⁴. Then together with (3.4), we have

a 2

Z

R^N|∇v_λ|²dx ≤NC₂+6λNκ⁴. Since J_λ,κ⁰ (v_λ) =_{0, we have}

akvλk²+λhκ(vλ)kvλk⁴+^λN

4κ²h⁰_κ(vλ)kvλk⁶

=

Z

R^N f(v_λ)v_λdx≤ ε Z

R^Nv²_λdx+C_ε Z

R^Nv²_λ^∗dx. (3.5) Therefore, byD^1,2(_R^N)⊂ L^2∗(_R^N)and Sobolev’s inequality,

(a−ε)kv_λk²≤Cε

Z

R^Nv²_λ^∗dx− ^λN

4κ²h⁰_κ(v_λ)kv_λk⁶

≤C₃ Z

R^N|∇v_λ|²dx+8λκ⁴

≤C₄(NC2+6λκ⁴)^2∗/2+8λκ⁴.

(3.6)

Arguing by contradiction, supposekvλk ≥κ. Then, by (3.6), we have κ²≤ kv_λk² ≤C₅(NC₂+6λκ⁴)^2∗/2+8C₆λκ⁴,

which is impossible withκ large and λ small. Sokv_λk ≤κ, similarly, we get ku_λk ≤ κ. The proof is finished.

In what follows, we start to prove Theorem1.1.

Proof of Theorem1.1. Let κ and λbe large and small, respectively. By Theorem 3.1, we know that J_λ,κ possesses a least energy critical pointu_λat levelec_λ,κand a least energy sigh-changing critical point v_λ at level c_λ,κ, and according to Lemma 3.3, we obtain ku_λk,kv_λk ≤ κ, then J_λ,κ = Jλ and uλ,vλ are critical point critical of Jλ at level ecλ and cλ, respectively. There- fore, equation (1.1) has a least energy signed solution u_λ and a least energy sigh-changing solutionv_λ.

Finally, we will see the energy of sign-changing solution is strictly more than the least energy. FromJ_λ,κ = Jλ and Theorem3.1, we have

c_λ = J_λ(v_λ)> J_λ(u_λ) =_ec_λ >0.

Thus the proof is complete.

References

[1] C. O. Alves, F. J. S. A. Corrêa, T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,Comput. Math. Appl. Mat.49(2005), No. 1, 85–93.MR2123187;

url

[2] C. J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems, available online onarXiv:1501.05733, 2015.

[3] S. Bernstein, Sur une classe d’équations fonctionnelles aux dérivées partielles (in Russian), Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4(1940), 17–26.

MR0002699

[4] F. J. S. A. Corrêa, G. M. Figueiredo, On an elliptic equation of p-Kirchhoff type via variational methods,Bull. Austral. Math.74(2006), No. 2, 263–277.MR2260494;url

[5] G. M. Figueiredo, R. G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation,Math. Nachr.288(2014), No. 1, 48–60.MR3310498;url

[6] X. He, W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation inR³,J. Differential Equations252(2012), No. 2, 1813–1834.MR2853562;url [7] N. Ikoma, Existence of ground state solutions to the nonlinear Kirchhoff type equations

with potentials,Discrete Contin. Dyn. Syst.35(2015), No. 3, 943–966.MR3277180;url [8] L. Jeanjean, S. LeCoz, An existence and stability result for standing waves of nonlinear

Schrödinger equations,Adv. Differential Equations11(2006), No. 7, 813–840.MR2236583 [9] H. Kikuchi, Existence and stability of standing waves for Schrödinger–Poisson–Slater

equation,Adv. Nolinear Stud.7(2007), No. 3, 403–437.MR2340278 [10] G. Kirchhoff,Mechanik, Teubner, Leipzig, 1883.

[11] Y. Li, F. Li, J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,J. Differential Equations253(2012), No. 7, 2285–2294.MR2946973;

url

[12] G. Li, H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations inR³,J. Differential Equations257(2014), No. 2, 566–600.MR3200382;url

[13] J.-L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North- Holland Math. Stud., Vol. 30, North-Holland, Amsterdam, New York, 1978, pp. 284–346.

MR519648

[14] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I,Ann. Inst. H. Poincaré Anal. Non Linéaire 1(1984), No. 2, 109–145.

MR778970;url

[15] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II,Ann. Inst. H. Poincaré Anal. Non Linéaire1(1984), No. 2, 223–283.

MR778974;url

[16] S. S. Lu, Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains,J. Math. Anal. Appl.432(2015), No. 2, 965–982.MR3378403;url

[17] T. F. Ma, J. E. Muñoz Rivera, Positive solutions for a nonlinear elliptic transmission problem,Appl. Math. Lett.16(2003), No. 2, 243–248.MR1962323;url

[18] A. Mao, Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition,Nonlinear Anal.70(2009), No. 3, 1275–1287.MR2474918;url [19] D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four,

J. Differential Equations257(2014), No. 4, 1168–1193.MR3210026;url

[20] J. Nie, Existence and multiplicity of nontrivial solutions for a class of Schrödinger–

Kirchhoff-type equations,J. Math. Anal. Appl.417(2014), No. 1, 65–79.MR3191413;url [21] K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang in-

dex,J. Differential Equations221(2006), No. 1, 246–255.MR2193850;url

[22] S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) 138(1975), No. 1, 152–166MR0369938

[23] W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains,J. Differential Equations259(2015), No. 4, 1256–1274.MR3345850;url

[24] J. Sun, T. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well,J. Differential Equations 256(2014), No. 4, 1771–1792.MR3145774;url [25] M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their

Applications, Vol. 24, Birkhäuser, Boston, 1996.MR1400007;url

[26] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger–

Kirchhoff-type equations in R^N, Nonlinear Anal. Real World Appl. 12(2011), No. 2, 1278–

1287.MR2736309;url

[27] H. Ye, The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations inR^N,J. Math. Anal. Appl.431(2015), No. 2, 935–954.url

[28] Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow,J. Math. Anal. Appl.317(2006), No. 2, 456–463.MR2208932;url