Multiple solutions for Kirchhoff type problems involving super-linear and sub-linear terms

Multiple solutions for Kirchhoff type problems involving super-linear and sub-linear terms

Xiaofei Cao and Junxiang Xu

Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, P. R. China Received 14 December 2015, appeared 24 March 2016

Communicated by Dimitri Mugnai

Abstract. In this paper, we consider the multiplicity of solutions for a class of Kirch- hoff type problems with concave and convex nonlinearities on an unbounded domain.

With the aid of Ekeland’s variational principle, Jeanjean’s monotone method and the Pohožaev identity we prove that the Kirchhoff problem has at least two solutions.

Keywords: Kirchhoff type problem, Pohožaev identity, variational method.

2010 Mathematics Subject Classification: 35A01, 35A15.

1 Introduction

This paper concerns the multiplicity of solutions for the following Kirchhoff type problem

−

a+b Z

R³|∇u|²dx

4u+u= f(u) +g(x)|u|^q−2u, x∈_R³, (1.1) where a,bare positive constants, 1< q<2, g(x)is a continuous function and f is a superlin- ear, subcritical nonlinearity.

Kirchhoff type problems were proposed by Kirchhoff in 1883 [14] as an extension of the classical D’Alembert’s wave equation for free vibration of elastic strings. Kirchhoff’s model takes into account the changes in the length of the string produced by transverse vibrations.

It is related to the stationary analogue of the equation (utt−(a+bR

Ω|∇u|²dx)4u=h(x,u) inΩ,

u=0 on ∂Ω, (1.2)

where u denotes the displacement, h(x,u)the external force andbthe initial tension whilea is related to the intrinsic properties of the string (such as Young’s modulus). Such problems are often viewed as nonlocal because the presence of the integral term R

Ω|∇u|²dx which implies that the problem (1.2) is no longer a pointwise identity. This phenomenon causes some mathematical difficulties making the study of such problems particularly interesting.

BCorresponding author. Email: xujun@seu.edu.cn

Besides, a similar nonlocal problem also appears in other fields such as physical and biological systems, whereudescribes a process that depends on its average, for example, the population density.

The case of Kirchhoff problems where the nonlinear term is super-triple or super-linear has been investigated in the last decades by many authors, for example [4–10,13,17–20] and references therein. Here, we are interested in the case of Kirchhoff problems where the non- linearity includes super-linear and sub-linear terms. Recently, Chen and Li [3] considered the following nonhomogeneous Kirchhoff equation

−

a+b Z

R^N|∇u|²dx

4u+V(x)u= f(u) +g(x) inR^N, (1.3) wherea,b>_0, N≤3. Under the conditionsg(x)∈L²(_R^N)_and

(V) V(x) ∈ C(_R^N,R), inf_RNV(x) ≥ C₁ > 0 and for each M > 0, meas({x ∈ _R^N : V(x) ≤ M}) < _{∞, where} C₁ is a positive constant and “meas” denotes the abbrevia- tion of Lebesgue measure inR^N;

(f₁) ^f(_s^s) →0 ass→0;

(f₂⁰) f ∈C(_R^N,R)and for some 2< p <2^∗ = _N^2N₋₂,C₂ >0,|f(s)| ≤C₂(1+|s|^p−1); (f₃⁰) there existsµ>4 such thats f(s)≥µF(s):=µRs

0 f(z)dz;

(f₄⁰) inf_x∈RN,|s|=1F(x,s)>0,

they proved that (1.3) has at least two solutions whenkgk_L2(_R^N) is small.

Jiang, Wang and Zhou [12] studied the following nonhomogeneous Schrödinger–Maxwell system

(−4u+u+λφ(x)u=|u|^p−2u+g(x) inR³,

−4φ=u² inR³, (1.4)

where λ > _0, p ∈ (_{2, 6}) _{and 0} ≤ g(x) = g(|x|) ∈ L²(_R³). By using a cut-off functional to obtain a bounded Palais–Smale sequence ((PS) sequence in short), they proved that there is a constant C_p > 0 such that (1.4) has at least two solutions for p ∈ (2, 6) provided that kgk_L2 ≤ C_p; however, for p∈(_{2, 3})they needed to assume in addition thatλ>0 is small.

Li, Li and Shi [15] considered the following Kirchhoff type problem

a+λ Z

R^N(|∇u|²+_bu²)_dx

(−4u+_bu) = _f(_u) _inR^N_{, (1.5)} whereN ≥3, a,b>0 and the parameterλ≥0. Under the conditions(f₁)and

(f) f ∈C(_R^N,R)and for some 2< p <2^∗ = _N^2N₋₂,C>0,|f(s)| ≤C(|s|+|s|^p−1); (f₂) ^f(_s^s) →+_∞as s→+_∞,

they proved that there existsλ₀>0 such that for anyλ∈[0,λ₀), (1.5) has at least one positive solution. Moreover, they pointed out that it is not clear whether (1.5) has a solution for large λ>0.

Motivated by these papers [3,8,12,15,17], we consider the Kirchhoff problem (1.1) with super-linear and sub-linear terms on the whole spaceR³. By the fact that the nonlocal term

(R

R³|∇u|²dx)² is homogeneous of degree 4 and that the nonlinearity is a combination of super-linearity and sub-linearity, we are unable to use the method in [3] to obtain a bounded (PS) sequence. Here, we overcome the difficulties with the aid of Jeanjean’s monotone method and the Pohožaev identity.

Theorem 1.1. Assume that in the problem (1.1), f(u) = |u|^p−2u with 2 < p < ₆ and g(x) is a nonnegative function with the following property:

(g₁) 0≤ g(x) =g(|x|)6=0and g(x)∈C¹(_R³)∩L^q∗(_R³), where q^∗ = ₂₋²_q; (g₂) h∇g(x)_,xi ∈L^q∗(_R³).

There existsσ>0such that if|g|_q^∗ ∈(0,σ), the problem(1.1)has two positive solutions, one of which has a positive energy and the other a negative energy.

Theorem 1.2. Assume that in the problem(1.1), f ∈C(_R³,R)satisfies the conditions(f₁),(f₂)and (f₃) there exists C₁,C₂ >0such that|^f(s)

s⁵ |<C₁if|s| ≥C₂;

(f₄) there existsµ>2such that f(s)s≥µF(s)>0,∀s6=0,where F(s) =Rs

0 f(z)dz.

Moreover, assume that g(x)is a nonnegative function satisfying the conditions(g₁),(g₂)and (g₃) g(x)− h∇g(x),xi ∈L^q(_R³), where q= ^µ

µ−q ∈ (2, 6).

There existsσ>0, which depends on f , such that if|g|_q^∗ ∈(_0,σ), the problem(1.1)has two solutions, one of which has a positive energy and the other a negative energy. Moreover, if, in addition, f(u)is odd, then the solutions are positive.

Remark 1.3. For example, f(u) = |u|^p−2u (₂ < p < ₆) satisfies the conditions(f₁)_–(f₄)_and g(x)∈C₀^∞(_R³)satisfies the conditions(g₁)–(g₃).

Remark 1.4. In the previous papers, because of the nonlocal term (R

R³|∇u|²dx)² with 4- degrees, the Kirchhoff problem (1.1) is usually considered under the condition (f₃⁰)_{, which} implies that f(u)is super-triple. In other way, the nonlinear condition (f₃⁰)demands N ≤ 3, thus the corresponding Kirchhoff type problem is usually studied in R^N with N ≤3. In the spirit of [3,8,12,15,17], we consider the Kirchhoff problem (1.1) under the condition(f₄)_{, which} implies the nonlinearity f(u)is a super-linear term. This nonlinear condition (f₄)can allow the dimension N ≥ 3. However, in this paper, for simplicity, we still consider the problem (1.1) inR³.

Remark 1.5. Consider the problem (1.1) with q=1:

−(a+b Z

R³|∇u|²dx)4u+u= f(u) +g(x), x∈_R³, (1.6) where 0 ≤ g(x) = g(|x|) ∈ L²(_R³) _and f satisfies the conditions (f₁)_–(f₄). By a similar method we can prove that there is a constant Cp > 0 such that if kgk_L2 ≤ Cp, (1.6) has two solutions with different signs of the energies.

Remark 1.6. We can also consider the following Kirchhoff problem:

−

a+b Z

R³|∇u|²dx

4u+V(x)u= f(u) +g(x)|u|^q−2u, x∈_R³, (1.7)

where 1≤ q< 2. Suppose thatV(x)satisfies the condition (V) andV(x) +h∇V(x),xisatis- fies suitable condition;g(x)is a continuous function and satisfies the conditions(g₁)–(g2)or (g₁)–(g₃); f is a superlinear and subcritical nonlinearity, which satisfies the conditions (f₁)–(f₄). With a similar method we can obtain similar results for (1.7).

This paper is organized as follows: Section 2 is dedicated to the abstract framework and some preliminary results. Sections 3 and 4 are concerned with the proofs of Theorems1.1and 1.2, respectively.

Throughout this paper, C or C_i is used in various places to denote distinct constants.

L^p(_R^N)denotes the usual Lebesgue space endowed with the standard norm

|u|_p = Z

R^N|u|^pdx ¹_p

for 1 ≤ p < _∞. When it causes no confusion, we still denote by {u_n}a subsequence of the original sequence{un}.

2 Preliminary results

In this section, we will recall some preliminaries and establish the variational setting for our problem. Since gis radially symmetric, we consider the problem in the radial space H_r¹(_R³), whose compactness is very important to our proof. LetE= H¹_r(_R³)be the subspace ofH¹(_R³) consisting of the radial functions and equipped with the norm

kuk² =

Z

R³(a|∇u|²+u²)dx, which is equivalent to the usual one fora >0.

The energy functional corresponding to (1.1) is I(u) = ¹

2 Z

R³(a|O^u|²+u²)dx+ ^b 4

_Z

R³|O^u|²dx 2

−

Z

R³ F(u)dx−¹ q

Z

R³g(x)|u|^qdx, where F(u) = Ru

0 f(s)ds. It is well known that a weak solution of problem (1.1) is a critical point of the functional I. In the following, we are devoted to finding critical points of I.

First we give the following lemma.

Lemma 2.1 ([1,22]). The embedding E ,→ L^p(_R³) is continuous for p ∈ [2, 2^∗] and compact for p∈(2, 2^∗). Denote by S_pthe best Sobolev constant for the embedding E,→L^p(_R³), which is given by

Sp= inf

u∈E\{0}

kuk²

|u|²_p >0.

In particular,

|u|_p ≤S⁻_p¹²kuk, ∀u∈ E. (2.1) In what follows, we recall the following two lemmata, which play an important role in obtaining a bounded (PS) sequence of I.

Lemma 2.2 ([11]). Let(X,k · k)be a Banach space and J ⊂ _R⁺be an interval. Consider the family of C¹functionals on X of the form

I_λ(u) =A(u)−λB(u), λ∈ J,

where B(u)≥0and either A(u)→_∞or B(u)→_∞askuk →∞. Assume that there are two points v₁,v₂∈ X such that

c_λ = inf

γ∈_Γmax

t∈[0,1]I_λ(γ(t))>max{I_λ(v₁),I_λ(v₂)}, λ∈ J, where

Γ={γ∈ C([_{0, 1}]_,X)|γ(₀) =_v1, γ(₁) =_v2}. Then, for almost everyλ∈ J, there is a sequence{vn} ⊂X such that

(i) {vn}is bounded;

(ii) I_λ(vn)→c_λ;

(iii) I_λ⁰(v_n)→0in the dual X⁻¹of X.

Furthermore, the mapλ7→c_λ is continuous from the left and non-increasing.

Lemma 2.3(Pohožaev identity [2,12,16]). Let u∈ H¹(_R³)be a weak solution to the problem(1.1), then we have the following Pohožaev identity:

0=¹ 2

Z

R³a|O^u|²dx+ ³ 2

Z

R³u²dx+ ^b 2

_Z

R³|O^u|²dx ₂

−3 Z

R³ F(u)dx−³ q

Z

R³g(x)|u|^qdx

−¹ q

Z

R³h∇g(x),xi|u|^qdx=:P(u).

(2.2)

3 Proof of Theorem 1.1

In this section, we are devoted to the proof of Theorem1.1, so we suppose that the assumptions of Theorem1.1hold throughout this section. First, we prove some useful preliminary results.

Lemma 3.1. There existsσ>₀such that if|g|_q^∗ ∈ (_0,σ), then there existα>₀andρ>₀such that I(u)|_kuk=α ≥ρ>0,

where

σ=qS

q

22C_p,q=qS

q

22





(₂−q)pS

p

p2

2(p−q)





2−q p−2

· ^p−2 2(p−q)^. Proof. By(g₁), the Hölder inequality and Lemma2.1, we have

I(u) = ¹

2kuk²+ ^b 4

_Z

R³|O^u|²dx 2

− ¹ p

Z

R³|u|^pdx−¹ q

Z

R³g(x)|u|^qdx

≥ ¹

2kuk²− ¹ pS⁻

p

p2kuk^p− ¹

q|g|_q^∗S⁻

q 2

2 kuk^q

=kuk^q 1

2kuk^2−q− ¹ pS⁻

p

p2kuk^p−q−¹

q|g|_q^∗S⁻

q

2 2

.

(3.1)

Setl(t) = ¹₂t^2−q− ¹_pS⁻

p

p2t^p−qfort>0. Direct calculations yield that

maxt>0 l(t) =l(α) =





(2−q)pS

p

p2

2(p−q)





2−q p−2

· ^p−2

2(p−q) =:Cp,q, where

α=





(2−q)pS

p

p2

2(p−q)





1 p−2

.

Then it follows from (3.1) that, if |g|_q^∗ < σ, I(u)|_kuk=α ≥ ρ > 0, where σ = qS^q/2₂ C_p,q and ρ=α^q l(α)− ¹_q|g|_q^∗S₂^−q/2

>0.

Lemma 3.2. If {u_n} ⊂ E is a bounded (PS) sequence of I, then {u_n} has a strongly convergent subsequence in E.

Proof. By Lemma2.1, going if necessary to a subsequence, we have un*u inE,

un→u in L^p(_R³), p ∈(2, 2^∗). Note that

(I⁰(u_n)−I⁰(u),u_n−u) = (I⁰(u_n),u_n−u)−(I⁰(u),u_n−u)

=

a+b Z

R³|O^un|²dx _Z

R³|O(u_n−u)|²dx+

Z

R³|u_n−u|²dx

−b Z

R³|O^u|²dx−

Z

R³|O^un|²dx Z

R³O^uO(u_n−u)dx

−

Z

R³ g(x)(|u_n|^q−2u_n− |u|^q−2u)(u_n−u)dx

−

Z

R³(|un|^p−2un− |u|^p−2u)(un−u)dx, then

ku_n−uk² ≤(I⁰(u_n)−I⁰(u),u_n−u) +b

_Z

R³|O^u|²dx−

Z

R³|O^un|²dx _Z

R³O^uO(u_n−u)dx +

Z

R³g(x)(|u_n|^q−2u_n− |u|^q−2u)(u_n−u)dx +

Z

R³(|u_n|^p−2u_n− |u|^p−2u)(u_n−u)dx.

From the boundedness of {u_n} in E and Lemma 2.1, {u_n}is bounded in L^p(_R³), p ∈ [2, 6). By twice using the Hölder inequality we obtain

Z

R³g(x)(|u_n|^q−2u_n− |u|^q−2u)(u_n−u)dx

≤ Z

R³|g|^q∗dx

_q¹∗ Z

R³||un|^q−2un− |u|^q−2u|^pq|un−u|^pqdx ^q_p

≤C|g|_q^∗|u_n|^q_p⁻¹+|u|^q_p⁻¹|u_n−u|_p →0 asn→_∞,

whereCis a positive constant. Similarly, we have

Z

R³(|u_n|^p−2u_n− |u|^p−2u)(u_n−u)dx

→0 asn→_∞, Combining with

b _Z

R³|O^u|²dx−

Z

R³|O^un|²dx _Z

R³O^uO(u_n−u)dx→0 asn→_∞, and

(I⁰(un)−I⁰(u),un−u)→0 asn→_∞, we haveku_n−uk →0 asn→∞. This completes the proof.

Lemma 3.3. There exists u₁ ∈E such that

I(u₁) =inf{I(u):u∈ B_α}<0, where Bα= {u∈E:kuk ≤α}andαis given in Lemma3.1.

Proof. We choose a functionv ∈ E such that g(x)v(x) 6= 0, then fort > 0 small enough, we have

I(tv)≤ ^a 2t²

Z

R³|O^v|²dx+^b 4t⁴

Z

R³|O^v|²dx 2

+¹ 2t²

Z

R³|v|²dx− ¹ qt^q

Z

R³g(x)|v|^qdx <0.

This shows that c₁ := inf{I(u) : u ∈ Bα} < 0. By Ekeland’s variational principle [21], there exists {u_n} ⊂ Bα which is a bounded (PS) sequence of I. Then, by Lemma 3.2, there exists u₁∈ Esuch thatu_n→u₁asn→_∞in E. Hence I(u₁) =c₁ <0 andI⁰(u₁) =0.

In order to apply Lemma2.2to get another solution, we introduce the following approxi- mation problem:

−

a+b Z

R³|∇u|²dx

4u+u=λ|u|^p−2u+g(x)|u|^q−2u, λ∈ [¹₂, 1]. (3.2) Define I_λ :E→_Rby

I_λ(u) =A(u)−λB(u), λ∈ [¹₂, 1], where B(u) = ¹_pR

R³|u|^pdxand A(u) = ¹

2 Z

R³(a|O^u|²+u²)dx+^b 4

_Z

R³|O^u|²dx 2

− ¹ q

Z

R³g(x)|u|^qdx.

Then {I_λ}_λ∈J is a family of C¹-functionals on E associated with the problem (3.2), where J = [¹₂, 1]. Obviously, we have B(u)≥0,∀u∈ E, and

A(u)≥ ¹

2kuk²−¹

q|g|_q^∗S⁻

q

2 2kuk^q→_∞ askuk →_∞.

In the following lemma, we show that the family of functionals{I_λ}satisfies the assump- tions of Lemma2.2.

Lemma 3.4. If|g|_q^∗ <σ, then for anyλ∈ J the following conclusions hold.

(i) There existη,ξ >0and e∈ E withkek>ξ such that

I_λ(u)≥η>0 with kuk=ξ and I_λ(e)<0.

(ii)

cλ := inf

γ∈_Γmax

t∈[0,1]Iλ(γ(t))>max{Iλ(0),Iλ(e)}, λ∈ J, whereΓ={γ∈ C([0, 1],E)|γ(0) =0, γ(1) =e}.

Proof. (i) Since I_λ ≥ I₁for all u∈ Eandλ∈ [¹₂, 1], by Lemma3.1there exist η>0 andξ > 0, which are independent ofλ∈ [¹₂, 1], such that I_λ(u)≥ η>0 withkuk=ξ.

Letv∈ E\{0}and setv_t(x) =tv(t⁻²x)fort >0, then we have I_λ(vt)≤ ^a

2t⁴ Z

R³|O^v|²dx+^b 4t⁸

_Z

R³|O^v|²dx 2

+¹ 2t⁸

Z

R³|v|²dx− ¹ pt^p+6λ

Z

R³|v|^pdx.

Noting that p ∈ (2, 6), there exists t₀ > 0 large enough, which is independent of λ ∈ [¹₂, 1], such thatIλ(vt0)<0 for all λ∈ [¹₂, 1]. Thus, by takinge=vt0(x), (i) holds.

(ii) By (i) and the definition ofc_λ,

c_λ ≥c₁ ≥η>0 for all λ∈[¹₂, 1]. SinceI_λ(₀) =_0,I_λ(e)<_{0 for all}λ∈[¹_{2, 1}], (ii) holds.

Then, thanks to Lemmata 2.2, 3.2 and 3.4, there exists {(λ_n,u_n)} ⊂ [¹₂, 1]×E such that λ_n→1 asn→_∞and

0< η≤ I_λn(u_n) =c_λn ≤c1

2, I_λ⁰_n(u_n) =0 for alln≥1. (3.3) In view of Lemma3.2, if the sequence{un} ⊂Egiven above is bounded, there existsu26=0 such thatI⁰(u₂) =0. In particular,u₂ is a non-trivial positive solution of the problem (1.1).

To complete the proof of Theorem1.1, we just require that{u_n} ⊂Eis bounded. Let A_n =

Z

R³|O^un|²dx, B_n=

Z

R³|u_n|²dx, C_n=λ_n Z

R³|u_n|^pdx and

D_n=

Z

R³ g(x)|u_n|^qdx, E_n=

Z

R³hO^g(x),xi|u_n|^qdx.

From (3.3) and Lemma2.3, we have











1

2(aA_n+B_n) + ^b₄(A_n)²− ¹_pC_n−¹_qD_n =c_λn, aA_n+B_n+b(A_n)²−C_n−D_n=0,

a

2An+³₂Bn+ ^b₂(An)²− ³_pCn− ³_qDn−¹_qEn=0.

(3.4)

Lemma 3.5. {u_n}is bounded in E.

Proof. We prove the lemma by the following two steps.

Step 1. {|u_n|₂}is bounded.

By contradiction, we assume that|u_n|₂→+_∞asn→_{∞. Let}v_n= _|^un

un|₂ and Xn =a

Z

R³|∇vn|²dx, Yn= b|un|²_{2 Z}

R³|∇vn|²dx 2

, Zn=λn|un|₂^p−2

Z

R³|vn|^pdx.

Using(g₁)and(g₂), and multiplying (3.4) by _|u¹

n|²₂, we see that











1

2Xn+ ¹₄Yn− ¹_pZn=−¹₂ +on(1), Xn+Yn−Zn= −1+on(1),

1

2X_n+ ¹₂Y_n− ³_pZ_n=−³₂ +o_n(1),

(3.5)

whereon(1)→0 asn →_{∞. For} p∈(2, 6), solving (3.5), we have Xn= ²−p

6−p +on(1).

This is a contradiction for n large enough, since X_n ≥ 0 for all n ∈ _{N. Thus,} {|u_n|₂} is bounded.

Step 2. {|∇u_n|₂}is bounded.

Similarly to the proof of Step 1, arguing by contradiction, if|∇u_n|₂ → _∞ as n → _{∞. Let} w_n= _|∇^un

un|₂,M_n =b|∇u_n|²₂(R

R³|∇w_n|²dx)²_andN_n =λ_n|∇u_n|^p₂⁻²R

R³|w_n|^pdx. Using(g₁)_,(g₂) and Step 1, and multiplying (3.4) by _|∇¹_u

n|²₂, we see that











1

4Mn− ¹_pNn= −^a₂+on(1), Mn−Nn= −a+on(1),

1

2M_n− ³_pN_n= −^a₂+o_n(1).

(3.6)

From the first two equations of (3.6),we have M_n= ^2a(p−2)

4−p +o_n(1), N_n= ^ap

4−p +o_n(1).

This together with the third equation of (3.6) implies that p = 6+on(1). So, if p 6= 6, (3.6) is impossible to hold. Thus,{|∇u_n|₂}is bounded.

Now we are in a position to give the proof of Theorem1.1.

Proof of Theorem1.1. By Lemma 3.3, we find a solutionu₁ of the equation (1.1) with negative energy. By Lemma3.4, due to the Mountain Pass Theorem [21], we get a critical point u₂ of I corresponding to positive energy. Because u₁ andu₂ have different energies, it follows that u₁ 6= u₂. Moreover, by strong maximum principle, u₁ and u₂ are positive. Thus, we obtain two positive solutions u₁ and u₂, one of which corresponds to positive energy and another one negative energy.

4 Proof of Theorem 1.2

At first, we assume that the assumptions of Theorem1.2always hold in this section. Since f is a nonhomogeneous nonlinearity, the method of Lemma3.5is not available. However, by the condition(g₃), we can still obtain a bounded (PS) sequence. Before proving Theorem1.2, we give some useful preliminary results.

Lemma 4.1. There existsσ>0, which depends on f, such that if|g|_q^∗ ∈(0,σ), then there existα>0 andρ>0such that

I(u)|_kuk=α ≥ρ>0.

Proof. From(f₁)and(f₃), for alle>0, there isC_e>0 such that

|f(u)| ≤e|u|+C_e|u|⁵. (4.1) Then, by Lemma2.1, we have

I(u) = ¹

2kuk²+ ^b 4

_Z

R³|O^u|²dx 2

−

Z

R³F(u)dx− ¹ q Z

R³ g(x)|u|^qdx

≥ ¹

2kuk²− ^e

2|u|²₂−^Ce

6 |u|⁶₆− ¹

q|g|_q^∗S⁻

q 2

2 kuk^q

≥ ¹

2kuk²− ^e

2S⁻₂¹kuk²− ^Ce

6 S₆⁻³kuk⁶−¹

q|g|_q^∗S⁻

q 2

2 kuk^q

≥ kuk^{q 1}−S₂⁻¹e

2 kuk^2−q− ^Ce

6 S₆⁻³kuk^6−q−¹

q|g|_q^∗S⁻

q 2

2

! .

(4.2)

We fixC_e withe= ^S₂². Then setl(t) = ¹₄t^2−q− ^C₆^eS⁻₆³t^6−qfort > 0. By direct calculations, it yields

maxt>0 l(t) =l(α) =

3S³₆(2−q) 2Ce(6−q)

2−q 4

· ¹

6−q =:C_q, where

α=

3S₆³(2−q) 2C_e(₆−q)

¹₄ .

Taking σ = qS₂^q/2Cq, then it follows from (4.2) that, if |g|_q^∗ < σ, I(u)|_kuk=α ≥ ρ > 0, where ρ=α^q l(α)− ¹_q|g|_q^∗S₂^−q/2

>0. Note thatCedepends on f, so doesσ.

Lemma 4.2. If {un} ⊂ E is a bounded (PS) sequence of I, then {un} has a strongly convergent subsequence in E.

Proof. Going if necessary to a subsequence, we have u_n * u in E. Inequality (4.1) and Lemma 3.2 of [18] imply thatR

R³F(u_n)dx = R

R³F(u_n−u)dx+R

R³ F(u)dx+o(₁), then, simi- larly to Lemma3.2, we can prove the result.

Lemma 4.3. There exists u₁∈ E such that

I(u₁) =inf{I(u): u∈B_α}<0, where B_α ={u∈ E:kuk ≤α}andαis given in Lemma4.1.

Proof. Since the proof is similar to Lemma3.3, we omit its details here.

In the same way as the previous section, we introduce the following approximation prob- lem:

−

a+b Z

R³|∇u|²dx

4u+u=λf(u) +g(x)|u|^q−2u, λ∈ [¹₂, 1]. (4.3) Define I_λ :E→_Rby

I_λ(u) = A(u)−λB(u), λ∈ [¹₂, 1] where B(u) =R

R³ F(u)dxand A(u) = ¹

2 Z

R³(a|O^u|²+u²)dx+^b 4

_Z

R³|O^u|²dx ₂

− ¹ q

Z

R³g(x)|u|^qdx.

Then {I_λ}_λ∈J is a family ofC¹-functionals on Ecorresponding to (4.3), where J = [¹₂, 1]. It is easy to see thatB(u)≥0, ∀u∈Eand

A(u)≥ ¹

2kuk²−¹

q|g|_q^∗S⁻

q 2

2 kuk^q→_∞ askuk →_∞.

Similarly to Lemma3.4, in the following lemma we want to show that {I_λ}satisfies the assumptions of Lemma2.2.

Lemma 4.4. If|g|_q^∗ <σ, then for anyλ∈ J, the following conclusions hold.

(i) There existη,ξ >0and e∈ E withkek>ξ such that

I_λ(u)≥η>0 with kuk=ξ and I_λ(e)<0.

(ii)

c_λ := inf

γ∈_Γmax

t∈[0,1]I_λ(γ(t))>max{I_λ(0),I_λ(e)}, λ∈ J, whereΓ={γ∈C([_{0, 1}]_,E)|γ(₀) =_0,γ(₁) =e}.

Proof. (i) Since Iλ ≥ I₁ for allu ∈ E andλ ∈ [¹₂, 1], by Lemma4.1 there existη,ξ > 0, which are independent of λ ∈ [¹₂, 1], such that I_λ(u) ≥ η > 0 with kuk = ξ. Let v ∈ C^∞₀ (_R³)such that 0 ≤ v ≤ 1,v(x) = 1 for|x| ≤1, v(x) = 0 for |x| ≥2, |O^v| ≤C and set v_t(x) = tv(t⁻²x) fort >0, then we have

I_λ(vt)≤ ^a 2t⁴

Z

R³|O^v|²dx+ ^b 4t⁸

_Z

R³|O^v|²dx 2

+¹ 2t⁸

Z

R³|v|²dx−t⁶λ Z

R³F(tv)dx

= ^a 2t⁴

Z

|x|≤2

|O^v|²dx+^b 4t⁸

Z

|x|≤2

|O^v|²dx 2

+¹ 2t⁸

Z

|x|≤2

|v|²dx

−t⁸λ Z

|x|≤2

F(tv) t²v² v²dx.

From the condition (f₂), F(tv)/(t²v²) → _∞ as t → ∞, so there exists t₀ > 0 large enough, which is independent of λ ∈ [¹₂, 1], such that I_λ(v_t0) < 0 for all λ ∈ [¹₂, 1]. Thus, by taking e=vt0(x), (i) holds.

(ii) By (i) and the definition ofc_λ,

c_λ ≥ c₁ ≥η>0, for all λ∈[¹₂, 1]. Since I_λ(₀) =_0,I_λ(e)<_{0 for all}λ∈[¹_{2, 1}], (ii) holds.

Then, thanks to Lemmata 2.2, 4.2 and 4.4, there exists {(λ_n,u_n)} ⊂ [¹₂, 1]×E such that λ_n→1 asn→_∞and

0< η≤ I_λn(u_n) =c_λn ≤c1

2, I_λ⁰_n(u_n) =0 for alln≥1. (4.4) In view of Lemma4.2, if the sequence {un} ⊂ E given above is bounded, there existsu₂ 6= 0 such thatI⁰(u₂) =0. In particular,u₂ is a non-trivial solution of problem (1.1).

To complete the proof of Theorem1.2, it is sufficient to prove that{u_n}is bounded inE.

Lemma 4.5. {u_n}is bounded in E.

Proof. From the condition (f₄), we can deduce that F(u) ≥ C|u|^µ with C = F(1). Let β ∈ (¹_µ,¹₂), then by (2.1), (4.4) and Lemma2.3, we have

4I_λn(u_n)−βI_λ⁰_n(u_n)u_n−P_λn(u_n)

= 3

2−β _Z

R³a|O^un|²+ 1

2 −β _Z

R³|un|²dx+ 1

2 −β

b _Z

R³|O^un|²dx 2

+λ_n Z

R³(βf(u_n)u_n−F(u_n))dx−

Z

R³

1 q−β

g(x)− ¹

qh∇g(x),xi

|u_n|^qdx

≥ 1

2−β

ku_nk²+λ_n(µβ−₁)C|u_n|^µ_µ−¹

q|u_n|^q_µ|g(x)− h∇g(x)_,xi|_q, whereq= ^µ

µ−q ∈(2, 6). Sinceq∈(1, 2),µ>2, {u_n}is bounded inE.

Now we are in a position to prove Theorem1.2.

Proof of Theorem1.2. By Lemma 4.3, we find a solution u₁ of the equation (1.1) with negative energy. By Lemma4.4, due to the Mountain Pass Theorem [21], we get a critical pointu₂ of I, whose energy is positive. Thus, u₁ and u₂ are two different solutions with their energies having different signs. If, in addition, f(u)is odd, the corresponding functional is even, then the solutionsu₁ andu₂are positive.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version. This work was supported by Natural Sci- ence Foundation of China (Nos: 11371090, 11571140), Natural Science Foundation of Jiangsu Province (Nos: BK 20140525, BK 20150478) .

References

[1] T. Bartsch, Z. Q. Wang, Existence and multiplicity results for some superlinear ellip- tic problems on R^N, Comm. Partial Differential Equations 20(1995), No. 9–10, 1725–1741.

MR1349229;url

[2] H. Berestycki, P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,Arch. Rational Mech. Anal.82(1983), No. 4, 313–345.MR0695535

[3] S. J. Chen, L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation onR^N, Nonlinear Anal. Real World Appl.14(2013), No. 3, 1477–1486.MR3004514

[4] B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems,J. Math. Anal. Appl.394(2012), No. 2, 488–495.MR2927472 [5] B. T. Cheng, X. Wu, Existence results of positive solutions of Kirchhoff type problems,

Nonlinear Anal.71(2009), No. 10, 4883–4892.MR2548720

[6] G. W. Dai, R. F. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math.

Anal. Appl.359(2009), No. 1, 275–284.MR2542174

[7] G. W. Dai, J. Wei, Infinitely many non-negative solutions for ap(x)-Kirchhoff-type prob- lem with Dirichlet boundary condition, Nonlinear Anal. 73(2010), No. 10, 3420–3430.

MR2680035

[8] Z. J. Guo, Ground states for Kirchhoff equations without compact condition,J. Differential Equations259(2015), No. 7, 2884–2902.MR3360660

[9] X. M. He, W. M. Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser.26(2010), No. 3, 387–394.MR2657696

[10] X. M. He, W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation inR³,J. Differential Equations252(2012), No. 2, 1813–1834.MR2853562 [11] L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on R^N, Proc. Roy. Soc. Edinburgh Sect. A129(1999), No. 4, 787–809.MR1718530

[12] Y. S. Jiang, Z. P. Wang, H. S. Zhou, Multiple solutions for a nonhomogeneous Schrödinger–Maxwell system inR³,Nonlinear Anal.83(2013), 50–57.MR3021537

[13] J. H. Jin, X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in R^N, J. Math. Anal. Appl.369(2010), No. 2, 564–574.MR2651702

[14] G. Kirchhoff,Mechanik, Teubner, Leipzig, 1883.

[15] Y. H. Li, F. Y. Li, J. P. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253(2012), No. 7, 2285–2294.

MR2946973

[16] Y. H. Li, F. Y. Li, J. P. Shi, Existence of positive solutions to Kirchhoff type problems with zero mass,J. Math. Anal. Appl.2014, No. 1, 361–374.MR3109846

[17] G. B. Li, H. Y. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations inR³,J. Differential Equations410(2014), No. 2, 566–600. MR3200382 [18] Z. S. Liu, S. J. Guo, Existence of positive ground state solutions for Kirchhoff type prob-

lems,Nonlinear Anal.120(2015), 1–13.MR3348042

[19] A. M. Mao, Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type prob- lems without the P.S. condition,Nonlinear Anal.70(2009), No. 3, 1275–1287.MR2474918 [20] J. Wang, L. X. Tian, J. X. Xu, F. B. Zhang, Multiplicity and concentration of positive solu-

tions for a Kirchhoff type problem with critical growth,J. Differential Equations253(2012), No. 7, 2314–2351.MR2946975

[21] M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996.MR1400007

[22] W. M. Zou, M. Schechter, Critical point theory and its applications, Springer, New York, 2006.MR2232879