# Multiplicity Results for a Class of Kirchhoff-Schrödinger-Poisson System Involving Sign-Changing Weight Functions

Authors: Dandan Yang; Chuanzhi Bai
Journal: Journal of Function Spaces
 (2019)

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## Abstract

This paper deals with the Kirchhoff-Schrödinger-Poisson system involving sign-changing weight functions. We prove the existence and multiplicity of solutions to the system. Our main results are based on the method of Nehari manifold.


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## Body

## 1. Introduction

In this paper, we consider the following Kirchhoff-Schrödinger-Poisson system with sign-changing weight functions:(1)a+b∫R3∇u2+Vxu2-Δu+Vxu+μlxϕu=λfxup-2u+gxuq-2u,x∈R3,-Δϕ=μlxu2,x∈R3,where a>0, b≥0, μ≥0, λ>0, 1≤p<2, and 4<q<6.We may assume that the weight functionsf(x) and g(x) satisfy the following conditions:(H1)f+=max⁡f,0≢0 and f∈Lνp(R3) where νp=p/6-p, and in addition f(x)≥0 a.e. in R3 in case p=1.(H2)g+=max⁡{g,0}≢0, and g∈Lνq(Ω), where νq=q/6-q.Whenμ=0, system (1) is the following Kirchhoff-type equation: (2)a+b∫R3∇u2+Vxu2-Δu+Vxu=λfxup-2u+gxuq-2u,x∈R3.The Kirchhoff-type equation was first put forward by Kirchhoff [1] as an extension of the classical D’Alembert wave equation for free vibrations of elastic string. Since the Kirchhoff-type problems arise in various models of physical and biological systems, many researchers have studied these problems in recent years; see [2–8] and references therein.Recently, in [9], using the symmetric mountain pass theorem, Zhao, Zhu, and Li investigated the existence of infinitely many solutions to system (1) with the nonlinearity λf(x)|u|p-2u+g(x)|u|q-2u being replaced by f(x,u), in which f(x,u) has sublinear growth in u.Whena=1, b=0, and the nonlinearity λf(x)|u|p-2u+g(x)|u|q-2u is replaced by f(x,u), system (1) reduces to the following Schrodinger-Poisson system: (3)-Δu+Vxu+μlxϕu=fx,u,x∈R3,-Δϕ=μlxu2,x∈R3,which has been first introduced in [10] as a physical model describing solitary waves interacting with its own electrostatic field in quantum mechanics. For more physical background of the system (3), we refer the readers to [11–13] and the references therein. In recent years, a great deal of work has been done in the study of (3) with μ=1 and l≡1 via variational methods and critical point theory under various hypotheses on the potential V and the nonlinear term f; see [14–20] and the references cited there.Motivated and inspired by [9, 21, 22] and the aforementioned works, in this paper, we investigate the existence and multiplicity of solutions to system (1) involving sign-changing weight functions. The main results we get are based on Nehari manifold. This article is organized as follows. In Section 2, we give some notations and preliminaries. Section 3 is devoted to the proof of our theorems.


## 2. Preliminaries

LetH1(R3) be the Sobolev space equipped with the norm uH1=∫R3[|∇u|2+V(x)u2]1/2. Let D1,2(R3)=u∈L6(R3):|∇u|∈L2(R3) with the norm uD1,2=∫R3[|∇u|2]1/2.Define(4)E=u∈H1R3:∫R3∇u2+Vxu2<∞.For V, l, the following hypotheses hold:(H3)V∈C(R3), V≥0. For every α>0, the set {x∈R3:V(x)≤α} has finite measure.(H4)l∈L2(R3), l≥0 and l≠0.It is known that from (H3)E is a Hilbert space with the inner product and endowed with the norm (5)u,vE=∫R3∇u·∇v+Vuv,u=u,uE1/2.Lemma 1 (see [9]).
By conditions (H3) and (H4),E is continuously embedded into H1(R3). Thus, the embedding E↪Lp(R3) is continuous for p∈[2,6], and then there exists μp>0 such that (6)uLp≤μpu,u∈E.Lemma 2 (see [9]).
For anyu∈E, there exists a unique ϕ=ϕu∈D1,2(R3) which solves the following equation: (7)-Δϕ=lxu2.Moreover, for any u∈E, ϕu can be expressed in the following form: (8)ϕux=14π∫R3lyu2yx-ydy,x∈R3.Lemma 3 (see [9]).
For anyu∈E, we have (i)
ϕu≥0 and ϕ-u=ϕu.(ii)
ϕuD1,22≤Cu4.(iii)
Ifun⇀u in E, then un2⇀u2 in L2(R3) and ϕun⇀ϕu in D1,2(R3).Now, we define the energy functionalIλ:E→R associated with problem (1) by (9)Iλu=a2u2+b4u4+μ4∫R3lϕuu2dx-λp∫R3fxupdx-1q∫R3gxuqdx.By [9], we know that Iλ is well defined and Iλ∈C1(E,R) with (10)Iλ′u,v=a+bu2∫R3∇u·∇v+Vuv+μ∫R3lϕuuvdx-λ∫R3fxup-2uvdx-∫R3gxuq-2uvdx,∀u,v∈E.Obviously, if u∈E is a critical point of Iλ, then the pair (u,ϕu) is a solution of system (1).The best Sobolev constantS is defined as follows: (11)S=infu∈D1,2R3,u≠0⁡∫R3∇u2dx∫R3u6dx1/3.The Nehari manifold forIλ is defined as (12)Nλ=u∈E∖0:Iλ′u,u=0=u∈E∖0∣au2+bu4+μ∫R3lϕuu2dx-λ∫R3fxupdx-∫R3gxuqdx=0.The Nehari manifoldNλ is closed linked to the behavior of functions of the form hλ:t→Iλ(tu) for t>0 named fibering maps [23]. If u∈E, we have (13)hλt=a2t2u2+b4t4u4+μ4t2∫R3lϕtuu2dx-λptp∫R3fxupdx-tqq∫R3gxuqdx=a2t2u2+b4t4u4+μ4t4∫R3lϕuu2dx-λptp∫R3fxupdx-tqq∫R3gxuqdx,hλ′t=atu2+bt3u4+μt3∫R3lϕuu2dx-λtp-1∫R3fxupdx-tq-1∫R3gxuqdx,and (14)hλ′′t=au2+3bt2u4+3μt2∫R3lϕuu2dx-p-1λtp-2∫R3fxupdx-q-1tq-2∫R3gxuqdx.Clearly, (15)thλ′t=at2u2+bt4u4+μt4∫R3lϕuu2dx-λtp∫R3fxupdx-tq∫R3gxuqdx=Iλ′tu,tu,which implies that for u∈E∖{0} and t>0, hλ′(t)=0 if and only if tu∈Nλ; i.e., positive critical points of hλ correspond to points on the Nehari manifold. In particular, hλ′(1)=0 if and only if u∈Nλ. Therefore, define (16)Nλ+=u∈Nλ:hλ′′1>0,Nλ0=u∈Nλ:hλ′′1=0,Nλ-=u∈Nλ:hλ′′1<0.For eachu∈Nλ, we have(17)hλ′′1=au2+3bu4+3μ∫R3lϕuu2dx-λp-1∫R3fxupdx-q-1∫R3gxuqdx=2-qau2+4-qbu4+4-qμ∫R3lϕuu2dx+λq-p∫R3fxupdx(18)=2-pau2+4-pbu4+4-pμ∫R3lϕuu2dx+p-q∫R3gxuqdx.Ifu∈Nλ0, then hλ′′(1)=0, and it follows from (17) and (18) that (19)2-qau2+4-qbu4+4-qμ∫R3lϕuu2dx+λq-p∫R3fxupdx=0,and (20)2-pau2+4-pbu4+4-pμ∫R3lϕuu2dx+p-q∫R3gxuqdx=0.Lemma 4.
If (H1)-(H4) hold, then the energy functionalIλ is coercive and bounded below on Nλ.Proof.
Foru∈Nλ, we have by Holder and Sobolev inequalities that (21)Iλu=a12-1qu2+b14-1qu4+μ14-1q∫R3lϕuu2dx-λ1p-1q∫R3fxupdx≥a12-1qu2+b14-1qu4-λ1p-1q∫R3fxupdx≥a12-1qu2+b14-1qu4-λ1p-1qfLνpS-p/2up,where νp=6/6-p. Thus Iλ is coercive and bounded below on Nλ.Lemma 5.
Assume that (H1)-(H4) hold. There existsλ1>0 such that, for any λ∈(0,λ1), we have Nλ0=∅.Proof.
If not, that isNλ0≠∅ for each λ>0, then by (19) and the Holder and Sobolev inequalities we have for u0∈Nλ0 that (22)λq-p∫R3fxu0pdx=aq-2u02+bq-4u04+μq-4∫R3lϕu0u02dx≥aq-2u02,which implies that (23)u02≤λq-paq-2∫R3fxu0pdx≤λq-paq-2fLνpS-p/2u0p,and so (24)u0≤λq-paq-2fLνpS-p/21/2-p.On the other hand, we obtain by (20) and the Holder and Sobolev inequalities that (25)a2-pu02≤a2-pu02+b4-pu04+μ4-p∫R3lϕu0u02dx=q-p∫R3gxu0qdx≤q-pgLνqS-q/2u0q,which implies that (26)u0≥a2-pSq/2q-pgLνq1/q-2.If λ is sufficiently small, then (24) contradicts with (26). Thus, we conclude that there exists λ1>0 such that Nλ0=∅ for λ∈(0,λ1).Let(27)cλ=infu∈Nλ⁡Iλu.From Lemma5, for λ∈(0,λ1), we have Nλ=Nλ+∪Nλ- and define (28)cλ+=infu∈Nλ+⁡Iλuandcλ-=infu∈Nλ-⁡Iλu.Lemma 6.
(i)
Ifu∈Nλ+, then ∫R3f(x)|u|pdx>0.(ii)
Ifu∈Nλ-, then ∫R3g(x)|u|qdx>0.The proof is an immediate result from (19) and (20).Define the functionku:R+→R as follows: (29)kut=at2-pu2+bt4-pu4+μt4-p∫R3lϕuu2dx-tq-p∫R3gxuqdx,t>0.Clearly, tu∈Nλ if and only if ku(t)=λ∫R3f(x)|u|pdx. Furthermore, (30)ku′t=a2-pt1-pu2+b4-pt3-pu4+μ4-pt3-p∫R3lϕuu2dx-q-ptq-p-1∫R3gxuqdx,which implies that tp-1ku′(t)=hλ′′(t) for tu∈Nλ. That is, u∈Nλ+ (or Nλ-) if and only if ku′(t)>0 (or <0).Set(31)M=aq-2q-pa2-pSq/2q-pgLνq2-p/q-2+bq-4q-pa2-pSq/2q-pgLνq4-p/q-2.Lemma 7.
Assume that (H1)-(H4) hold. Letλ2=MSp/2/fLνp. Then for each u∈E∖{0} and λ∈(0,λ2), we have the following.
(1) If∫R3f|u|pdx≤0, then there exists a unique t-=t-(u)>tmax(u) such that t-u∈Nλ- and (32)Iλt-u=supt≥0⁡Iλtu>0.
(2) If∫R3f|u|pdx>0, then there exists a unique 0<t+=t+(u)<tmax(u)<t- such that t+u∈Nλ+, t-u∈Nλ- and (33)Iλt+u=inf0≤t≤tmaxu⁡Iλtu,Iλt-u=supt≥0⁡Iλtu.Proof.
From (29) and (30), we have that ku(0)=0, ku(t)→-∞ as t→∞, limt→0+ku′(t)>0 and limt→∞ku′(t)<0. Thus there exists a unique tmax(u)≔tmax>0 such that ku(t) is increasing on (0,tmax) and decreasing on (tmax,∞) and ku′(tmax)=0. Moreover, tmax is the root of (34)a2-pu2+b4-ptmax2u4+μ4-ptmax2∫R3lϕuu2dx-q-ptmaxq-2∫R3gxuqdx=0,which implies that (35)q-ptmaxq-2∫R3gxuqdx≥a2-pu2.From (35), we obtain (36)tmax≥a2-pu2q-p∫Ωguqdx1/q-2≥1ua2-pSq/2q-pgLνq1/q-2≔t∗.Thus, we have by (35), (36), and the Holder and Sobolev inequalities that (37)kutmax=tmax2-pau2+btmax2u4+μtmax2∫R3lϕuu2dx-tmaxq-2∫R3gxuqdx=aq-2q-ptmax2-pu2+bq-4q-ptmax4-pu4+q-4q-ptmax4-p∫R3gxuqdx≥aq-2q-ptmax2-pu2+bq-4q-ptmax4-pu4≥aq-2q-pt∗2-pu2+bq-4q-pt∗4-pu4≥aq-2q-pa2-pSq/2q-pgLνq2-p/q-2up+bq-4q-pa2-pSq/2q-pgLνq4-p/q-2up=Mup.
Case 1 [∫Ωf|u|pdx≤0]. Then ku(t)=λ∫Ωf|u|pdx has unique solution t->tmax and ku′(t-)<0. On the other hand, we have (38)a2-pt-u2+b4-pt-u4+μ4-p∫R3lϕt-ut-u2dx-q-p∫R3gxt-uqdx=t-p+1ku′t-<0,and (39)Iλ′t-u,t-u=at-2u2+bt-4u4+μt-4∫R3lϕuu2dx-λt-p∫R3fxupdx-t-q∫R3gxuqdx=t-pkut--λ∫Ωfupdx=0.Therefore, t-u∈Nλ- or t-=1. For t>tmax, we obtain (40)a2-ptu2+b4-ptu4+μ4-p∫R3lϕtutu2dx-q-p∫R3gxtuqdx<0,d2dt2Iλtu<0,ddtIλtu=atu2+bt3u4+μt3∫R3lϕuu2dx-λtp-1∫R3fupdx-tq-1∫R3guqdx=0for t=t-. Thus, Iλ(u)=supt≥0Iλ(tu). Furthermore, we have (41)Iλu≥Iλtu≥a2t2u2+b4t4u4-1qtq∫R3guqdx,t≥0.Let (42)hut=a2t2u2+b4t4u4-1qtq∫R3guqdx,t≥0.Similar to the argument in the function ku(t), we get that hu(t) achieves its maximum at tm≥au2/∫R3g|u|qdx1/q-2>0. Thus, we have (43)Iλu≥hutm≥12-1qatm2u2+14-1qbtm4u4>0.
Case 2 [∫Ωf|u|pdx>0]. By (37), (44)ku0=0<λ∫Ωfupdx≤λfLνpS-p/2up<λ2fLνpS-p/2up=Mup≤kutmax,forλ∈0,λ2.Then there exist t+ and t- such that 0<t+<tmax<t-, (45)kut+=λ∫R3fupdx=kut-.Moreover, we have ku′(t+)>0 and ku′(t-)<0. Thus, there are two multiplies of u lying in Nλ, that is, t+u∈Nλ+ and t-u∈Nλ-, and Iλ(t-u)≥Iλ(tu)≥Iλ(t+u) for each t∈[t+,t-] and Iλ(t+u)≤Iλ(tu) for each t∈[0,t+]. Hence, t-=1 and (46)Iλu=supt≥0⁡Iλtu,Iλt+u=inf0≤t≤tmax⁡Iλtu.Let(H5)1≤p<2, and 4<q<6.Remark 8.
Ifq>9/2, then 4(q-4)>2>p, which implies that 1/4-q-4/p<0.Lemma 9.
Assume that conditions (H1)-(H5) are satisfied; then we havecλ≤cλ+<0.Proof.
Foru∈Nλ+, we obtain (47)λq-p∫R3fxupdx>q-2au2+q-4bu4+μq-4∫R3lϕuu2dx.Hence, we have (48)Iλu=aq-22qu2+bq-44qu4+μq-44q∫R3lϕuu2dx-λq-ppq∫R3fxupdx<aq-22qu2+bq-44qu4+μq-44q∫R3lϕuu2dx-1pqq-2au2+q-4bu4+μq-4∫R3lϕuu2dx=aq-2q12-1pu2+bq-4q14-1pu4+μq-4q14-1p∫R3lϕuu2dx<0,which implies that cλ≤cλ+<0.


## 3. Main Results

Using the idea of Ni-Takagi [24], we obtain the following Lemma.Lemma 10.
For eachu∈Nλ, there exist ϵ>0 and a differentiable function ξ:B(0;ϵ)⊂E→R+ such that ξ(0)=1, the function ξ(v)(u-v)∈Nλ, and (49)ξ′0,v=W2-pau2+4-pbu4+4-pμ∫R3lϕuu2dx+p-q∫R3gxuqdx,for all v∈E, where (50)W=2a+2bu2∫R3∇u·∇v+Vuv+4μ∫R3lϕuuvdx-λp∫R3fxup-2uvdx-q∫R3gxuq-2uvdx.Proof.
Foru∈Nλ, we define a function F:R×E→R by (51)Fuξ,w=Iλ′ξu-w,ξu-w=aξ2u-w2+bξ4u-w4+μξ4∫R3lϕu-wu-w2dx-λξp∫R3fxu-wpdx-ξq∫R3gxu-wqdx.Then Fu(1,0)=〈Iλ′(u),u〉=0 and (52)ddξFu1,0=2au2+4bu4+4μ∫R3lϕuu2dx-λp∫R3fxupdx-q∫R3gxuqdx=2-pau2+4-pbu4+4-pμ∫R3lϕuu2dx+p-q∫R3gxuqdx≠0.From the implicit function theorem, we know that there exist ϵ>0 and a differentiable function ξ:B(0;ϵ)⊂E→R such that ξ(0)=1, (53)ξ′0,v=W2-pau2+4-pbu4+4-pμ∫R3lϕuu2dx+p-q∫R3gxuqdx,where W is as in (50), and (54)Fuξv,v=0for  allv∈B0;ϵwhich is equivalent to (55)Iλ′ξvu-v,ξvu-v=0for  allv∈B0;ϵ,which implies that ξ(v)(u-v)∈Nλ.By means of the similar technique used in Lemma10, we have the following consequence.Lemma 11.
For eachu∈Nλ-, there exist ϵ>0 and a differentiable function ξ-:B(0;ϵ)⊂E→R+ such that ξ-(0)=1, the function ξ-(v)(u-v)∈Nλ-, and (56)ξ-′0,v=W2-pau2+4-pbu4+4-pμ∫R3lϕuu2dx+p-q∫R3gxuqdx,for all v∈E, where W is as in (50).Let(57)λ3=a2-pq-pq-p/q-2·q-44-p·1fLνpgLνq2-p/2-qSq-p/q-2.Lemma 12.
Assume that (H1)-(H5) hold. IfΓ0=min⁡λ1,λ2,λ3, then for λ∈(0,Γ0),
(i) there exists a minimizing sequence{un}⊂Nλ such that (58)Iλun=cλ+o1,Iλ′un=o1inE∗;
(ii) there exists a minimizing sequence{un}⊂Nλ- such that (59)Iλun=cλ-+o1,Iλ′un=o1inE∗.Proof.
By the Ekeland variational principle [11] and Lemma 5, there exists a minimizing sequence {un}⊂Nλ such that (60)Iλun<cλ+1nand(61)Iλun<Iλw+1nw-un∀w∈Nλ.Let n be large enough; by Lemma 9, we get (62)Iλun=aq-22qun2+bq-44qun4+μq-44q∫R3lϕunun2dx-λq-ppq∫R3fxunpdx<cλ+1n<cλ2,which implies that (63)fLνpS-p/2unp≥∫R3fxunpdx>-pqλq-pcλ2>0.This implies un≠0 and by using (61), (63), and the Holder inequality, we obtain (64)un>-pqλq-pcλ2fLνp-1Sp/21/pand (65)un<2λq-papq-2fLνpS-p/21/2-p.In the following, we will prove that (66)Iλ′unE∗→0asn→∞.In fact, by using Lemma 10 with un to get the functions ξn:B(0;ϵn)→R+ for some ϵn>0, such that ξn(w)(un-w)∈Nλ, fixed n∈N and we choose 0<ρ<ϵn. Let u∈E with u≠0 and let wρ=ρu/u. Set ηρ=ξn(wρ)(un-wρ); since ηρ∈Nλ, we deduce from (61) that (67)Iληρ-Iλun≥-1nηρ-un,∀w∈Nλ,and by the Mean Value Theorem, we get (68)Iλ′un,ηρ-un+oηρ-un≥-1nηρ-un.Therefore, (69)Iλ′un,-wρ+ξnwρ-1Iλ′un,un-wρ≥-1nηρ-un+oηρ-un.By ξn(wρ)(un-wρ)∈Nλ and (69) it follows that (70)-ρIλ′un,uu+ξnwρ-1Iλ′un-Iλ′ηρ,un-wρ≥-1nηρ-un+oηρ-un.Hence, (71)Iλ′un,uu≤1nρηρ-un+1ρoηρ-un+ξnwρ-1ρIλ′un-Iλ′ηρ,un-wρ.Since (72)ηρ-un≤ρξnwρ+ξnwρ-1unand(73)limn→∞⁡ξnwρ-1ρ≤ξn′0,taking limit ρ→0 in (71), we obtain (74)Iλ′un,uu≤Cn1+ξn′0for some constant C>0, independent of ρ. In the following, we will show that ξn′(0) is uniformly bounded in n. From (49), (65), and the Holder inequality, we obtain for some κ>0 that (75)ξn′0,v≤κv2-paun2+4-pbun4+4-pμ∫R3lϕunun2dx+p-q∫R3gxunqdx.We only need to prove that (76)2-paun2+4-pbun4+4-pμ∫R3lϕunun2dx+p-q∫R3gxunqdx>cfor some c>0 and n large enough. If (76) fails, then there exists a subsequence {un} such that (77)2-paun2+4-pbun4+4-pμ∫R3lϕunun2dx+p-q∫R3gxunqdx=o1.Combining (77) with (64), we may find a suitable constant d>0 such that (78)∫R3gxunqdx≥dfornsufficiently  large.By (77) and un∈Nλ, we have (79)λ∫R3fxunpdx=aun2+bun4+μ∫R3lϕunun2dx-∫R3gxunqdx=14-pa4-pun2+b4-pun4+4-pμ∫R3lϕunun2dx-∫R3gxunqdx≥14-pa2-pun2+b4-pun4+4-pμ∫R3lϕunun2dx-∫R3gxunqdx≥q-p4-p∫R3gxunqdx-∫R3gxunqdx+o1≥q-44-p∫R3gxunqdx+o1.Furthermore, we obtain by (77) and (79) that (80)2-paun2≤2-paun2+4-pbun4+4-pμ∫R3lϕunun2dx=q-p∫R3gxunqdx+o1≤λq-p4-pq-4∫R3fxunpdx+o1≤λq-p4-pq-4fLνpS-p/2unp+o1,which implies that (81)un≤λq-p4-pa2-pq-4fLνpS-p/21/2-p+o1.Let (82)Jλu=Kp,qu2q-1∫R3gxunqdx1/q-2-λ∫R3fxunpdx,where(83)Kp,q=a2-pq-pq-1/q-2q-44-p.In view of (77), it is easy to know that (84)un2≤q-pa2-p∫R3gxunqdx.Thus, (85)Jλun≤a2-pq-pq-1/q-2q-44-pq-p/a2-pq-1∫R3gxunqdxq-1∫R3gxunqdx1/q-2-q-44-p∫R3gxunqdx+o1=o1.But, by (78), (81), and λ<λ3, (86)Jλun≥Kp,qun2q-1gLνqS-q/2unq1/q-2-λfLνpS-p/2unp=unpKp,qgLνq1/2-qSq/2q-2un1-p-λfLνpS-p/2≥unpKp,qgLνq1/2-qSq/2q-2λq-p4-pa2-pq-4fLνpS-p/21-p/2-p-λfLνpS-p/2>0,which contradicts (85). Hence, we obtain (87)Iλ′un,uu≤Cn.This completes the proof of (i). Similarly, we can prove (ii) by using Lemma 11.Theorem 13.
Assume that (H1)-(H5) hold. For each0<λ<Γ0 (Γ0 is as in Lemma 12), the functional Iλ has a minimizer uλ+ in Nλ+ satisfying that (1)
Iλ(uλ+)=cλ+=cλ;(2)
uλ+ is a solution of (1).Proof.
By Lemma12 (i), there exists a minimizing sequence {un}⊂Nλ for Iλ on Nλ such that (88)Iλun=cλ+o1,Iλ′un=o1inE∗.From Lemma 9 and the compact imbedding theorem, we know that there exist a subsequence {un} and uλ+∈E such that (89)un⇀uλ+weakly  inEand (90)un→uλ+strongly  inLηR3for1<η<6.
In the following we will prove that∫R3f(x)|uλ+|pdx≠0. In fact, if not, by (90) and the Holder inequality we can get that (91)∫R3fxunpdx→∫R3fxuλ+pdx=0as n→∞. Hence, (92)aun2+bun4+μ∫R3lϕunun2dx=∫R3gxunqdx+o1and (93)Iλun=a12-1qun2+b14-1qun4+μ14-1q∫R3lϕunun2dx+o1,which contradicts Iλ(un)→cλ<0 as n→∞. Moreover, (94)o1=Iλ′un,ϕ=Iλ′uλ+,ϕ+o1for  allϕ∈E.Hence, uλ+∈Nλ is a nonzero solution of (1) and Iλ(uλ+)≥cλ. Next, we will prove that Iλ(uλ+)=cλ. By Lemma 5, we get that (95)un⇀uλ+inL6R3,un2⇀uλ+2inL3R3,ϕun⇀ϕuλ+inL6R3.By using (3.7) in [9], we obtain that (96)∫R3lϕunun2dx→∫R3lϕuλ+uλ+2dx.Combining Fatou’s lemma and (96), we have (97)Iλuλ+=a12-1quλ+2+b14-1quλ+4+μ14-1q∫R3lϕuλ+uλ+2dx+λ1q-1p∫R3fxuλ+pdx≤lim⁡infn→∞⁡a12-1qun2+b14-1qun4+μ14-1q∫R3lϕunun2dx+λ1q-1p∫R3fxunpdx=lim⁡infn→∞⁡Iλun=cλ.Hence, Iλ(uλ+)=cλ. Moreover, we have uλ+∈Nλ+. In fact, if uλ+∈Nλ-, by Lemma 7, there are unique t+ and t- such that t+uλ+∈Nλ+ and t-uλ+∈Nλ-; we have tλ+<tλ-=1. Since (98)ddtIλtλ+uλ+=0andd2dt2Iλtλ+uλ+>0,there exists tλ+<t∗≤tλ- such that Iλ(tλ+uλ+)<Iλ(t∗uλ+). By means of Lemma 4, we get (99)Iλtλ+uλ+<Iλt∗uλ+≤Iλtλ-uλ+=Iλuλ+,which is a contradiction. Since Iλ(uλ+)=Iλ(|uλ+|) and |uλ+|∈Nλ+, we have that uλ+ is a solution of (1) in virtue of Lemma 6.Similarly, we can obtain the theorem of existence of a local minimum forIλ on Nλ- as follows.Theorem 14.
Assume that (H1)-(H5) hold. For each0<λ<Γ0, the functional Iλ has a minimizer uλ- in Nλ- satisfying that (1)
Iλ(uλ-)=cλ-;(2)
uλ- is a solution of (1).Finally, we give the main result of this paper as follows.Theorem 15.
Suppose that conditions (H1)-(H5) hold. Then there existsΓ0>0 such that for λ∈(0,Γ0), (1) has at least two solutions.Proof.
From Theorems13 and 14, we have that (1) has two solutions uλ+ and uλ- such that uλ+∈Nλ+, uλ-∈Nλ-. Since Nλ+∩Nλ-=∅, we get that uλ+ and uλ- are different, which completes the proof.



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*Source: 6059459-2019-06-02.xml*