# Multiplicity of Solutions for Schrödinger Equations with Concave-Convex Nonlinearities

Authors: Dong-Lun Wu; Chun-Lei Tang; Xing-Ping Wu
Journal: International Journal of Analysis
 (2016)

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## Abstract

We study the multiplicity of solutions for a class of semilinear Schrödinger equations:-Δu+V(x)u=gx,u,forx∈RN;u(x)→0,asu→∞, where V satisfies some kind of coercive condition and g involves concave-convex nonlinearities with indefinite signs. Our theorems contain some new nonlinearities.


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## Body

## 1. Introduction and Main Results

In this paper, we consider the multiplicity of solutions for the following semilinear Schrödinger equations:(1)-Δu+Vxu=gx,u,forx∈RN,ux⟶0,asu⟶∞.Equation (1) has many applications in mathematical physics. For instance, in finding the standing wave solutions for the following nonlinear Schrödinger equation(2)iħ∂φ∂t=-ħ22mΔφ+Wxφ-bx,φφ,we can see that a standing wave solution of (2) is a solution of the form (3)φx,t=uxe-iEt/ħ,where i=-1. The function φ(x,t) solves (2) if and only if u(x) solves (1) with V(x)=W(x)-E and g(x,u)=b(x,u)u.The existence and multiplicity of solutions for problem (1) have been studied by many mathematicians in last two decades [1–36]. In 1992, Coti Zelati and Rabinowitz [7] obtained the existence of infinitely many solutions for problem (1) when V(x) and g(x,u) are both periodic in x and g(x,u) is supposed to satisfy the following so-called Ambrosetti-Rabinowitz superlinear condition.(AR)
there existsμ>2 such that (4)tgx,t≥μGx,t>0,∀x∈RN,t∈R∖0,whereGx,t=∫0tgx,vdv.Condition (AR) provided a global growth condition of g at both origin and infinity, which plays an important role in showing the boundedness of Palais-Smale sequences and the geometrical structure for the corresponding functional. But (AR) is so strict that many functions do not satisfy this condition. An usual and weaker superlinear condition is(SQ)
G(x,u)/u2→∞ as u→∞ uniformly in x∈RN,which is first introduced by Liu and Wang [20] to obtain multiple solutions for superlinear elliptic equations and has been used by many mathematicians. Via a Nehari-type argument, Li et al. [19] obtained a ground state solution for problem (1) with the help of the following Nehari type assumption:(Ne)
u→g(x,u)/u is strictly increasing on (-∞,0) and (0,+∞).In a recent paper, (Ne) is weakened by Liu [17] when the author treated a class of periodic Schrödinger equations. He made the following assumption:(WN)
u→g(x,u)/u is increasing on (-∞,0) and (0,+∞).After then, there are some papers [28, 35, 36] that obtained existence and multiplicity of nontrivial solutions for problem (1) with condition (WN). Recently, Tang [26] introduced a new superlinear condition.(Ta)
there exists aτ0∈(0,1) such that (5)1-τ22tgx,t≥∫τttgx,sds=Gx,t-Gx,τt,∀τ∈0,τ0,x,t∈RN×R.With (Ta), Tang obtained the existence of ground state solutions for a class of superlinear Schrödinger equation involving some new nonlinearities. Motivated by the above works, in this paper, we shall study the multiplicity of solutions of problem (1) with concave-convex nonlinearities and the superlinear term satisfies some different growth assumptions from above. There are only few papers considering the concave-convex nonlinearities for problem (1). In [30], Wu considered problem (1) in a bounded domain with concave-convex nonlinearities and obtained two positive solutions when the weight function is indefinite in sign. After then, Wu [31] considered problem (1) in the entire space with sign-changing weight and obtained multiple positive solutions for problem (1). The results on multiple solutions for problem (1) with concave-convex nonlinearities can be also found in [10, 13]. But in [10, 13, 30, 31], the authors only considered the specific nonlinearities. In this paper, we consider a more general case. The potential V(x) satisfies the following coercive condition which is introduced by Bartsch and Wang in [4]:(V)
V∈C(RN,R), infx∈RN⁡V(x)>0. There exists r¯>0 such that (6)limy→∞⁡meas⁡x∈RN:x-y≤r¯,Vx≤M=0,∀M>0.The main purpose of this paper is to obtain multiplicity of solutions for problem (1) with some new nonlinearities. The nonlinear term g is considered to satisfy the following form:(7)gx,t=λfx,t+kx,t.Let F(x,t)=∫0tf(x,v)dv and K(x,t)=∫0tk(x,v)dv. Now we state our main results.Theorem 1.
Suppose that(V), (7), and the following conditions hold:(g1)
F(x,t)∈C1(RN×R,R) and F(x,0)=0.(g2)
There exitx¯∈RN, r0∈(1,2), and b0>0 such that F(x¯,t)>b0tr0 for all t∈R.(g3)
For any(x,t)∈RN×R, there exist r1,r2∈(1,2) such that(8)fx,t≤b1xtr1-1+b2xtr2-1,where bi(x)∈Lβi(RN,R+) and βi∈22∗/(2∗(2-ri)+2∗-2),2/(2-ri) for i=1,2.(g4)
K(x,t)=a(x)ts, where 2<s<2∗ and a(x)∈L∞(RN,R).(g5)
There exitsΘ⊂R such that a(x)>0 in Θ with meas⁡Θ>0.
Then, there existsλ1>0 such that for any λ∈(0,λ1), problem (1) possesses at least two solutions.Remark 2.
Sinceri>1, we can see that βi>22∗/(2∗(2-ri)+2∗-2)>2∗/(2∗-ri), which implies that riβi∗<2∗, where 1/βi+1/βi∗=1 for i=1,2.Remark 3.
It is easy to see that(g4) does not satisfy (AR), (SQ), (WN), and (Ta) since a(x) can change sign.Remark 4.
In 2005, Liu and Wang [21] also considered problem (1) with concave-convex nonlinearities. But in their theorems, the nonlinear term was assumed to be a specific form, which is different from our theorem. Furthermore, it was required that ∫RN(V(x))-1dx<+∞ in [21], which is not needed in (V).Theorem 5.
Suppose that(V), (7), (g1), (g3)–(g5), and the following condition hold:(g6)
F(x,-t)=F(x,t) for all (x,t)∈RN×R.
Then, for anyλ≥0, problem (1) possesses infinitely many solutions.Remark 6.
It is easy to see thatF(x,t) and K(x,t) are both indefinite in signs. The sign-changing nonlinear terms have been studied by Tang [25]. But in [25], the author only considered the case λ=0 and K(x,t) is positive when t is large enough which is different from (g5).Theorem 7.
Suppose that(V), (7), (g1), (g2), (g3), (g5), and the following conditions hold:(g7)
K∈C1(RN×R,R), K(x,0)=0 for all x∈RN.(g8)
K(x,t)/t2→+∞ as t→∞ uniformly in x.(g9)
There existγ>2 and d1,ρ∞>0 such that (9)tkx,t-γKx,t≥-d1t2,∀x∈RN,t≥ρ∞.(g10)
There existζ∈(2,2∗) and d2>0 such that (10)kx,t≤d2t+tζ-1,∀x,t∈RN×R.(g11)
k(x,t)=o(t) as t→0 uniformly in x.
Then, there existsλ2>0 such that for any λ∈(0,λ2), problem (1) possesses at least two solutions.Remark 8.
There are functions satisfying the conditions of(g7)–(g11), but not the condition (g4). For example, let (11)Kx,t=1+t21+t2t3.Theorem 9.
Suppose that(V), (7), (g1), (g3), (g6)–(g10), and the following condition hold:(g12)
There existsd3>0 such that K(x,t)≥-d3t2 for all x∈RN.(g13)
K(x,-t)=K(x,t) for all (x,t)∈RN×R.
Then, for anyλ≥0, problem (1) possesses infinitely many solutions.Remark 10.
In Theorem9, we only need (g9) to hold when t is large enough, which is different from the results in [25], in which the author required (g9) to hold in the entire space.Remark 11.
Obviously, it can be, respectively, deduced from (AR), (WN), and (Ta) that(WSQ)
tg(x,t)-2G(x,t)≥0 for all (x,t)∈RN×R.However, (WSQ) cannot be deduced from the conditions of our theorems and there are functions to show this difference. For example, let λ=0 and(12)Kx,t=-t4+t3,fort≤45,x-4+41/354+64-44/3625,fort≥45.It is easy to see that (12) satisfies the conditions (g7)–(g12), but not (WSQ).
In this paper, we will use the variational methods to prove our theorems. First, we introduce the definition of the(PS)∗ condition and (C) condition.Definition 12.
LetE be a Hilbert space. A functional I∈C1(E,R) is said to satisfy the (PS)∗ condition with respect to Ej, j=1,2,…, if any sequence xj∈Ej satisfying (13)Ixj<∞,I′Ejxj⟶0implies a convergent subsequence, where Ej is a sequence of linear subspace of E with finite dimensional.Definition 13.
LetE be a Hilbert space. A functional I∈C1(E,R) is said to satisfy the (C) condition if for any sequence {un}⊂E satisfying {I(un)} which is bounded and I′(un)(1+un)→0 as n→∞ possesses a convergent subsequence.
In our proof, the Mountain Pass Theorem and the following critical points theorems are employed.Lemma 14 (Lu [37]).
LetX be a real reflexive Banach space and Ω⊂X be a closed bounded convex subset of X. Suppose that φ:X→R is a weakly lower semicontinuous (w.l.s.c. for short) functional. If there exists a point x0∈Ω∖∂Ω such that (14)φx>φx0,∀x∈∂Ω,then there must be a x∗∈Ω∖∂Ω such that (15)φx∗=infx∈Ω⁡φx.Lemma 15 (Chang [6]).
Suppose thatI∈C1(E,R) is even with I(0)=0 and that(C1)
there are constantsϱ,α>0 and a finite dimensional linear subspace X such that IX⊥∩Sϱ≥α,(C2)
there is a sequence of linear subspaceX~m, dim⁡X~m=m, and there exists rm>0 such that (16)Iu≤0,onX~m∖Brm,m=1,2,….
If, further,I satisfies the (PS)∗ condition with respect to {X~m∣m=1,2,…}, then I possesses infinitely many distinct critical points corresponding to positive critical values.


## 2. Preliminaries

In this paper, we let(17)E≔u∈H1RN:∫RN∇u2+Vxu2dx<∞with the inner product (18)u,v=∫RN∇u·∇v+Vxuvdxand the norm u=u,u1/2. Then, E is a Hilbert space. For any 1≤p<∞, we denote (19)up=∫RNupdx1/p,u∞=esssup⁡ux:x∈RN.The embedding theorem shows that E↪Lp(RN) continuously for p∈[2,2∗], which implies that there exists a constant Cp>0 such that(20)up≤Cpufor all u∈E. The corresponding functional is defined on E as(21)Iu=12∫RN∇u2+Vxu2dx-∫RNGx,udx=12u2-λ∫RNFx,udx-∫RNKx,udx.With condition (V), we have the following compact embedding theorem.Lemma 16 (see [33]).
Under assumption(V), the embedding from E into Ls(RN) is compact for 2≤s<2∗.Lemma 17.
Suppose that(V), (g3), (g7), and (g10) hold; then, the functional I is well defined and of C1 class with(22)I′u,v=u,v-ψ′u,v-κ′u,v,for all v∈E, where ψ(u)=∫RNF(x,u)dx and κ(u)=∫RNK(x,u)dx. Moreover, the critical points of I in E are solutions for problem (1).Proof.
By(g3), (g7), and (g10), we have(23)Fx,t≤1r1b1xtr1+1r2b2xtr2,(24)Kx,t≤d2t2+tζfor all (x,t)∈RN×R. It follows from (g3) and (20) that there exists M1>0 such that(25)∫RNFx,udx≤M1ur1+ur2.Then, we can deduce that (26)∫RNGx,udx≤∫RNλFx,u+Kx,udx≤λ∫RNFx,udx+∫RNKx,udx≤λM1ur1+ur2+d2∫RNu2+uζdx≤λM1ur1+ur2+d2C22u2+Cζζuζ<∞,which implies that I is well defined. Similar to the proof of Proposition 2.2 in [34], we can see that ψ∈C1(E,R) and ψ′:E→E∗ is compact. Obviously, κ is also of C1 class and κ′ is compact, which means I is of C1 class and (22) holds. Finally, since E is continuously embedded into H1(RN), a standard argument shows that all critical points of I on E are solutions of (1). We finish the proof of this lemma.Remark 18.
Lemma17 still holds with (V) and (g4) since the functions in (g4) satisfy (g7) and (g10).
By Lemma17, we can easily obtain(27)I′u,u=u2-λ∫RNfx,uudx-∫RNkx,uudx.


## 3. Proof of Theorem1

Subsequently, we showI possesses the conditions of the Mountain Pass Theorem.Lemma 19.
Suppose the conditions of Theorem1 hold; then, there exist λ1,ϱ1,α1>0 such that I∂Bϱ1≥α1 for all λ∈(0,λ1), where Bϱ1={u∈E:u≤ϱ1}.Proof.
It follows from (21), (25), (g3), (g4), and (20) that(28)Iu=12u2-λ∫RNFx,udx-∫RNKx,udx≥12u2-λM1ur1+ur2-∫RNaxusdx≥12u2-λM1ur1+ur2-Cssa∞us≥12-λM1ur1-2+ur2-2-Cssa∞us-2u2.It is easy to see that there exist positive constants λ1, ϱ1, and α1 such that I|∂Bϱ1≥α1 for all λ∈(0,λ1). We finish the proof of this lemma.Lemma 20.
Suppose the conditions of Theorem1 hold; then, there exists e1∈E such that e1>ϱ and I(e1)≤0, where ϱ is defined in Lemma 19.Proof.
By Lusin’s Theorem and(g5), there exists Σ⊂Θ such that a(x) is continuous in Σ with measΣ>(1/2)measΘ>0 with infx∈Σ⁡a(x)>0. We choose φ1∈C0∞(Σ,R)∖{0}. Then, by (21), (25), (g3), and (g4), for any ξ>0, we obtain (29)Iξφ1=ξ22∫Σφ˙12dx-λ∫ΣFx,ξφ1dx-ξs∫Σaxφ1sdx≤ξ22∫Σφ˙12dx+λM1ξr1φ1r1+ξr2φ1r2-ξsa0∫Σφ1sdx,where a0=infx∈Σ⁡a(x), which implies that (30)I1ξφ1⟶-∞,asξ⟶+∞.Therefore, there exists ξ1>0 such that I1(ξ1φ1)<0. Let e1=ξ1φ1, we can see I(e1)<0, which proves this lemma.Lemma 21.
Suppose the conditions of Theorem1 hold; then, I satisfies the (C) condition.Proof.
Assume that{un}n∈N⊂E is a sequence such that {I(un)} is bounded and I′(un)(1+un)→0 as n→∞. Then, there exists a constant M2>0 such that(31)Iun≤M2,I′un1+un≤M2.Subsequently, we show that {un} is bounded in E. Arguing in an indirect way, we assume that un→∞ as n→∞. It follows from (31), (27), (21), (23), (g3), and (g4) that there exist M3,M4>0 such that (32)o1=s+1M2un2≥sIun+I′un1+unun2≥sIun-I′un,unun2=s2-1-λ∫RNsFx,un-fx,unundxun2≥s2-1-λM3∫RNb1xunr1+b2xunr2dxun2≥s2-1-λM4unr1-2+unr2-2⟶s2-1,asn⟶∞,which is a contradiction. Hence, {un} is bounded in E. Then, there exists a subsequence, still denoted by {un}, such that un⇀u in E. Therefore, (33)I′un-I′u,un-u⟶0,asn⟶+∞.Let σi=2/(ri-1) and ηi>0 satisfying 1/βi+1/σi+1/ηi=1, where i=1,2. By (g3), we can see that ηi∈[2,2∗) for i=1,2. It follows from (20) and Lemma 16 that (34)∫RNfx,un-fx,u,un-udx≤∫RNfx,un-fx,uun-udx≤∫RNb1xunr1-1+ur1-1+b2xunr2-1+ur2-1un-udx≤b1β1un2r1-1+u2r1-1un-uη1+b2β2un2r2-1+u2r2-1un-uη2⟶0,asn⟶∞.Similarly, we have (35)∫RNkx,un-kx,u,un-udx≤∫RNkx,un-kx,uun-udx=s∫RNaxuns-1+us-1un-udx≤sa∞unss-1+uss-1un-us⟶0,asn⟶∞.It follows from (27) that (36)I′un-I′u,un-u=un-u2-λ∫RNfx,un-fx,u,un-udx-∫RNkx,un-kx,u,un-udx,which implies that un-u→0 as n→+∞. Then, I satisfies the (C) condition.Lemma 22.
Suppose that the conditions of Theorem1 hold; then, there exists a critical point of I corresponding to negative critical value.Proof.
By Lemma19, we can see that there exists a local minimizer of I in Bϱ1, the following proof is to show this minimizer is not zero. By (g1) and (g2), there exists σ3>0 such that(37)Fx¯,t>12b0tr0for all x∈Υσ3(x¯) and t∈R, where Υσ3(x¯)={x∈RN:x-x¯≤σ3}. Choosing φ2∈C0∞(Υσ3(x¯),R)∖{0}, it follows from (21), (37), and (g4) that (38)Iθφ2=θ22φ22-λ∫RNFx,θφ2dx-θs∫RNaxφ2sdx≤θ22φ22-θr02λb0∫Υσ3x¯φ2r0dx+θsa∞∫Υσ3x¯φ2sdx<0for θ>0 small enough. By Lemmas 19 and 14, there exists U0∈Bϱ1∖∂Bϱ1 such that (39)IU0=infu∈Bϱ1⁡Iu<0<α1,I′U0=0.The proof of this lemma is finished.From Lemmas19–22, we can see that problem (1) possesses at least two solutions.


## 4. Proof of Theorem5

Lemma 23.
Suppose the conditions of Theorem5 hold; then, I satisfies (C1).Proof.
Let{vj}j=1∞ be a completely orthogonal basis of E and Xk=⨁j=1kSj, where Sj=span{vj}. For any q∈[2,2∗), we set(40)hkq=supu∈Xk⊥,u=1⁡uq.It follows from Lemma 2.10 in [25] that hk(q)→0 as k→∞ for any q∈[2,2∗). Set(41)Hk=λr1hkr1r1β1∗b1β1+λr2hkr2r2β2∗b2β2+hkssa∞.Then, there exists k0>0 such that Hk≤1/4 for all k≥k0. Then, for any u∈Xk0⊥∩∂Bϱ with 0<ϱ≤1, it follows from (21), (g3), (23), (g4), and (40) that(42)Iu=12u2-λ∫RNFx,udx-∫RNaxusdx≥12u2-λr1∫RNb1xur1dx-λr2∫RNb2xur2dx-a∞∫RNusdx≥12u2-λr1hk0r1r1β1∗b1β1ur1-λr2hk0r2r2β2∗b2β2ur2-hk0ssa∞us≥12u2-Hk0u≥12u2-14u.Hence, (42) shows that there exist α2>0 and ϱ2∈(0,1) such that I|Xk0⊥∩∂Bϱ2≥α2. We finish the proof of this lemma.Lemma 24.
Suppose the conditions of Theorem5 hold; then, I satisfies (C2).Proof.
LetΣ and a0 be as defined in Lemma 20. Then, it is easy to see that W01,2(Σ,R)⊂E and W01,2(Σ,R) is a Hilbert space. We can choose a sequence completely orthogonal basis {ej}j=1∞⊂W01,2(Σ,R). Let Rj=span{ej} and X~m=⨁j=1mRj. Then, for any um∈X~m, we have supp⁡um⊂Σ, where supp⁡um={x∈RN:um(x)≠0}¯. Since dim⁡X~m=m, there exists a constant Tm>0 such that(43)us≥Tmufor all u∈X~m. We can deduce from (21), (25), (g4), and (43) that (44)Ium=12um2-λ∫RNFx,umdx-∫RNKx,umdx≤12um2+λM1umr1+umr2-∫Σaxumsdx≤12um2+λM1umr1+umr2-Tmsa0ums.Then, there exists r(m)>0 such that I(um)≤0 for all um∈X~m∖Br(m), which proves this lemma.The proof of the following lemma is similar to Lemma21; we omit it here.Lemma 25.
Suppose the conditions of Theorem5 hold; then, I satisfies the (PS)∗ condition.Then, by Lemma15, we can deduce that I possesses infinitely many critical points, which implies that problem (1) has infinitely many solutions.


## 5. Proof of Theorem7

Lemma 26.
Suppose the conditions of Theorem7 hold; then, there exist λ2,ϱ3,α3>0 such that I∂Bϱ3≥α3 for all λ∈(0,λ2).Proof.
By(g7), (g10), and (g11), for any ε>0, there exists Dε>0 such that(45)Kx,t≤εt2+Dεtζ,∀x,t∈RN×R.It follows from (21), (45), (25), (g3), and (20) that(46)Iu=12u2-λ∫RNFx,udx-∫RNKx,udx≥12u2-λM1ur1+ur2-ε∫RNu2dx-Dε∫RNuζdx≥12u2-λM1ur1+ur2-εC22u2-DεCζζuζ=12-εC22-λM1ur1-2+ur2-2-DεCζζuζ-2u2.Letting ε<1/2C22, there exist positive constants λ2, ϱ3, and α3 such that I∂Bϱ3≥α3 for all λ∈(0,λ2).Lemma 27.
Suppose the conditions of Theorem7 hold; then, there exists e2∈E such that e2>ϱ3 and I(e2)≤0, where ϱ3 is defined in Lemma 26.Proof.
Sete3∈C0∞(Υ1(0),R) such that e3=1, where Υ is defined in Lemma 22. For M5>2∫Υ1(0)e32dx-1, it follows from (g8) that there exist Q>0 such that (47)Kx,t≥M5t2for all x∈Υ1(0) and t>Q. It follows from (g7) and (g11) that there exists ρ1>0 such that(48)Kx,t≤t2for all |t|≤ρ1 and x∈RN. It follows from (g8) and (48) that there exists d4>0 such that(49)Kx,t≥-d4t2for all (x,t)∈RN×R. Then, we can deduce from (47) and (49) that(50)Kx,t≥M5t2-Q2-d4Q2for all (x,t)∈Υ1(0)×R. By (21), (50), (20), and (25), for every η∈R+, we have (51)Iηe3=η22e32-λ∫RNFx,ηe3dx-∫RNKx,ηe3dx≤η22+λM1ηr1e3r1+ηr2e3r2-∫Υ10M5ηe32-Q2-d4Q2dx≤12-M5∫Υ10e32dxη2+λM1ηr1e3r1+ηr2e3r2+M5+d4Q2meas⁡Υ10,which implies that (52)Iηe3⟶-∞,asη⟶+∞.Therefore, there exists η1>0 such that I(η1e3)<0 and η1e3>ϱ3. Let e2=η1e3, we can see I(e2)<0, which proves this lemma.Lemma 28.
Suppose the conditions of Theorem7 hold; then, I satisfies the (PS) condition.Proof.
Assume that{un}n∈N⊂E is a sequence such that (53)Iun<∞,I′un⟶0,asn⟶∞.Then, there exists a constant M6>0 such that(54)Iun≤M6,I′unE∗≤M6.Subsequently, we show that {un} is bounded in E. Set (55)K~x,t=tkx,t-γKx,t,where γ is defined in (g9). Arguing in an indirect way, we assume that un→+∞ as n→∞. Set zn=un/un; then, zn=1, which implies that there exists a subsequence of {zn}, still denoted by {zn}, such that zn⇀z0 in E and zn→z0 uniformly on RN as n→∞. The following discussion is divided into two cases.
Case 1 ( z 0 ≢ 0 ). Let Ω={x∈RN∣z0x>0}. Then, we can see that meas(Ω)>0. Since un→+∞ as m→∞ and un=zn·un; then, we have un→+∞ as n→∞ for a.e. x∈Ω. On one hand, it follows from (21), (25), and (54) that (56)∫RNKx,unun2dx-12=Iunun2+λ∫RNFx,unun2dx≤M6un2+λM1unr1+unr2un2⟶0,asn⟶∞,which implies that(57)limn→∞⁡∫RNKx,unun2dx=12.On the other hand, by (g8), (49), and Fatou’s Lemma, we can obtain (58)limn→∞⁡∫RNKx,unun2dx≥limn→∞⁡∫ΩKx,unun2dx-d4limn→∞⁡∫RN∖Ωzn2dx≥limn→∞⁡∫ΩKx,unun2zn2dx-d4C22=+∞,which contradicts (57).
Case 2 ( z 0 ≡ 0 ). By (g10), we can deduce that (59)tkx,t≤d2t2+tζfor all (x,t)∈RN×R, which implies that(60)K~x,t=tkx,t-γKx,t≤d21+γt2+tζfor all (x,t)∈RN×R. It follows from (54), (21), (25), (g9), (g3), (20), (60), and Sobolev’s embedding theorem that (61)o1=γM6+M6unun2≥γIun-I′un,unun2≥γ2-1-λγmax⁡1/r1,1/r2+1un2Cr1β1∗r1b1β1unr1+Cr2β2∗r2b2β2unr2+1un2∫RNK~x,undx=γ2-1+1un2∫x∈RN∣un≤ρ∞K~x,undx+∫x∈RN∣un>ρ∞K~x,undx+o1≥γ2-1-1un2d21+γ∫x∈RN∣un≤ρ∞un2+unζdx+∫x∈RN∣un>ρ∞d1un2dx+o1≥γ2-1-1un2d21+γ1+ρ∞ζ-2∫x∈RN∣un≤ρ∞un2dx+∫x∈RN∣un>ρ∞d1un2dx+o1≥γ2-1-d21+γ1+ρ∞ζ-2+d1∫RNzn2dx+o1⟶γ2-1,asn⟶∞,which is a contradiction. Hence, {un} is bounded in E. The following proof is similar to Lemma 21. Then, I satisfies the (C) condition.It follows from the Mountain Pass Theorem that there exists a critical pointu^0 such that I(u^0)≥α3 and I′(u^0)=0, where α3 is defined in Lemma 26. Subsequently, we look for the second critical point of I by Lemma 14.Lemma 29.
Suppose that the conditions of Theorem7 hold; then, there exists a critical point of I corresponding to negative critical value.Proof.
Since we have (45), the proof of this lemma is similar to Lemma 22.Then, problem (1) possesses at least two solutions. The proof of Theorem 7 is finished.


## 6. Proof of Theorem9

In this section, we use Lemma15 to prove Theorem 9.Lemma 30.
Suppose the conditions of Theorem9 hold; then, I satisfies (C1).Proof.
LetXk and hk(q) be as defined in Lemma 23. For any u∈Xk⊥∩∂Bϱ with 0<ϱ≤1, it follows from (21), (23), (45), (20), and (40) that (62)Iu=12u2-λ∫RNFx,udx-∫RNKx,udx≥12u2-λr1∫RNb1xur1dx-λr2∫RNb2xur2dx-d2∫RNu2+uζdx≥12u2-λr1hkr1r1β1∗b1β1ur1-λr2hkr2r2β2∗b2β2ur2-d2hk22u2+hkζζuζ≥12u2-λr1hkr1r1β1∗b1β1+λr2hkr2r2β2∗b2β2+d2hk22+d2hkζζu.The following proof is similar to Lemma 23. Hence, I satisfies (C1). We finish the proof of this lemma.Lemma 31.
Suppose the conditions of Theorem9 hold; then, I satisfies (C2).Proof.
SetX~m=⨁j=1mSj, where Sj is defined in Lemma 23. For any u∈X~m∖{0} and ϑ>0, set (63)Γϑu=x∈RN:u≥ϑu.Similar to Lemma 2.4 in [34], there exists ϑ0>0 such that(64)meas⁡Γϑ0u≥ϑ0for all u∈X~m. By (g8), there exist ξ>0 such that(65)Kx,u≥1/2+d3C22+1ϑ03u2≥1/2+d3C22+1ϑ0u2for all u∈X~m and x∈Γϑ0(u) with u≥ξ, where d3 is defined in (g12). Choosing ςm>ξ, then for any u∈X~m∖Bςm, it follows from (21), (20), (g12), (25), (64), (g3), and (65) that (66)Iu=12u2-λ∫RNFx,udx-∫RNKx,udx≤12u2-λ∫RNFx,udx-∫Γϑ0uKx,udx+d3∫RN∖Γϑ0uu2dx≤12u2+λM1ur1+ur2-1/2+d3C22+1ϑ0meas⁡Γϑ0uu2+d3C22u2≤-u2+λM1ur1+ur2.Since 1<r1,r2<2, there exists r(m)>ξ such that I(um)≤0 for all u∈X~m∖Br(m), which proves this lemma.Lemma 32.
Suppose the conditions of Theorem9 hold; then, I satisfies the (PS)∗ condition.Proof.
Since we have(g12), the proof is similar to Lemma 28, we omit it here.Proof of Theorem9.
By Lemmas30–32 and 15, I possesses infinitely many distinct critical points corresponding to positive critical values.



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*Source: 8350396-2016-11-22.xml*