# Positive Solutions for Schrödinger-Poisson Systems with Sign-Changing Potential and Critical Growth

Authors: Pengfei He; Hongmin Suo
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we study the existence of positive solutions for Schrödinger-Poisson systems with sign-changing potential and critical growth. By using the analytic techniques and variational method, the existence and multiplicity of positive solutions are obtained.


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## Body

## 1. Introduction and Main Result

In this paper, we consider the existence and multiplicity of positive solutions of the Schrödinger-Poisson systems with critical nonlinearity of the following form:(1)−Δu−ϕu=λfλxuq−2u+u4u,inΩ,−Δϕ=u2,inΩ,u=ϕ=0,on∂Ω,where Ω is a smooth bounded domain in ℝ3, 1<q<2, fλ=λf++f−,λ>0, and f±=±max±f,0. When N=3, 2∗=2N/N−2=6, the right hand side of the system of (1) has the critical term u4u. It is pointed out that system (1) is related to the following Schrödinger-Poisson system:
(2)−Δu+Vxu+kxϕuq−2u=fx,u,inℝ3,−Δϕ=kxuq,inℝ3,where the potentials Vx, kx satisfy some mild conditions. The existence, uniqueness, and multiplicity of positive solutions of systems like (2) have been extensively studied in the last few decades, such as [1–14]. Furthermore, the Schrödinger-Poisson systems involving critical growth have been attracted many researchers, e.g., [15–24]. But there is little literature for liking the system (1) that considers the critical growth on bounded domains, such as [15–17].In [24], the authors considered a Schrödinger-Poisson system with concave-convex nonlinearities of the form
(3)−Δu+λϕu=λuq−1+u5,inΩ,−Δϕ=u2,inΩ,u=ϕ=0,on∂Ω,where Ω is a smooth bounded domain in ℝ3, 1<q<2 and λ>0. They obtained two positive solutions when λ is enough small by using the concentration-compactness principle.Recently, in [25], the authors considered a Schrödinger-Newton system with critical nonlinearities of the form
(4)−Δu=fλxuq−2u+ϕu3u,inΩ,−Δϕ=u5,inΩ,u=ϕ=0,on∂Ω,where Ω is a smooth bounded domain in ℝ3, 1<q<2, fλ=λf++f−,λ>0, and f±=±max±f,0. The authors studied the existence and multiplicity of positive solutions by the variational method and analytic techniques.Motivated by results found in the above mentioned papers, in the present paper, we intend to study the existence of multiple positive solutions for system (1). Since system (1) contains a critical term and a nonlocal term, the critical value level becomes very difficult to estimate. The main objective of this paper is to look for new estimates and establish two positive solutions by analytic techniques.The main result is as follows:Theorem 1.
Assume that1<q<2 and fλ∈L∞Ω is a possibly sign-changing function. Then, there exists λ∗>0 such that for any λ∈0,λ∗, system (1) has at least two different positive solutions. Additionally, system (1) admits a positive ground state solution.
Throughout this paper, we use the following notations:(i)
The spaceH01Ω is equipped with the norm ∥u∥2=∫Ω∇u2dx. The norm in LpΩ is denoted by ⋅p(ii)
C,C1,C2,⋯ denote various positive constants, which may vary from line to line(iii)
We denote byB0,r (respectively, ∂B0,r) the closed ball (respectively, the sphere) of center zero and radius r, i.e., B0,r=u∈H01Ω:∥u∥≤r, ∂B0,r=u∈H01Ω:∥u∥=r(iv)
LetS be the best Sobolev embedding constant for the embedding H01Ω↣L6Ω, namely,(5)S=infu∈H01Ω\0∫Ω∇u2dx∫Ωu6dx1/3


## 2. Existence of a First Positive Solution of System (1)

For any givenu∈H01Ω, by the Lax-Milgram theorem, the Dirichlet boundary problem −Δϕ=u2 in Ω has a unique solution ϕu∈H01Ω. Substitute ϕu in the first equation of system (1), system (1) transformed into the following variable equations:
(6)−Δu−ϕu=λfλxuq−2u+u4u,inΩ,−Δϕ=u2,on∂Ω.The formal energy functional corresponding to the system (6) is the following:
(7)Iλu=12∥u∥2−14∫Ωϕuu2dx−1q∫Ωfλxuqdx−16∫Ωu6dx.If a functionu∈H01Ω satisfies
(8)∫Ω∇u,∇vdx−∫Ωfλxuq−1vdx−∫Ωϕuuvdx−∫Ωu5vdx=0,∀v∈H01Ω,where v∈H01Ω, we can prove that u,ϕ∈H01Ω×H01Ω is a solution of system (1) if and only if ϕ=ϕu, and u∈H01Ω is called a weak solution of (6).To prove our main theorem1, some preliminary results are needed, let us first collect some properties.Lemma 2 (see [17]).
For eachu∈H01Ω, there exists a unique solution ϕu∈H01Ω of
(9)−Δϕ=u2,inΩ,ϕ=0,onΩ.
Moreover,(1)
ϕu≥0 for x∈Ω(2)
For eacht≠0, ϕtu=t2ϕu(3)
∫Ωϕuu2dx=∫Ω∇ϕu2dx≤S−3∥u∥4(4)
Suppose thatun⇀u in H01Ω, then(10)∫Ωϕunun2dx−∫Ωϕun−uun−u2dx=∫Ωϕuu2dx+o1Lemma 3 (see [18]).
Letr,s>1, ψ∈LsΩ, and ψ+=maxψ,0≠0, then there exists φ0∈C0∞Ω such that ∫Ωψφ0rdx>0.Lemma 4.
There existα,ρ,Λ0>0, such that every λ∈0,Λ0, we have
(11)Iλu>0foru∈∂B0,ρ,Iλu<0foru∈B0,ρ.Proof.
According to the Sobolev and Hölder inequalities and Lemma2, we have
(12)Iλu=12∥u∥2−14∫Ωϕuu2dx−1q∫Ωfλxuqdx−16∫Ωu6d≥12∥u∥2−14S−3∥u∥4−λqΩ6−q/6S−q/2f+∞∥u∥q−16S−3∥u∥6=∥u∥q12∥u∥2−q−14S−3∥u∥4−q−16S−3∥u∥6−q−λqΩ6−q/6S−q/2f+∞.
Settinggt=1/2t2−q−1/4S−3t4−q−1/6S−3t6−q for t>0, there exists a positive constant
(13)ρ=1/24−qS−3−1/44−q2S−1+4/36−q2−qS−32/36−qS−31/2>0,such that maxt>0gt=gρ>0. Letting Λ0=qSq/2/2f+∞Ω6−q/6gρ. Consequently, it follows that Iλ∥u∥=ρ≥α≜gρ/6ρq for any λ∈0,Λ0.
Additionally, by Lemma3, there exists φ0∈C0∞Ω⊂H01Ω such that
(14)limt→0+Iλtutq=1q∫Ωfλxφ0qdx.
Foru small enough, we have Iλu<0. The lemma is proved.We recall that forc∈ℝ, a sequence un⊂H01Ω is called a PSc sequence for I if Iun=c+o1 and I′un=o1 strongly in the dual space of H01Ω as n→∞. That is to say, I satisfies the PSc condition in H01Ω if every PSc sequence in H01Ω for I contains a convergent subsequence.Lemma 5.
Ifun⊂H01Ω is a PSc sequence for Iλ with c<1/3S3/2−Dλ2/2−q where D=D∣Ω∣,S,q,f+∞>0. Then, there exists u∈H01Ω such that un→u∗ in H01Ω and ∫Ωun6dx→∫Ωu6dx.Proof.
Letu∈H01Ω be such that
(15)Iλun→c,Iλ′un→0.
We claim thatun is bounded in H01Ω. Indeed, by (15), we have that
(16)1+c+o1∥un∥≥Iλun−14Iλ′un,un≥14∥un∥2−1q−14∫Ωfλxunqdx≥14∥un∥2−λ1q−14Ω6−q/6S−q/2f+∞∥un∥q,which implies that un is bounded in H01Ω. Thereby, there exists a subsequence, still denoted by un, and a function u∗∈H01Ω, such that
(17)un⇀u,weaklyinH01Ω,un→u,stronglyinLsΩ1≤s<6,unx→ux,a.e.inΩ,as n→∞. Set wn=un−u, then ∥wn∥→0. Otherwise, there exists a subsequence (still denoted by wn) such that
(18)limn→∞∥wn∥=l>0.
By (15), letting n→∞, for every φ∈H01Ω, it follows
(19)∫Ω∇u,∇φdx−∫Ωfλxuq−2uφdx−∫Ωϕuuφdx−∫Ωu4uφdx=0.
In particular, taking the test functionφ=u in (19), we have
(20)∥u∥2−∫Ωfλxuqdx−∫Ωϕuu2dx−∫Ωu6dx=0.
By Brézis–Lieb (see [26]) and Lemma 2, one has
(21)∥un∥2=∥wn∥2+∥u∥2+o1,∫Ωun6dx=∫Ωwn6dx+∫Ωu6dx+o1.
Therefore, according toIλ′un,un→0 as n→∞, we have
(22)∥u∥2+∥wn∥2+∫Ωϕuu2dx−∫Ωu6dx−∫Ωwn6dx−∫Ωfλxuqdx=o1.
It follows from (20) and (22) implies that
(23)∥wn∥2−∫Ωwn6dx=o1.
Note that∫Ωwn6dx≤S−3∥wn∥6, we deduce that
(24)l2≥S3/2.
On the one hand, by using the Young inequality and (20), there holds
(25)Iλu∗=12∥u∥2−14∫Ωϕuu∗2dx−16∫Ωu6dx−1q∫Ωfλxuqdx≥14∥u∥2−1q−14∫Ωfλxuqdx≥14∥u∥2−λ1q−14Ω6−q/6S−q/2f+∞∥u∥q≥−Dλ2/2−q,where D=D∣Ω∣,S,q,f+∞>0 is a constant.
On the other hand, note thatc<1/3S3/2−Dλ2/2−q, by (23), one has
(26)Iλu∗=Iλun−12∥wn∥2−16∫Ωwn6dx+o1=Iλun−13∥wn∥2+o1≤c−13l2<13S3/2−Dλ2/2−q−13S3/2=−Dλ2/2−q.
From the above analysis, this is a contradiction. Therefore,l=0, it implies that un→u in H01Ω, and
(27)0≤∫Ωun6dx−∫Ωu6dx=∫Ωwn6dx+o1≤S−3∥wn∥6+o1⟶0.
So∫Ωun6dx→∫Ωu6dx as n→∞. The lemma is proved.Theorem 6.
Assume1<q<2 and fλ∈L∞Ω. Then, system (1) has a positive solution u∗,ϕu∗∈H01Ω×H01Ω which satisfies Iλu∗<0.Proof.
From Lemma4 and applying Ekeland’s variational principle in Bρ0¯, there exists a minimizing sequence un⊂Bρ0¯ such that
(28)Iλun≤infun∈Bρ0¯Iλu+1q,Iλv≥Iλun−1n∥v−un∥,∀un∈Bρ0¯.
Therefore,(29)Iλ′un→0,Iλun<c.
Sinceun is bounded in H01Ω, there exist u∗∈Bρ0¯ and a subsequence still denoted by un such that un⇀u∗ in H01Ω as n→∞.
Note thatIλun=Iλ∣un∣ and Lemma 5, we can obtain un⇀u∗ in H01Ω and u∗≥0 and u∗≠0. Since u∗∈H01Ω, the embedding theorem means u∗∈L6Ω. Note that fλ∈L∞Ω, by the regularity of weak solutions, one has u∗∈W2,6/qΩ. Thereby, according to the classical bootstrap argument, we have u∗∈W1,α for r∈1,+∞. By the embedding theorem again, it holds that u∗∈C1,αΩ for 0<α<1. Since ϕu∗≥0, we have
(30)−Δu∗=u∗5+fλxu∗q−1+ϕu∗u∗≥−f−xu∗q−1≥−Cu∗for some C>0. Thereby, by the strong maximum principle, we obtain u∗>0 in Ω, then u∗,ϕu∗ is a positive solution of system (1) with Iλu∗<0 (by (11)). The theorem is proved.
Foru∈H01Ω, we have
(31)Iλu=12∥u∥2−14∫Ωϕuu2dx−16∫Ωu6dx−1q∫Ωfλxuqdx≤12∥u∥2−16∫Ωu6dx−1q∫Ωfλxuqdx.
Now, if we consider the following problem:(32)−Δu=fλxuq−2u+u4u,inΩ,u>0,inΩ,u=0,on∂Ω.
Then, we find the weak solutions of problem (32) correspond to the critical points of the functional Jλ, where
(33)Jλu=12∥u∥2−1q∫Ωfλxuqdx−16∫Ωu6dx.
Obviously, one has(34)Iλu≤Jλu,foru∈H01Ω.Remark 7.
Problem (32) has a positive local minimizer solution v0∈Bρ with Jλv0<0 for each λ∈0,Λ0 where ρ,Λ0 are as in Lemma 4.
Indeed, by Lemma4, we have
(35)12ρ2−q−14S−3ρ4−q−λqΩ6−q/6S−q/2f+∞>16S−3ρ6−q.
Therefore, when∥u∥=ρ, there holds
(36)Jλu=12∥u∥2−1q∫Ωfλxuqdx−16∫Ωu6dx≥12∥u∥2−S−36∥u∥6−λΩ6−q/6qS−q/2f+∞∥u∥q=∥u∥q12∥u∥2−q−S−36∥u∥6−q−λΩ6−q/6qS−q/2f+∞≥16S−3ρ6−qρq=16S−3ρ6.
Similar to Lemma4, we have
(37)limt→0+Jλtutq=1q∫Ωfλxφ0qdx.
So, we getJλu<0. Similar to Theorem 6, problem (32) has a positive local minimizer solution v0∈Bρ with Jλv0<0 for each λ∈0,Λ0.


## 3. Existence of a Second Positive Solution of System (1)

As usual, we consider the following function:(38)vε=3ε21/4ε2+x21/2,where ε is a positive constant. Moreover, we know that vε is a positive solution of problem −Δu=u4u in ℝ3 and ∫Ω∇vε2dx=∫Ωvε6dx=S3/2. Let ζ be a smooth cut-off function ζ∈C0∞Ω such that 0≤ζx≤1 in Ω, ζx=1 near x=0 and it is radially symmetric. Set uεx=vεxζx for x∈Ω. Then, we haveLemma 8.
Assume1<q<2, there holds
(39)supt≥0Iλv0+tuε<13S3/2−Dλ2/2−q.Proof.
According to the definition and from [27], there hold
(40)∥uε∥2≤S3/2+Oε,∫Ωuε6dx≥S3/2+Oε.
It is well known that the following inequality(41)a+b6≥a6+b6+6a5b+6ab5holds for a,b≥0. Since v0∈Bρ1 is a positive solution of the problem (32), and Jλ<0 for each λ∈0,Λ1. Obviously, there exist two constants n,M>0 such that m≤v0x≤M, using (41), there holds
(42)Jλv0+tuε=12∥v0∥2+t2∥uε∥2+∫Ω∇v0,∇tuεdx−16∫Ωv0+tuε6dx−1q∫Ωfλxv0+tuλqdx≤Jλv0+t22∥uε∥2−16∫Ωv0+tuε6−v06−6v05tuεdx+1q∫Ωf−∞∫0tuεv0+ηq−1−v0q−1dηdx≤t22∥uε∥2−16∫Ωtuε6+6v0tuε5dx+1q∫Ωf−∞∫0tuεηq−1dx≤t22∥uε∥2−t26∫Ωuε6dx−C1t5∫Ωuε5dx+C2tq∫Ωuεqdx,where C1,C2>0 (independent of ε,λ). Let
(43)ht=t22∥uε∥2−t26∫Ωuε6dx−C1t5∫Ωuε5dx+C2tq∫Ωuεqdx.
Sincelimt→0ht=0,limt→∞ht=−∞, it is similar to [25] that we prove that there exist tε and positive constants t1,t2 independent of ε,λ, such that
(44)0<t0≤tε≤t2<+∞,supt≥0ht≤13S3/2+C3ε−C1t15∫Ωuε5dx+C2t2q∫Ωuεqdx.
Note that∫Ωuε5dx=Cε1/2+Oε1/2, ∫Ωuεqdx=Cεq/2. Then,
(45)supt≥0ht≤13S3/2+C3ε−C4ε1/2+C5εq/2≤13S3/2+C3+C5εq/2−C4ε1/2.where C3,C4,C5>0 (independent of ε,λ). Let ε=λ4/q2−q, 0<λ<Λ1≔C4/C3+C5+D2/2q−1, there holds
(46)C3+C5εq/2−C4ε1/2=C3+C5λ2/2−q−C4λ2/q2−q=λ2/2−qC3+C5−C4λ2/q2−q<−Dλ2/2−q.
Recall that(47)Iλu≤Jλufor all u∈H01Ω. Therefore, from the above analysis, we conclude that (39) holds true when λ<Λ1. The lemma is proved.Theorem 9.
Assume1<q<2 and fλ∈L∞Ω. Then, the system (1) admits a positive solution u∗,ϕu∗∈H01Ω×H01Ω with Iλu∗>0.Proof.
Letλ∗=minΛ0,Λ1,1, then Lemma 4 and Theorem 6 and Lemma 8 hold for 0<λ<λ∗. By Remark 7, we can choose a large enough T0>0 such that Iλv0+T0uε<0. Consequently, Iλ satisfies the mountain geometry structure. Applying the mountain-pass lemma [28], there exists a sequence un⊂H01Ω such that
(48)Iλ′un→0andIλun→c>0asn→∞,where
(49)c=infγ∈Γmaxt∈0,1Iλγt,Γ=γ∈C0,1,H01Ω:γ0=v0,γ1=v0+T0uε.
From Lemma5 and Lemma 8, un has a convergent subsequence (still denoted by un) and there exists u∗∗∈H01Ω such that un→u∗∗ in H01Ω. Moreover, we can obtain u∗∗ is a nonnegative weak solution of system (1) and
(50)Iλu∗∗=limn→∞Iλun=c>0.
Therefore, we infer thatu∗∗≡0. Similar to Theorem 6, we obtain u∗∗>0 in Ω, and so u∗∗,ϕu∗∗ is a positive solution of system (1). The theorem is proved.


## 4. Existence of a Ground State Solution of System (1)

Define(51)N=infIλu:u∈H01Ω,u≠0,Iλ′un=0.By the definition ofN, there exists u∈H01Ω such that un≠0, and
(52)Iλun→N,Iλ′un→0asn→∞.Obviously, we can easily deduce thatun is bounded in H01Ω. There then exist a nonnegative subsequence of unIλun=Iλ∣un∣, still denoted by itself, and uλ∈H01Ω such that un⇀uλ weakly in H01Ω. We claim that uλ≠0. Arguing by contradiction, un⇀0 weakly in H01Ω and un→0 strongly in LpΩ1≤p<6. Then, we have
(53)∫Ωfλxunqdx=o1.Consequently, from (53), it holds that
(54)∥un∥2−∫Ωun6dx=o1.Setlimn→∞∥un∥=A, note that ∫Ωun6dx≤S−3∥un∥6, it follows
(55)A2≥S3/2or
(56)A=0.So it follows from (53) and (54) that
(57)N=limn→∞12∥u∗∥2−14∫Ωϕunun2dx−16∫Ωun6dx−1q∫Ωfλxunqdx=limn→∞14∥u∗∥2=14A2.IfA=0, then N=0, this contradicts to Theorem 6. Consequence, N≥1/3S3/2. According to Lemma 8, we have 1/3S3/2≤N<1/3S3/2−Dλ2/2−q, this is a contradiction. Consequently un⇀uλ≠0 in H01Ω. Moreover, by Lemma 5, we can prove that un⇀uλ in H01Ω, and uλ,ϕuλ is a positive solution of system (1) with Iλun≥N. Next, we prove that Iλun≤N. Indeed, since Iλ′uλ=0, it follows
(58)Iλuλ=12∥uλ∥2−14∫Ωϕuλuλ2dx−16∫Ωuλ6dx−1q∫Ωfλxuλqdx−14∥uλ∥2−∫Ωϕuλuλ2dx−∫Ωuλ6dx−∫Ωfλxuλqdx=14∥uλ∥2−1q−14∫Ωfλxuλqdx.In addition, it follows from (53) and the weak lower semicontinuity of the norm; it holds that
(59)N=limn→∞12∥un∥2−14∫Ωϕunun2dx−16∫Ωun6dx−1q∫Ωfλxunqdx−14∥un∥2−∫Ωϕunun2dx−∫Ωun6dx−∫Ωfλxunqdx=limn→∞12∥uλ∥2−1q−14∫Ωfλxuλqdx≥14∥uλ∥2−1q−14∫Ωfλxuλqdx.Consequently,uλ≠0 and Iλuλ=N, which implies that uλ,ϕuλ is a positive ground state solution of system (1). Hence, we finish the proof of Theorem 1.



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