# Existence Solutions for a Class of Schrödinger-Maxwell Systems with Steep Well Potential

Authors: Guocui Yang; Shengzhong Duan
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we are concerned with the system of the Schrödinger-Maxwell equations−Δu+λVxu+bKxϕu=up−2u,inR3,−Δϕ=Kxu2,inR3, where λ,b>0 are constants, and 3<p<6. Under appropriate assumptions on V and K, we prove the existence of positive solutions in the case 3<p<4 via the truncation technique. Moreover, suppose that V may change sign, we also obtain the multiplicity of solutions for the case 4<p<6.


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## Body

## 1. Introduction and Main Results

Consider the following system of Schrödinger-Maxwell equations:(1)−Δu+λVxu+bKxϕu=up−2u,inR3,−Δϕ=Kxu2,inR3,where λ,b>0 are constants, and 3<p<6. Such a system is also called the Schrödinger-Poission equation which is obtained while looking for the existence of standing waves for nonlinear Schrödinger equations interacting with an unknown electrostatic field. For more details on the physical aspects, we refer the reader to [1] and the references therein.On the potentialV, we make the following assumptions:(V1) V∈CR3,R and V is bounded below.(V2) there exists a constant c>0 such that the set x∈R3:Vx≤c is nonempty and measx∈R3:Vx≤c<+∞, where meas denote the Lebesgue measure in R3.(V3)Ω=intV−10 is nonempty and has smooth boundary and Ω¯=V−10.In recent years, system (1) has been widely studied under various conditions on V and K. The greatest part of the literature focuses on the study of the system for V and K being constants or radially symmetric functions. We refer the reader to [2–9].When the potentialVx is neither a constant nor radially symmetric, in [10, 11], the existence of ground state solutions is proved for 3<p<6. In [12–14], the existence of nontrival solution is obtained via the variational techniques in a standard way under the following condition:(V4) V∈CR3,R, infx∈R3Vx≥a0>0, where a0>0 is a constant. Moreover, for any M>0, measx∈R3:Vx≤M<+∞, where meas denote the Lebesgue measure in R3.It is worth mentioning that conditions (V4) were first introduced by Bartsch and Wang [15] to guarantee the compact embedding of the functional space. If replacing V4 by more general assumptions V1 and V2, the compactness of the embedding fails and this situation becomes more complicated. Recently, [16, 17] considered this case. The authors studied the following problem
(2)−Δu+λVxu+Kxϕu=fx,u,inR3,−Δϕ=Kxu2,inR3,where λ>0 is a parameter, the potential V may change sign and f is either the superlinear or sublinear in u as u⟶∞. Very recently, Liu and Mosconi [18] considered the following system with a coercive sign-changing potential and a 3-sublinear nonlinearity:
(3)−Δu+Vxu+λϕu=fu,inR3,−Δϕ=u2,inR3.By using a linking theorem, the authors obtained the existence of nontrivial solutions. Nextly, Gu, Jin, and Zhang [19] investigated the existence of sign-changing solutions for system (3). By using the method of invariant sets of descending flow, the multiple radial sign-changing solutions are obtained in the subquadratic case as λ small. For more results about the Schrödinger-Poisson systems, we refer the reader to [20–23] and the reference therein.Here, we should point out that for the power-type nonlinearityfu=up−2u, in order to get the boundedness of a (PS) sequence, the methods heavily rely on the restriction p∈4,6. Meanwile, the condition V1−V3 cannot guarantee the compactness of the embedding of H1R3 into the Lebesgue spaces LsR3, s∈2,6. This prevents from using the variational techniques in a standard way. Motivated by the works mentioned above and [24–28], in the present paper, we are mostly interested in sign-changing potentials and consider system (1) with more general potential V, K, and the range of p. Our main results are as follows:Theorem 1.
Suppose thatV≥0, (V1)- V3, K∈L3R3∩Lloc∞R3∪L∞R3, Kx≥0, and 3<p<4 hold. Then, system (1) possesses at least a nontrivial solution for b small and λ large.Remark 2.
It is known that it is difficult to get the boundedness of a (PS) sequence when dealing with the casep∈3,4. To overcome the difficulty, motivated by [24, 25], we use the truncation technique to obtain a bounded Cerami sequence for b small. In this case, the conditions V4 and K1 of Theorem 1.3 in [23] cannot be used. Moreover, in the process of proving the convergence of a bounded Cerami sequence, we use the observation that the condition K∈L3R3∪L∞R3 makes the less strong influence of the nonlocal term Kxϕu (The conclusions remain valid if K∈L2R3). In this sense, Theorem 1 can be viewed as an improvement of Theorem 1.3 in Zhao et al. [23].Theorem 3.
Suppose that (V1), V2, K∈L3R3∪L∞R3, Kx≥0, and 4<p<6 hold. Then, system (1) possesses infinitely many distinct pairs of nontrivial solutions whenever λ>0 is sufficiently large.Remark 4.
In Theorem3, V is allowed to be sign-changing for p∈4,6. We obtain the multiplicity of solutions for (1).


## 2. Preliminaries

Let(4)H1R3=u∈L2R3:∇u∈L2R3is the usual Sobolev space with the standard inner product and norm
(5)u,v=∫R3∇u∇v+uvdx,u=∫R3∇u2+u2dx12.In our problem, we work in the space defined by(6)Eλ=u∈H1R3:∫R3∇u2+λV+xu2dx<+∞with the inner product and the norm
(7)u,vEλ=∫R3∇u·∇v+λV+xuvdx,uEλ=u,uEλ1/2,where V±x=max±Vx,0, Vx=V+x−V−x. Obviously, it follows from V1,V2 that the embedding Eλ↪H1R3 is continuous.As in [26], let
(8)Fλ=u∈Eλ:suppu⊂V−10,∞and denote the orthogonal complement of Fλ in Eλ by Fλ⊥. Consider the eigenvalue problem
(9)−Δu+V+xu=μV−xu,u∈Fλ⊥.In view ofV1,V2, the quadratic form u↦∫R3V−xu2dx is weakly continuous. We have the following proposition.Proposition 5 ([26], Lemma 8).
SupposeV1, V2, and V≠0. Then, for each fixed j,
(i)
μjλ⟶0 as λ⟶+∞(ii)
μjλ is a nonincreasing continuous function of λ, where μjλ=infdimM≥j,M⊂Fλ⊥supuEλ2:u∈M,∫R3λV−xu2dx=1j=1,2,3⋯ is sequence of positive eigenvalues of problem P satisfying μ1λ≤μ2λ≤⋯≤μjλ⟶∞ as j⟶∞ and the corresponding eigenfunctions ejλj=1∞.
Let(10)Eλ−=spanejλ:μjλ≤1,Eλ+=spanejλ:μjλ>1.
Then,(11)Eλ=Eλ−⊕Fλ⊕Eλ−.
Moreover dimEλ−<+∞ for every fixed λ>0.In the sequel, we denote the usualLp-norm by ·p, and C stands for different positive constants. For any R>0, BRx denotes the open ball of radius R centered at x. Since the continuity of the following embedding
(12)H1R3↪LsR3,2≤s≤6,D1,2R3↪L6R3,there are constants as>0 and a>0 such that
(13)us≤asuEλ,ϕ6≤aϕD1,2,∀u∈Eλ,∀ϕ∈D1,2R3.It is well known that system (1) is the Euler-Lagrange equation of the functional J:Eλ×D1,2R3⟶R defined by
(14)Ju,ϕ=12uEλ2−12∫R3λV−xu2dx−14∫R3∇ϕ2dx+b2∫R3Kxϕu2dx−1p∫R3updx.Evidently, the action functionalJ belongs to C1Eλ×D1,2R3,R and its critical points are the solutions of system (1). It is easy to know that J exhibits a strong indefiniteness, namely, it is unbounded both from below and from above on infinitely dimensional subspaces. This indefiniteness can be removed using the reduction method described in [29], by which we are led to study a one variable functional that does not present such a strongly indefiniteness.Actually, considering for allu∈Eλ, the linear functional Lu defined in D1,2R3 by
(15)Luv=∫R3Kxu2vdx.IfK∈L∞R3, the Hölder inequality and the Sobolev inequality imply
(16)Luv≤∫R3Kxu2vdx≤KL∞u12/52v6≤CKL∞u12/52vD1,2,while for K∈L3R3, we have
(17)Luv≤∫R3Kxu2vdx≤K3u42v6≤CK3u42vD1,2.Hence, by the Lax-Milgram theorem, there exists a uniqueϕu∈D1,2R3 such that
(18)∫R3∇ϕu∇vdx=∫R3Kxu2vdx,∀v∈D1,2R3.Moreover, we can write an integral expression forϕu in the form:
(19)ϕux=14π∫R3Kyx−yu2ydy,∀u∈Eλ.So, we can consider the functionalIλ:Eλ⟶R defined by Iλu=Ju,ϕu. Then,
(20)Iλu=12uEλ2−12∫R3λV−xu2dx+b4∫R3Kxϕuu2dx−1p∫R3updx.It follows from (16), (17), (18), and the Sobolev inequality that
(21)ϕuD1,2=Lu≤CKL∞u1252,∫R3Kxu2ϕuxdx≤CKL∞2uEλ4,(22)ϕuD1,2=Lu≤CK3uEλ2,∫R3Kxu2ϕuxdx≤CK32uEλ4.Thus,I is a well-defined C1 functional with derivative given by
(23)<Iλ′u,v>=∫R3∇u∇vdx+∫R3λVxuvdx+b∫R3Kxϕuuvdx−∫R3up−2uvdx.By the proposition 2.3 in [12], we know that u,ϕ∈Eλ×D1,2R3 is a critical point of J if and only if u is a critical point of I and ϕ=ϕu.To complete the proof of our theorem, we need the following results.Theorem 6 (see [30]).
LetX be a real Banach space with its dual space X∗, and suppose that J∈C1X,R satisfies
(24)maxJ0,Je≤μ<η≤inf∥u∥X=ρJu,for some μ<η,ρ>0 and e∈X with eX>ρ. Let c≥η be characterized by
(25)c=infγ∈Γmaxt∈0,1Jγt,where Γ=γ∈C0,1,X:γ0=0,γ1=e is the set of continuous paths joining 0 and e. Then, there exists a sequence un⊂X such that
(26)Jun⟶c≥η,1+unJ′unX∗⟶0asn⟶∞.Theorem 7 see ([31], Theorem 9.12).
LetE be an infinite dimensional Banach space and let I∈C1E,R be even, satisfy (PS), and I0=0. If E=V⊕X, where V is finite dimensional and I satisfies.
I1 there are constants ρ,α>0 such that I∂Bρ∩X≥α, and
I2 for each finite dimensional subspace E~⊂E, there is an R=RE~ such that I≤0 on E~\BRE~,
then,I possesses an unbounded sequence of critical values.Lemma 8 ([17], Lemma 2.3).
The functionϕu possess the following properties:
(1)
The mappingu⟶ϕu maps bounded sets of Eλ into bounded sets of D1,2R3;(2)
Ifun⇀u in Eλ, then ϕun⇀ϕu in D1,2R3;(3)
The mappingu⟶ϕu:Eλ⟶D1,2R3 is continuous.


## 3. Proof Of Theorem 1

In this section, we give the proof of Theorem2 by using Theorem 6. Let η∈C′0,∞,R be a cut-off function satisfying 0≤η≤1, ηt=1 if 0≤t≤1, ηt=0 if t≥2, maxt>0η′t≤2 and η′t≤0 for each t>0. For every T>0 we consider the truncated functional J:Eλ⟶R defined by
(27)Ju=12uEλ2+b4ηuEλ2/T2∫R3Kxϕuu2dx−1p∫R3updx.It is easy to see thatJ is of C1. Moreover, for each u,v∈Eλ, we have
(28)<J′u,v>=u,vEλ+bηuEλ2/T2∫R3Kxϕuxuvdx+b2T2η′uEλ2/T2u,vEλ∫R3Kxϕuxuvdx−∫R3up−2uvdx.Lemma 9.
Suppose that3<p<4 and V1−V3,Vx≥0 and K∈L3R3∩Lloc∞R3∪L∞R3, Kx≥0 hold. Then,
(i)
there exists av∈Eλ\0,with Jv<0(ii)
there existsM>0 independent of T,λ and b such that cb,λT≤M, where cb,λT=infγ∈Γmaxt∈0,1Jγt, Γ=γ∈C0,1,Eλ:γ0=0,γ1=vProof.
For anyt>0, by (19), we have
(29)ϕt2utyx=14π∫R3Kyx−yt4u2tydy≤14πKL∞∫R31x−yt4u2tydy=14πKL∞∫R31x−z/tt4u2z1t3dz=t24πKL∞2∫R3Kztx−zu2zdz=t2KL∞2ϕutx.Letx0∈Ω and Br0x0⊂Ω for some r0>0. Let u∈C0∞Ω be such that suppu⊂Br00. Set vt=t2utx−x0, then suppv⊂Br0x0 for t>1. Hence, it is easy to see that
(30)Jvt≤t32∫Br00∇u2dx+t34KL∞2∫Br00ϕuu2dx−t2p−3p∫Br00updx.Sincep>3, we have that Jvt⟶−∞ as t⟶+∞. Taking v=vt for t large enough, we have Jv<0.(iii)
sincecb,λT≤maxt>0Jv, there exists a constant M>0 (independent of T, λ and b) such that cb,λT≤M.Lemma 10.
Suppose that3<p<4 and V1,V2 hold. Then, there exists α>0 such that cb,λT≥α>0.Proof.
For anyu∈Eλ, we have
(31)Ju=12uEλ2+b4ηuEλ2/T2∫R3Kxϕuu2dx−1p∫R3updx≥12uEλ2−1p∫R3updx=12uEλ2−1pupp≥uEλ212−CuEλp−2.
Sincep>3, we conclude that there exists ρ>0 such that Ju≥α>0 for all u∈Eλ with uEλ=ρ.
From Lemmas9 and 10 and Theorem 6, we thus deduce that there exist a Cerami sequence un⊂Eλ such that
(32)Jun⟶cb,λT,1+unEλJ′un⟶0.Lemma 11.
Let3<p<4, Vx≥0, V1−V2, Kx≥0, and K∈L3R3∪L∞R3 be satisfied. Then, there exists Λ>1 such that for each λ∈Λ,∞, if un⊂Eλ is sequence satisfying (32), then un has a convergent subsequence in Eλ for b small enough.Proof.
Letun be a Cerami sequence satisfying (32). Let T=2pM+1/p−2, we show that unEλ≤T. We first prove that unEλ≤2T. Suppose by contradiction that there exist a subsequence of un, still denoted by un, such that unEλ>2T, we obtain
(33)cb,λT=limn⟶∞Jun−1pJ′un,un=limn⟶∞12−1punEλ−bp−b4ηunEλT2∫R3Kxϕunxun2dx−b2pT2η′unEλT2unEλ∫R3Kxϕunxun2dx≥liminfn⟶∞12−1punEλ≥2M+1,
which is a contradiction by Lemma9. Suppose that there exists no subsequence of un which is uniformly bounded by T. Then, we deduce that TunEλ≤2T. We handle the case of K∈L3R3 (The conclusions remain valid if K∈L∞R3). By (22), we obtain
(34)cb,λT=limn⟶∞Jun−1pJ′un,un=limn⟶∞12−1punEλ−bp−b4ηunEλ/T2∫R3Kxϕunxun2dx−b2pT2η′unEλT2unEλ∫R3Kxϕunxun2dx≥liminfn⟶∞12−1punEλ2−bp−b4∫R3Kxϕunxun2dx≥liminfn⟶∞12−1p−bp−b4CK32unEλ4.≥M+1−44−ppbp−22M+12CK3.
This is contradiction by choosingb sufficiently small. Since unEλ≤T, passing to a subsequence if necessary, we can assume that there exists u∈Eλ such that un⇀u, inEλ . In view of the Sobolev embedding theorems and Lemma 8, un⇀u inEλ implies.
(35)aun⇀uinL6R3,bϕun⇀ϕuinD1,2R3,cϕun⇀ϕuinL6R3.
Furthermore, for anyφ∈Eλ, we have
(36)∫R3Kxϕunxunxφdx−∫R3Kxϕuxuφdx⟶0,asn⟶∞.
In fact, ifK∈L∞R3, we just need to show that
(37)∫R3ϕunxunxφdx−∫R3ϕuxuφdx⟶0,asn⟶∞.
Note that(38)∫R3ϕunxunxφdx−∫R3ϕuxuφdx=∫R3ϕunun−uφdx+∫R3ϕun−ϕuuφdx.The first limit on the right is 0 by the fact ϕunun−u⇀0inL2R3 and so is the second limit because ϕun−ϕu⇀0inL6R3 and uφ∈L6/5R3. While for K∈L3R3, on the one hand, since Ku2∈L6/5R3, using (35) (c), we have
(39)∫R3Kxϕunx−ϕuxu2dx⟶0,as n⟶∞. On the other hand, since K∈L3R3, for any ε1>0 there exists ρ1=ρ1ε1>0 such that
(40)K3,R3\Bρ10<ε1.
Moreover, in view of the Sobolev embedding theorem,un⇀u implies that
(41)un⟶uinLlocsR3,2≤s<6.
Hence, for largen, we obtain
(42)un−u4,Bρ10<ε1.
Consequently,(43)∫R3Kxϕunxunx−uxuxdx≤K3ϕun6u4ε1+ε1ϕun6u4un−u4≤Cε1.
From above inequality and (39), one has
(44)∫R3Kxϕunxunxuxdx−∫R3Kxϕuxu2dx⟶0,asn⟶∞.
Then,J′un⟶0 implies that
(45)∫R3∇u2dx+∫R3λVxu2dx+b∫R3Kxϕuxu2dx−μ∫R3updx=0.
Letvn≔un−u. It follows from V1, V2 that
(46)vn22=∫Vx≥cvn2dx+∫Vx<cvn2dx≤1λcvnEλ2+o1.
Moreover, let0<α<min6−p/2,1,2<p<6. Then, 2<2p−α/2−α<6. By the Sobolev inequalities and Hölder inequality, one has
(47)vnpp=∫R3vnαvnp−αdx≤∫R3vn2dxα2∫R3vn2p−α2−αdx2−α2=vn2αvn2p−α2−αp−α≤Cλc−α2vnEλp+o1.
We know(48)o1=J′un,un=unEλ2+b∫R3Kxϕunxunxdx−unpp−uEλ2−b∫R3Kxϕuxuxdx+upp=vnEλ2−vnpp+o1≥vnEλ2−vnpp−2vnp2+o1≥1−2apTp−2Cp−2pλc−αp−22pvnEλ2+o1.
LettingΛ>0 be so large that the term in the brackets above is positive when λ≥Λ, we get vn⟶0 in Eλ. Since vn=un−u and vn⟶0, it follows that un⟶u in Eλ. This completes the proof.Proof of Theorem 2.
Note thatun is also a Cerami sequence of Iλ satisfying unEλ≤T, the conclusion follows from Lemmas 9, 10, and 11 and Theorem 6.


## 4. Proof Of Theorem 3

In this section, whileV is sign-changing, we study the existence of solutions of (1) for the case 4<p<6 and give the proof of Theorem 3. Without loss of generality, we assume that b=1.Lemma 13.
LetV1,V2, Kx≥0, and K∈L3R3∪L∞R3 be satisfied. Then, there exist α,ρ>0 such that Iλu≥α for all u∈Eλ+⊕Fλ with uEλ=ρ.Proof.
By proposition5, for each fixed λ>Λ, there exists a positive integer nλ such that μjλ≤1 for j<nλ and μjλ>1 for j≥nλ. Thus, for any u=u1+u2∈Eλ+⊕Fλ, we have
(49)Iλu=12uEλ2−12∫R3λV−xu2dx+14∫R3Kxϕuu2dx−1p∫R3updx≥12∫R3∇u12+λVxu12dx+12u2Eλ2−1p∫R3updx≥121−1μnλλuEλ2−1pupp≥121−1μnλλuEλ2−CuEλp≥121−1μnλλ−CuEλp−2uEλ2,
for allu∈Bρ0¯, where Bρ0=u∈Eλ+⊕Fλ:uEλ<ρ. Since p>2, the conclusion follows by choosing ρ sufficiently small.Lemma 14.
LetV1,V2, Kx≥0, and K∈L3R3∪L∞R3 be satisfied. Then, for any finite dimensional subspace E~⊂Eλ, there is a large r>0 such that Iλu<0 on E~\Br0.Proof.
Since all norms are equivalent in a finite dimensional space, there is a constantb1>0 such that
(50)up≥b1uEλ,∀u∈E~⊂Eλ.
By (22), there is a constant CK>0 such that
(51)∫R3Kxϕuu2dx≤CKuEλ4,∀u∈Eλ.
Hence, for allu∈E~,
(52)Iλu=12uEλ2−12∫R3λV−xu2dx+14∫R3Kxϕuu2dx−1p∫R3updx≤12uEλ2+CK4uEλ4−1pupp≤12uEλ2+CK4uEλ4−CuEλp.
Sincep>4, there is a large r>0 such that Iu<0 on E~\Br0.Lemma 15.
LetV1,V2, Kx≥0, and K∈L3R3∪L∞R3 be satisfied. Then, there exists Λ>1 such that, for each λ∈Λ,∞, Iλ satisfies the PSc condition.Proof.
Letun be a PSc sequence, that is Iλun⟶c and Iλ′un⟶0. If un is unbounded in Eλ, up to a subsequence, we can assume that
(53)unEλ⟶+∞,Iλun⟶c,Iλ′un⟶0;as n⟶∞, after passing to a subsequence. Set wn=un/unEλ, we can assume that wn⇀w in Eλ and wnx⟶wx a.e.x∈R3.
Ifw=0, since u↦∫R3V−xu2dx is weakly continuous, we have
(54)o1=1unEλ2Iλun−1pIλ′un,un=12−1p−12−1p∫R3λV−xwn2dx+14−1p∫R3Kxϕunwn2dx.≥12−1p+o1,
a contradiction. Ifw≠0, then the set Ω=x∈R3:ωx≠0 has the positive Lebesgue measure. For x∈Ω, one hasunx⟶∞ as n⟶∞, Fatou’s lemma shows that ∫Ωunp−4wn4dx⟶∞ as n⟶∞. Thus, by (22), we obtain
(55)1p∫R3unp−4wn4dx=12unEλ2−12unEλ4∫R3λV−xun2dx+1punEλ4∫R3Kxϕunun2dx+o1.≤12unEλ2+1punEλ4∫R3Kxϕunun2dx+o1.≤Cp+o1.
This is a contradiction. This impliesun is bounded in Eλ. Going if necessary to a subsequence, we can assume that un⇀u in Eλ. The following proof is similar to the proof of Lemma 11. We omit details of this.Proof of Theorem 3.
Obviously,Iλ0=0. Furthermore, Iλ is even. The conclusion follows from Lemmas 13, 14, and 15, and Theorem 7.



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