# Existence of Positive Ground State Solution for the Nonlinear Schrödinger-Poisson System with Potentials

Authors: Lu Xiao; Haiqin Qian; Mengmeng Qu; Jun Wang
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

In the present paper we study the existence of positive ground state solutions for the nonautonomous Schrödinger-Poisson system with competing potentials. Under some assumptions for the potentials we prove the existence of positive ground state solution.


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## Body

## 1. Introduction

In this paper we are concerned with the existence of solution of the nonautonomous Schrödinger-Poisson system(1)-Δu+Axu+Bxϕxu=Qxup-1u,x∈R3,-Δϕ=Bxu2,u∈H1R3,where p∈(3,5), A(x), B(x), and Q(x) are positive functions such that limx→∞⁡A(x)=a∞>0, limx→∞⁡B(x)=b∞>0, and limx→∞⁡Q(x)=q∞>0. Here B:R3→R denotes the nonnegative measurable function which represents a nonconstant charge corrector to the density u2. A and Q are called the potentials of system (1). In the context of the so-called Density Functional Theory, variants of system (1) appear as mean field approximations of quantum many-body systems; see [1–3].This kind of system also arises in many fields of physics. Indeed, one considers the following system:(2)iħ∂ψ∂t=-ħ22mΔψ+kxψ+Bxϕψ-Qxψp-2ψ,x∈R3,t∈R,-ħ22mΔϕx=Bxψ2,u∈H1R3,where i is the imaginary unit, Δ is the Laplacian operator, and ħ is the Planck constant. A standing wave is a solution of (2) of the form ψ(x,t)=u(x)eiωt,ω>0 and t∈R. It is clear that ψ(x,t) solves (2) if and only if u(x) solves the so-called stationary system(3)-ħ22mΔu+Axu+Bxϕxu=Qxup-2u,x∈R3,-ħ22mΔϕ=Bxu2,x∈R3,where A(x)=k(x)+ωħ. Here we consider that the case ħ is constant. Without loss of generality we assume that ħ2/2m=1, then system (3) becomes (1). System (1) is also modeled in Abelian Gauge Theories; for instance, see [4–6] and reference therein. In fact, in order to describe the interaction of a nonlinear Schrödinger field with an electromagnetic field E-H, the gauge potentials A-F(4)A:R3×R→R3,F:R3×R→Rare related to E-H by the Maxwell equations (5)E=-∇F+∂A∂t,H=∇×A.If we are interested in finding standing waves (solutions of a field equation whose energy travels as a localized packet preserving this localization in time) and we consider the electrostatic case (when A=0), the Schrödinger field is described by a real function u:R3→R, which represents the matter (see [4, 7]). In this situation we need to consider the stationary states of Schrödinger-Maxwell system (1). The system is also used in quantum electrodynamics, semiconductor theory, nonlinear optics, and plasma physics; for more information on this direction, one can refer to [7–9] and the references therein.In recent years many papers focus on the existence, multiplicity, and concentration of positive solutions of (3) for the semiclassical case (ħ>0 sufficiently small). In this framework one is interested not only in existence of solutions but also in their asymptotic behavior as ħ→0. Typically, the solution tends to concentrate on critical points of A(B or Q). These solutions are called spikes. For more information on this direction. one can refer to [10–14] and the references therein.In the present paper we are interested in studying the case whenħ is constant. In [15], the authors studied the existence and nonexistence of solutions of (1) when A=B=Q=1. The existence of the multiple solutions of (1) has been found in the paper [16] in a radial setting. In the paper [17], the author considers that the case B=1,V,K is radial and satisfies(6)A01+xα≤Ax≤A1,0<Qx≤Q01+xβ,where α∈(0,2], β,A0, A1, and Q0 are positive constants. The author proved the existence of nontrivial positive classical mountain-pass solution of (1). Moreover, some generalizations of the last cases, with Q(x)up-1u replaced by a more generic function f(x,u), were considered in [18, 19]. It is well known that, dealing with system (1), one has to face different kinds of difficulties, which are related to the potential and to the unboundedness of the space R3. Many early studies were devoted to the autonomous case and to the case in which the coefficients are supposed to be radial (see [16, 20, 21]), just to overcome the lack of compactness-taking advantage of the compact embedding Hr1(R3)↪Lq(R3)(∀q∈(2,6)). More recently, many contributions to (1) have also been given looking at cases in which no symmetry assumptions are given on the coefficients appearing in (1); one can refer to the papers [22–24]. Furthermore, for more results on the existence of positive solutions, ground and bound states, one can see [18, 19, 21, 25–33] and references therein. Nearly, the paper [34] proves the existence of bound state solution of (1) under some decay condition on A, B, and Q. Precisely, assume that A(x)=V∞+W(x) and ∫R3W(x)(xe2V∞x)<∞, Q(x)=Q∞-β(x) and ∫R3β(x)(xe2V∞x)<∞, and (7)0≤Bx≤ce-σx,forxlarge  enough,where V∞ and Q∞ are positive constants. From this assumption one can easily deduce that lim|x|→∞⁡B(x)=0. In order to get the compactness, the paper [34] studies the case when the limit equation is Scalar Schrödinger equation, i.e.,(8)-Δu+a∞u=q∞up-1u,u∈H1R3.According to [35, 36], (8) has unique positive solution, and the energy level of any sign-changing solution is strictly greater than 2c∞, where c∞ is the least energy level for the positive solution of (8). This information is very important for proving the existence of positive bound solution (high energy) of (1.1) in [34].Motivated by [34, 37], in the present paper we shall study the case when the limit equation is the Schrödinger-Poisson system(9)-Δu+a∞u+b∞ϕxu=q∞up-1u,inR3,-Δϕ=b∞u2,u∈H1R3,and, under some conditions for the A, B, and Q, we prove the existence of positive ground state solution of (1).In order to state our main results, we shall give some assumptions. Forλ∈R+, we define A(x)=a∞+λa(x), B(x)=b∞-b(x), and Q(x)=q∞+q(x). Throughout the paper we need the following conditions.( A 1 )   a ∞ > 1, a∈L3/2(R3),0<a<1,a≠0, limx→∞a(x)=0.( A 2 )   b ∞ > 1, b∈L3/2(R3)∩L∞(R3),0≤b≤1,b≠0, limx→∞b(x)=0.( A 3 )   q ∞ > 1, q∈L∞(R3),0<q<1,q≠0, limx→∞q(x)=0.Clearly, system (1) becomes(10)-Δu+a∞+λaxu+b∞-bxϕxu=q∞+qxup-1u,inR3,-Δϕ=b∞-bxu2,u∈H1R3,where λ∈R+ and p∈(3,5). In the following we shall focus on system (10). According to [11, 12], we know that (9) has a positive radial symmetric solution w. Moreover, w is decreasing when the radial coordinate increases. Precisely, there exists a constant c∗>0 such that(11)limx→+∞⁡wxec∗x=Constant.Then we have the following main results.Theorem 1.
Assume thatp∈(3,5) and (A1)-(A3) hold. Moreover, if one of the following conditions holds
(1)limx→∞q(x)e(2τ/1+τ)c∗x=∞ and limx→∞q(x)e2τ/1+2τc∗x≤c0, limx→∞a(x)x1+τe(2τ/1-τ)c∗x≤c1, limx→∞b(x)e(2τ(1-τ)/(1+τ)(1+2τ))c∗|x|≥c2, where τ∈(0,1), c0,c1, and c2 are positive constants.
(2)lim|x|→∞q(x)e2τ/1+τc∗x=∞ and limx→∞a(x)ep-1τ/1-τc∗x≤c3, where τ∈(0,1) and c3 is positive constant.
Then system (10) has a positive ground state solution for each λ∈R+.Remark 2.
(i) From the assumptions onB, we know that limx→∞B(x)=b∞. Hence we need to consider the limit equation (9) to recover the compactness. This is quite different from the recent work of [34]. The main novelty here is that we shall compare the decay rate of A,B, and Q to recover the compactness and prove the existence of positive ground state solution.
(ii) Note that in condition (1) of Theorem1, we know that a(x) is decaying faster than b(x). In condition (2) of Theorem 1, we only need the decay condition for a(x) (whatever the decay speed of b(x) to 0 as x→∞) to prove the existence of positive solution. This is different phenomenon compared to [34].


## 2. Preliminary Results

We define the following notation:(i)
· is the norm of H1(R3) defined by u2=∫R3(∇u2+a∞u2);(ii)
·D1,2 is the norm of D1,2(R3) defined by uD1,22=∫R3∇u2;(iii)
(·,·) is the scalar product of H1(R3) defined by (u,v)=∫R3[∇u∇v+uv];(iv)
letC, c∗, c,ci(i=1,⋯) denote different positive constants.In this part we mainly give some basic knowledge which will be used later. We first show that the second equation of (10) can be solved. We consider, for all u∈H1(R3), the linear functional Ju defined in D1,2(R3) by (12)Juv=∫R3b∞-bxu2v.We infer from condition (A2) and Hölder inequality that(13)Juv≤Cu12/52vD1,2.By the Lax-Milgram theorem, we know that there exists unique ϕu∈D1,2(R3) such that(14)∫R3∇ϕu∇v=∫R3b∞-bxu2v∀v∈D1,2R3.So, ϕu is a weak solution of -Δϕ=u2 and the following formula holds(15)ϕux=∫R3b∞-byu2y4πx-ydy=14πx∗b∞-bu2.Moreover, ϕu>0 when u≠0.We recall the following classical Hardy-Littlewood-Sobolev inequality (see [38, Theorem 4.3]). Assume that f∈Lp(R3) and g∈Lq(R3). Then one has(16)∫R3∫R3fxgyx-ytdxdy≤cp,q,tfpgq,where 1<p,q<∞, 0<t<3, and 1/p+1/q+t/3=2. By (16) we know that(17)∫R3∫R3u2xu2yx-ydxdy≤cu12/54≤cu4.It is well known that solutions of (10) correspond to critical points of the energy functional(18)Φλu=12∫R3∇u2+a∞+λaxu2+14∫R3b∞-bxϕuu2-1p+1∫R3q∞+qxup+1.From (17), we know that Φλ is well defined and that(19)Φλ′uv=∫R3∇u∇v+a∞+λaxuv+∫R3b∞-bxϕuuv-∫R3q∞+qxup-2uv.We define the operator I:H1(R3)→D1,2(R3) as (20)Iu=ϕu.We infer from [34, Proposition 2.1-2.2] that I has the following properties.Lemma 3.
(1)I is continuous.
(2)I maps bounded sets into bounded sets.
(3)I[tu]=t2I[u] for all t∈R.
(4) Ifun⇀u∈H1(R3), then we have ∫R3b(x)ψun(x)un2→∫R3b(x)ψu(x)u2 and ∫R3b(x)ψun(x)unϕ→∫R3b(x)ψu(x)uϕ for each ϕ∈H1(R3), as n→∞, where ψu(x)=∫R3(b(y)u2(y)/4πx-y)dy.
(5) Ifun⇀u∈H1(R3), then we have I(un)→I(u) in D1,2(R3) and ∫R3b∞ϕ¯un(x)unϕ→∫R3b∞ϕ¯u(x)uϕ for each ϕ∈H1(R3), as n→∞, where ϕ¯u(x)=∫R3(b∞u2(y)/4πx-y)dy.Proof.
The conclusions (1)-(4) can be proved as in [34, Proposition 2.1-2.2]. Hence we only focus on the proof of (5). Since, by definition of I, for all u∈D1,2(R3) we have (21)IuD1,2R3=JuLD1,2R3,Rthen, in order to prove I(un)→I(u) in D1,2(R3), it suffices to prove that(22)Jun-JuLD1,2R3,R→0,asn→∞.Let ε>0 be fixed arbitrarily. Then there exists a positive number Rε>0 so large that vL6(BRεc)<ε for v∈D1,2(R3), where BRε=x∈R3:x≤Rε and BRεc=R3∖BRε. Hence we deduce that (23)Junv-Juv=∫R3b∞-bxun2-u2vdx=∫BRεb∞-bxun2-u2vdx+∫BRεcb∞-bxun2-u2vdx≤cv6un-uL12/5BRεun+uL12/5Rε+cvL6BRεcun2-u26/5≤cε+cun-uL12/5BRεun+uL12/5RεvD1,2.Since un→u in Llocp(R3)(p∈(2,6)), we know that (22) holds.
Next we prove the later conclusion. For any fixedR>0 large, we infer that for each x∈R(24)ϕ¯unx-ϕ¯ux=∫R3b∞un2y-u2y4πx-ydy=∫x-y≤1b∞un2y-u2y4πx-ydy+∫1≤x-y≤Rb∞un2y-u2y4πx-ydy+∫x-y≥Rb∞un2y-u2y4πx-ydy≤cun-uL4/3B1xun+uL4B1x∫x-y≤11x-y2dy+cun-uL8/3BRx∖B1xun+uL8/3BRx∖B1x∫1≤x-y≤R1x-y4dy+cun-uL8/3BRcxun+uL8/3BRcx∫x-y≥R1x-y4dyand, hence, we deduce that ϕ¯un(x)→ϕ¯u(x) a.e., in x∈R3 and ϕ¯un-ϕ¯uL∞(BR0)→0 for each R0>0, as n→∞. A direct computation shows that for R>0 large (25)∫R3b∞ϕ¯unxunϕ-∫R3b∞ϕ¯uxuϕ=∫R3b∞ϕ¯un-ϕ¯uunϕ+b∞ϕ¯uxun-uϕ≤c∫BRϕ¯un-ϕ¯uunϕ+c∫BRcϕ¯un-ϕ¯uunϕ+c∫BRϕ¯uxun-uϕ+c∫BRcϕ¯uxun-uϕ≤cϕ¯un-ϕ¯L∞BRϕ2un2+cϕ¯un-ϕ¯6ϕL12/5BRcun12/5+cϕ¯u6ϕ12/5un-uL12/5BR+cϕ¯uL6BRcϕL12/5BRcun-u12/5which proves ∫R3ϕ¯un(x)unϕ→∫R3ϕ¯u(x)uϕ for each ϕ∈H1(R3), as n→∞.It is very easy to verify that, whateverλ∈R is, the function Φλ is bounded either from above or from below. Hence, it is convenient to consider Φλ restricted to a natural constraint, the Nehari manifold. We set(26)Nλ≔u∈H1R3∖0:Φλ′uu=0.Next lemma contains the statement of the main properties of Nλ.Lemma 4.
Assume that(A1)-(A3) hold. Then the following conclusions hold.
(a)Nλ is a C1 regular manifold diffeomorphic to the sphere of H1(R3).
(b)Φλ is bounded from below on Nλ by a positive constant.
(c)u is a free critical point of Φλ if and only if u is a critical point of Φλ constrained on Nλ.Proof.
(a) Let u∈H1(R3)∖{0} be such that u=1. We claim that there exists a unique t∈(0,∞) such that tu∈Nλ. Let f(t)=Φλ(tu) for t>0. It is easy to verify, by (A1)-(A3), that f(0)=0, f(t)>0 for t>0 small and f(t)<0 for t large. Therefore, maxt>0⁡f(t) is achieved at a t=t(u)>0 so that f(tu)=0 and t(u)u∈Nλ. Assume that there exist t1>t2>0 such that t1u,t2u∈Nλ. Then we see(27)w2+∫R3λax)u2t12-w2+∫R3λax)u2t22=t1p-3-t2p-3∫R3q∞+qxup+1.This is a contradiction. So, we prove that f(t) admits a unique positive solution t(u)>0 and t(u)u∈Nλ. We infer from (16) and (A1)-(A3) that(28)c1up+12≤u2+∫R3λaxu2+∫R3b∞-bxϕuu2=∫R3q∞+qxup+1≤c1up+12.This implies that(29)u≥c>0.Let Gλ(u)≔Φλ′(u)[u]. Then Gλ∈C1(H1(R3),R) by the regularity of Φλ. Moreover, we infer from (29) that(30)Gλ′uu=-p-1∫R3∇u2+a∞+λaxu2-p-3∫R3b∞-bxϕuu2≤-p-3C<0.
(b) For all u∈Nλ,(31)Φλu=12-1p+1∫R3∇u2+a∞+λaxu2+14-1p+1∫R3b∞-bxϕuu2≥12-1p+1u2≥c>0.Here we use the fact that 3<p<5.
(c) If u≠0 is a critical point of Φλ, then we have Φλ′(u)=0,u∈Nλ. On the other hand, if u is a critical point of Φλ(u) constrained on Nλ, then there exists k∈R such that (32)0=Φλ′uu=Gλu=kGλ′uu.
One infers from (30) that k=0.Next we consider the limit functionalΦ∞:H1(R3)→R, defined as (33)Φ∞u=12∫R3∇u2+a∞u2+14∫R3b∞ϕ¯uxu2-1p+1∫R3q∞up+1.And we consider the corresponding natural constraint (34)N∞≔u∈H1R3∖0:Φ∞′uu=0.Critical points of Φ∞ are solutions of the limit problem at infinity(35)-Δu+a∞u+b∞ϕ¯uxu=q∞up-1u,x∈R3,-Δϕ¯u=b∞u2,u∈H1R3,and, clearly, the conclusions of Lemma 4 hold true for Φ∞ and N∞. Moreover, for any u∈H1(R3)∖0, it is easy to see that there exists unique t(u)>0 such that t(u)u∈N∞. Set(36)m∞≔inf⁡Φ∞u,u∈N∞.From [11, 12], we deduce that m∞ is achieved by a positive radially symmetric function w satisfying (11). In what follows, for any y∈R3, we use the translation symbol wy≔w(·-y). Set(37)mλ≔inf⁡Φλu,u∈Nλ.Then the following relations of mλ and m∞ hold true.Lemma 5.
Suppose that(A1)-(A3) hold. Then for each λ≥0 one has(38)0<mλ≤m∞.Proof.
Letλ≥0 be fixed. The first inequality of (38) is a straight consequence of (31). In order to show the second inequality we should construct a sequence un⊂Nλ and limn⁡Φλ(un)=m∞. To accomplish this, we take yn, with yn∈R3, yn→+∞, as n→+∞ and set un=tnwyn, where wyn=w(x-yn) and tn=tλ(wyn)>0 such that un=tnwyn∈Nλ. Here we recall that w is a radial solution of (35). A direct computation shows that(39)Φλun=tn22w2+∫R3λaxwyn2+tn44∫R3∫R3b∞-bxwyn2xb∞-bywyn2y4πx-ydxdy-q∞tnp+1p+1wynp+1p+1-tnp+1p+1∫R3qxwynp+1=tn22w2+∫R3λax+ynw2-tnp+1q∞p+1wp+1p+1-tnp+1p+1∫R3qx+ynwp+1+tn44∫R3∫R3b∞-bx+ynw2xb∞-by+ynw2y4πx-ydxdy.It is clear that(40)limn→∞∫R3ax+ynw2=0,limn→∞∫R3qx+ynwp+1=0.We claim that(41)limn→∞∫R3∫R3bx+ynw2xby+ynw2yx-ydxdy=0.In fact, we infer from Hardy-Littlewood-Sobolev inequality (17) that for R>0 large (42)∫R3∫R3bx+ynw2xby+ynw2yx-ydxdy≤c∫R3b6/5x+ynw12/5xdx≤cw6∫BRb2x+yndx+cb2wL6BRcand, from condition (A2), we know that the claim holds. On the other hand, we infer from (40)-(41) and tnwyn∈Nλ that (43)w2+tn2∫R3b∞ϕ¯ww2+otn2=tnp-1q∞wp+1p+1+otnp-1.Thus, we get σ0≤tn≤σ1 for σ0,σ1>0. Moreover, we deduce from w∈N∞ that (44)1tn2-1w2=tnp-3-1wp+1p+1+o1.Since p∈(3,5), we infer that tn→1 as n→∞. Finally, we let n→∞ in (39) and obtain(45)tn→1,Φλun→m∞.This finishes the proof.By applying the well-known concentration-compactness principle [39] and maximum principle [40], we have the following splitting lemma results. For the details of the proof, one can refer to [23, Lemma 4.1 and Corollary 4.2]Lemma 6.
If the strict inequality(46)mλ<m∞
holds, thenmλ is achieved by a positive function. Furthermore, all the minimizing sequences are relatively compact.Next we consider the special caseλ=0.Lemma 7.
Assume that(A1)-(A3) and λ=0 hold. Then (10) has a ground state positive solution.Proof.
Note that(47)Φ0u=12u2+14∫R3b∞-bxϕuxu2-1p+1∫R3q∞+qxup+1,and(48)Φ∞u=12u2+14∫R3b∞ϕ¯uxu2-1p+1∫R3q∞up+1,where ϕu(x)=∫R3((b∞-b(y))u2(y)/4πx-y)dy and ϕ¯u(x)=∫R3(b∞u2(y)/4πx-y)dy. Therefore, we have that (49)m0≤m∞.From (36) we know that Φ∞(w)=m∞. Similar to the proof of Lemma 4 (a), we infer that g(t)=Φ0(tw) has unique maximum t0>0 such that t0w∈N0. Then we infer the following from condition (A2).(50)Φ0t0w=t022w2-t0p+1p+1∫R3q∞+qxup+1+t044∫R3∫R3b∞-bxw2xb∞-byw2y4πx-ydxdy<Φ∞t0wMoreover, since w∈N∞ is the unique maximum point of g∞(t)=Φ∞(tw) for t>0, it follows that Φ∞(t0w)≤Φ∞(w). Combining the above arguments, we infer that(51)m0≤Φ0t0w<Φ∞t0w≤Φ∞w=m∞.From Lemma 6, we know that the conclusion holds.


## 3. Proof of Theorem1

In this section we shall give the proof of Theorem1. This can be accomplished by the following Lemma.Lemma 8.
Assume that(A1)-(A3) hold. Then for each λ∈R+, we have mλ<m∞.Proof.
We first observe that, by Lemma7, we know that m0<m∞. So, we consider the case λ>0 in the sequel. For fixed λ>0, we choose tn such that un=tnwyn∈Nλ, where yn and tn are chosen as in the proof of Lemma 5. Moreover, as in (45), we infer that 0<σ0≤tn≤σ1. Similar to (51), we know that Φ∞(tnw)≤m∞. Thus, we infer that(52)mλ≤Φλun=Φλtnwyn=tn22w+λ∫R3ax+ynw2-tnp+1p+1∫R3q∞+qx+ynwp+1+tn44∫R3∫R3b∞-bx+ynw2xb∞-by+ynw2y4πx-ydxdy=Φ∞tnw+tn22λ∫R3ax+ynw2+tn22∫R3∫R3bx+ynw2xby+ynw2y4πx-ydxdy-tn2∫R3bx+ynϕ¯ww2-2tnp-1p+1∫R3qx+ynwp+1≤m∞+tn22λ∫R3ax+ynw2+tn22∫R3∫R3bx+ynw2xby+ynw2y4πx-ydxdy-tn2∫R3bx+ynϕ¯ww2-2tnp-1p+1∫R3qx+ynwp+1.Since b∈L∞(R3), we prove the conclusion if we show that, for large n, (53)λ∫R3ax+ynw2+c∫R3bx+ynw2xφwxdx-c∫R3bx+ynϕ¯ww2-c∫R3qx+ynwp+1<0,where φw(x)=∫R3b(y+yn)w2(y)/4πx-ydy. This is equal to proving that, for large n,(54)
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2
.A direct computation shows that for R>0 large(55)φwx=∫x-y≤1by+ynw2y4πx-ydy+∫1≤x-y≤Rby+ynw2y4πx-ydy+∫x-y≥Rbx+ynw2y4πx-ydy≤cw42∫x-y≤11x-ydy+cw8/32∫1≤x-y≤R1x-ydy1/4+cw8/32∫x-y≥R1x-ydy1/4≤c.Now we are ready to give the estimate of the term A1. We infer from condition (A2), (11), and (55) that (56)A1≤∫R3∖Bτynλax+ynw2+c∫R3∖Bτynbx+ynφww2≤c∫R3∖Bτynax+yn3/22/3+∫R3∖Bτynbx+yn3/2∫R3∖Bτynw61/3≤ce-2c∗τynholds for n large.
Next we give the estimate for the second partA2. We first consider that the case a(x) decays faster than b(x). By condition (1) of Theorem 1, we know that, for all M>0, there exists n0≥1 such that, for all n≥n0 and for all x∈Bτyn,(57)Me-2τc∗yn≤qx+yn≤ce-2τ1-τ/1+2τc∗yn,ax+yn≤c1-τ-1+τyn-1+τe-2τc∗ynbx+yn≥ce-2τ1-τ/1+2τc∗yn.Moreover, by [41, Lemmas 2.3 and 2.6], we know that(58)ϕ¯wx=∫R3b∞w2y4πx-ydy~1x,asx→∞.Since x+yn≥1-τyn→∞ as n→∞, we infer from (58) and (A2) that(59)φwx=∫R3bx+ynw2y4πx-ydy~oϕ¯wasn→∞.Thus, we infer that, for n sufficiently large and for all x∈Bτyn,(60)cbx+ynϕ¯wqx+yn-cbx+ynφwqx+yn-λax+ynqx+yn>0.Hence we deduce that the inequality(61)A2≥∫Bτynqx+ynw2cbx+ynϕ¯wqx+yn+cwp-1-cbx+ynφwqx+yn-λax+ynqx+yn>c∫Bτynqx+ynwp+1>cMe-2τc∗yn∫B1wp+1>cMe-2τc∗ynholds for n sufficiently large. Thus, by the arbitrariness of M, we can conclude that A1<A2, as desired.
Finally, we consider the remaining case in Theorem1. By (2) of Theorem 1, for all ε>0 and M>0, there exists n0≥1 such that, for all n≥n0 and for all x∈Bτyn,(62)Me-2τc∗yn≤qx+yn,ax+yn≤ce-p-1τc∗yn,where p∈(3,5). Thus we get that, for n sufficiently large and for all x∈Bτyn,(63)cwp-1-λax+ynqx+yn>c2wp-1.Hence, we infer from (59) that(64)A2≥c∫Bτynbx+ynw2ϕ¯w-φw+∫Bτynqx+ynw2cwp-1-λax+ynqx+yn>c2∫Bτynqx+ynwp+1>c2Me-2τc∗yn∫B1wp+1>c2Me-2τc∗yn.Hence, by the arbitrariness of M, we can conclude that A1<A2. This finishes the proof.Proof of Theorem1.
By Lemmas6 and 8, we know that the conclusions of Theorem 1 hold.


## 4. Conclusion

In this paper, the authors prove the existence of positive ground state solutions for the nonautonomous Schrödinger-Poisson system. In condition (1) of Theorem1, we know that if a(x) decays faster than b(x), we find the existence of positive ground state solution of (10). In condition (2) of Theorem 1, we only need the decay condition for a(x)(whatever the decay speed of b(x) to 0 as x→∞) to prove the existence of positive solution. This is different phenomenon compared to the previous paper [34].



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*Source: 8653670-2018-12-13.xml*