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

where p ∈ (3, 5), A(x), B(x), and Q(x) are positive functions such that lim|x|󳨀→∞ A(x) = a∞ > 0, lim|x|󳨀→∞ B(x) = b∞ > 0, and lim|x|󳨀→∞Q(x) = q∞ > 0. Here B : R 󳨀→ 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: iħ∂ψ ∂t = − ħ 2 2mΔψ + k (x) ψ + B (x) φψ − Q (x) 󵄨󵄨󵄨󵄨ψ󵄨󵄨󵄨󵄨p−2 ψ, x ∈ R3, t ∈ R, − ħ2 2mΔφ (x) = B (x) 󵄨󵄨󵄨󵄨ψ󵄨󵄨󵄨󵄨2 , u ∈ H1 (R3) ,


Introduction
In this paper we are concerned with the existence of solution of the nonautonomous Schrödinger-Poisson system −Δ +  ()  +  ()  ()  =  () || −1 , where  ∈ (3,5), (), (), and () are positive functions such that lim ||→∞ () =  ∞ > 0, lim ||→∞ () =  ∞ > 0, and lim ||→∞ () =  ∞ > 0.Here  : R 3 → R denotes the nonnegative measurable function which represents a nonconstant charge corrector to the density  2 . and  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][2][3].This kind of system also arises in many fields of physics.Indeed, one considers the following system: where  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 (, ) = ()  ,  > 0 and  ∈ R. It is clear that (, ) solves (2) if and only if () solves the so-called stationary system where () = () + ℏ.Here we consider that the case ℏ is constant.Without loss of generality we assume that ℏ 2 /2 = 1, then system (3) becomes (1).System (1) is also modeled in Abelian Gauge Theories; for instance, see [4][5][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 A : R 3 × R → R 3 , ( 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  : R 3 → 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][8][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 ( or ).These solutions are called spikes.For more information on this direction.one can refer to [10][11][12][13][14] and the references therein.
In the present paper we are interested in studying the case when ℏ is constant.In [15], the authors studied the existence and nonexistence of solutions of (1) when  =  =  = 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  = 1, ,  is radial and satisfies where  ∈ (0, 2], ,  0 ,  1 , and  0 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 ()|| −1  replaced by a more generic function (, ), 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 R 3 .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  1  (R 3 ) → L  (R 3 ) (∀ ∈ (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][23][24].Furthermore, for more results on the existence of positive solutions, ground and bound states, one can see [18,19,21,[25][26][27][28][29][30][31][32][33] and references therein.Nearly, the paper [34] proves the existence of bound state solution of (1) under some decay condition on , , and .Precisely, assume that where  ∞ and  ∞ are positive constants.From this assumption one can easily deduce that lim ||→∞ () = 0.In order to get the compactness, the paper [34] studies the case when the limit equation is Scalar Schrödinger equation, i.e., According to [35,36], (8) has unique positive solution, and the energy level of any sign-changing solution is strictly greater than 2 ∞ , where  ∞ 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 and, under some conditions for the , , and , we prove the existence of positive ground state solution of (1).
Clearly, system (1) becomes where  ∈ R + and  ∈ (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 .Moreover,  is decreasing when the radial coordinate increases.Precisely, there exists a constant  * > 0 such that lim Then we have the following main results.
Then system (10) has a positive ground state solution for each  ∈ R + .Remark 2. (i) From the assumptions on , we know that lim ||→∞ () =  ∞ .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 , , and  to recover the compactness and prove the existence of positive ground state solution.
(ii) Note that in condition (1) of Theorem 1, we know that () is decaying faster than ().In condition (2) of Theorem 1, we only need the decay condition for () (whatever the decay speed of () to 0 as || → ∞) to prove the existence of positive solution.This is different phenomenon compared to [34].

Preliminary Results
We define the following notation: 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  ∈  1 (R 3 ), the linear functional   defined in  1,2 (R 3 ) by We infer from condition ( 2 ) and Hölder inequality that By the Lax-Milgram theorem, we know that there exists unique So,   is a weak solution of −Δ =  2 and the following formula holds Moreover,   > 0 when  ̸ = 0. We recall the following classical Hardy-Littlewood-Sobolev inequality (see [38,Theorem 4.3]).Assume that  ∈   (R 3 ) and  ∈   (R 3 ).Then one has where 1 < ,  < ∞, 0 <  < 3, and 1/ + 1/ + /3 = 2.By (16) we know that It is well known that solutions of (10) correspond to critical points of the energy functional From ( 17), we know that Φ  is well defined and that We define the operator  : We infer from [34, Proposition 2.1-2.2] that  has the following properties.
Next we prove the later conclusion.For any fixed  > 0 large, we infer that for each  ∈ R It is very easy to verify that, whatever  ∈ R is, the function Φ  is bounded either from above or from below.Hence, it is convenient to consider Φ  restricted to a natural constraint, the Nehari manifold.We set Next lemma contains the statement of the main properties of N  .Proof.(a) Let  ∈  1 (R 3 )\{0} be such that ‖‖ = 1.We claim that there exists a unique  ∈ (0, ∞) such that  ∈ N  .Let () = Φ  () for  > 0. It is easy to verify, by ( 1 )-( 3 ), that (0) = 0, () > 0 for  > 0 small and () < 0 for  large.Therefore, max >0 () is achieved at a  = () > 0 so that () = 0 and () ∈ N  .Assume that there exist  1 >  2 > 0 such that  1 ,  2  ∈ N  .Then we see This is a contradiction.So, we prove that () admits a unique positive solution () > 0 and () ∈ N  .We infer from ( 16) and ( 1 )-( 3 ) that This implies that Let ) by the regularity of Φ  .Moreover, we infer from (29) that Here we use the fact that 3 <  < 5.
(c) If  ̸ = 0 is a critical point of Φ  , then we have Φ   () = 0,  ∈ N  .On the other hand, if  is a critical point of Φ  () constrained on N  , then there exists  ∈ R such that One infers from (30) that  = 0.
Next we consider the limit functional Φ ∞ :  1 (R 3 ) → R, defined as And we consider the corresponding natural constraint Critical points of Φ ∞ are solutions of the limit problem at infinity and, clearly, the conclusions of Lemma 4 hold true for Φ ∞ and N ∞ .Moreover, for any  ∈  1 (R 3 ) \ {0}, it is easy to see that there exists unique () > 0 such that () ∈ N ∞ .Set From [11,12], we deduce that  ∞ is achieved by a positive radially symmetric function  satisfying (11).In what follows, for any  ∈ R 3 , we use the translation symbol   fl (⋅ − ).Set Then the following relations of   and  ∞ hold true.
Lemma 5. Suppose that ( 1 )-( 3 ) hold.Then for each  ≥ 0 one has Proof.Let  ≥ 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 {  } ⊂ N  and lim  Φ  (  ) =  ∞ .To accomplish this, we take {  }, with   ∈ R  (35).A direct computation shows that It is clear that We claim that lim In fact, we infer from Hardy-Littlewood-Sobolev inequality (17) that for  > 0 large and, from condition ( 2 ), we know that the claim holds.On the other hand, we infer from ( 40)- (41) and Thus, we get Since  ∈ (3, 5), we infer that   → 1 as  → ∞.Finally, we let  → ∞ in (39) and obtain 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 holds, then   is achieved by a positive function.Furthermore, all the minimizing sequences are relatively compact.
Combining the above arguments, we infer that From Lemma 6, we know that the conclusion holds.

Proof of Theorem 1
In this section we shall give the proof of Theorem 1.This can be accomplished by the following Lemma.
Proof.We first observe that, by Lemma 7, we know that  0 <  ∞ .So, we consider the case  > 0 in the sequel.
Next we give the estimate for the second part  2 .We first consider that the case () decays faster than ().By condition (1) of Theorem 1, we know that, for all  > 0, there exists  0 ≥ 1 such that, for all  ≥  0 and for all  ∈  |  | , Hence, by the arbitrariness of , we can conclude that  1 <  2 .This finishes the proof.
Proof of Theorem 1.By Lemmas 6 and 8, we know that the conclusions of Theorem 1 hold.

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

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Advances in Mathematical Physics are related to E − H by the Maxwell equations

Lemma 4 .
Assume that ( 1 )-( 3 ) hold.Then the following conclusions hold.(a)N  is a  1 regular manifold diffeomorphic to the sphere of  1 (R 3 ).(b)Φ  is bounded from below on N  by a positive constant.(c)  is a free critical point of Φ  if and only if u is a critical point of Φ  constrained on N  .