Advances in Mathematical Physics
Volume 2018 (2018), Article ID 8653670, 10 pages
https://doi.org/10.1155/2018/8653670

Research Article

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

Lu Xiao,¹ Haiqin Qian,¹ Mengmeng Qu,² and Jun Wang²

¹School of Management, Jiangsu University, Zhenjiang, Jiangsu, 212013, China
²Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013, China

Correspondence should be addressed to Jun Wang; wangmath2011@126.com

Received 11 September 2018; Revised 6 November 2018; Accepted 2 December 2018; Published 13 December 2018

Academic Editor: Pietro d’Avenia

Copyright © 2018 Lu Xiao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

In this paper we are concerned with the existence of solution of the nonautonomous Schrödinger-Poisson system

where , , , and are positive functions such that , , and . Here denotes the nonnegative measurable function which represents a nonconstant charge corrector to the density . 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 [1–3].

This kind of system also arises in many fields of physics. Indeed, one considers the following system:

where is the imaginary unit, is the Laplacian operator, and is the Planck constant. A standing wave is a solution of (2) of the form and . It is clear that solves (2) if and only if solves the so-called stationary system

where . Here we consider that the case is constant. Without loss of generality we assume that , 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 , the gauge potentials

are related to by the Maxwell equations

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 ), the Schrödinger field is described by a real function , 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 ( sufficiently small). In this framework one is interested not only in existence of solutions but also in their asymptotic behavior as . 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 [10–14] and the references therein.

In the present paper we are interested in studying the case when is constant. In [15], the authors studied the existence and nonexistence of solutions of (1) when . 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 is radial and satisfies

where , , , and 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 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 . 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 . 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 , , and . Precisely, assume that and , and , and

where and are positive constants. From this assumption one can easily deduce that . 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 , 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 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).

In order to state our main results, we shall give some assumptions. For , we define , , and . Throughout the paper we need the following conditions.

, , .

, , .

, , .

Clearly, system (1) becomes

where and . 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 such that

Then we have the following main results.

Theorem 1. Assume that and - hold. Moreover, if one of the following conditions holds
(1) and , , , where , , and are positive constants.
(2) and , where and is positive constant.
Then system (10) has a positive ground state solution for each .

Remark 2. (i) From the assumptions on , we know that . 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].

2. Preliminary Results

We define the following notation:

(i) is the norm of defined by ;(ii) is the norm of defined by ;(iii) is the scalar product of defined by ;(iv)let , , 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 , the linear functional defined in by

We infer from condition and Hölder inequality that

By the Lax-Milgram theorem, we know that there exists unique such that

So, is a weak solution of and the following formula holds

Moreover, when .

We recall the following classical Hardy-Littlewood-Sobolev inequality (see [38, heorem 4.3]). Assume that and . Then one has

where , , and . 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 is well defined and that

We define the operator as

We infer from [34, roposition 2.1-2.2] that has the following properties.

Lemma 3. (1) is continuous.
(2) maps bounded sets into bounded sets.
(3) for all .
(4) If , then we have and for each , as , where .
(5) If , then we have in and for each , as , where .

Proof. The conclusions (1)-(4) can be proved as in [34, roposition 2.1-2.2]. Hence we only focus on the proof of (5). Since, by definition of , for all we have

then, in order to prove in , it suffices to prove that

Let be fixed arbitrarily. Then there exists a positive number so large that for , where and . Hence we deduce that

Since in , we know that (22) holds.
Next we prove the later conclusion. For any fixed large, we infer that for each

and, hence, we deduce that a.e., in and for each , as . A direct computation shows that for large

which proves for each , as .

It is very easy to verify that, whatever is, the function is bounded either from above or from below. Hence, it is convenient to consider restricted to a natural constraint, the Nehari manifold. We set

Next lemma contains the statement of the main properties of .

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

Proof. (a) Let be such that . We claim that there exists a unique such that . Let for . It is easy to verify, by -, that , for small and for large. Therefore, is achieved at a so that and . Assume that there exist such that . Then we see

This is a contradiction. So, we prove that admits a unique positive solution and . We infer from (16) and - that

This implies that

Let . Then by the regularity of . Moreover, we infer from (29) that

(b) For all ,

Here we use the fact that .
(c) If is a critical point of , then we have . On the other hand, if is a critical point of constrained on , then there exists such that

One infers from (30) that .

Next we consider the limit functional , defined as

And we consider the corresponding natural constraint

Critical points of are solutions of the limit problem at infinity

and, clearly, the conclusions of Lemma 4 hold true for and . Moreover, for any , it is easy to see that there exists unique such that . Set

From [11, 12], we deduce that is achieved by a positive radially symmetric function satisfying (11). In what follows, for any , we use the translation symbol . Set

Then the following relations of and hold true.

Lemma 5. Suppose that - hold. Then for each one has

Proof. Let 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 and . To accomplish this, we take , with , , as and set , where and such that . Here we recall that is a radial solution of (35). A direct computation shows that

It is clear that

We claim that

In fact, we infer from Hardy-Littlewood-Sobolev inequality (17) that for large

and, from condition , we know that the claim holds. On the other hand, we infer from (40)-(41) and that

Thus, we get for . Moreover, we deduce from that

Since , we infer that 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, emma 4.1 and orollary 4.2]

Lemma 6. If the strict inequality

holds, then is achieved by a positive function. Furthermore, all the minimizing sequences are relatively compact.

Next we consider the special case .

Lemma 7. Assume that - and hold. Then (10) has a ground state positive solution.

Proof. Note that

and

where and . Therefore, we have that

From (36) we know that . Similar to the proof of Lemma 4 (a), we infer that has unique maximum such that . Then we infer the following from condition .

Moreover, since is the unique maximum point of for , it follows that . Combining the above arguments, we infer that

From Lemma 6, we know that the conclusion holds.

3. Proof of Theorem 1

In this section we shall give the proof of Theorem 1. This can be accomplished by the following Lemma.

Lemma 8. Assume that hold. Then for each , we have .

Proof. We first observe that, by Lemma 7, we know that . So, we consider the case in the sequel. For fixed , we choose such that , where and are chosen as in the proof of Lemma 5. Moreover, as in (45), we infer that . Similar to (51), we know that . Thus, we infer that

Since , we prove the conclusion if we show that, for large ,

where . This is equal to proving that, for large ,

A direct computation shows that for large

Now we are ready to give the estimate of the term . We infer from condition , (11), and (55) that

holds for large.
Next we give the estimate for the second part . We first consider that the case decays faster than . By condition (1) of Theorem 1, we know that, for all , there exists such that, for all and for all ,

Moreover, by [41, emmas 2.3 and 2.6], we know that

Since as , we infer from (58) and that

Thus, we infer that, for sufficiently large and for all ,

Hence we deduce that the inequality

holds for sufficiently large. Thus, by the arbitrariness of , we can conclude that , as desired.
Finally, we consider the remaining case in Theorem 1. By (2) of Theorem 1, for all and , there exists such that, for all and for all ,

where . Thus we get that, for sufficiently large and for all ,

Hence, we infer from (59) that

Hence, by the arbitrariness of , we can conclude that . This finishes the proof.

Proof of Theorem 1. By Lemmas 6 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 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].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors contributed equally to this article. They have all read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants 11571140, 11671077, 71704066), the Fellowship of Outstanding Young Scholars of Jiangsu Province (BK20160063), the Major Projects of the National Social Science Foundation (18VSJ038), the Six Big Talent Peaks Project in Jiangsu Province (XYDXX-015), and the Natural Science Foundation of Jiangsu Province (Nos. BK20150478, BK20170542).

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