XU Na

（School of Science，Tianjin University of Technology and Education，Tianjin 300222，China）



Ground State Solutions for Schr¨odinger-Poisson Systems

XU Na

（School of Science，Tianjin University of Technology and Education，Tianjin 300222，China）

This paper deals with a class of Schr¨odinger-Poisson systems.Under some conditions，we prove that there exists a ground state solution of the system.The proof is based on the compactness lemma for the system.Our results here improve some existing results in the literature.

Schr¨odinger-Poisson system；ground state solution；compactness

2000 MR Subject Classification:35J20，35J50

Article ID:1002—0462（2016）01—0009—10

Chin.Quart.J.of Math.

2016，31（1）：9—18

§1.　Introduction

In this paper we are concerned with the Schr¨odinger-Poisson system

where 1＜p＜3，λ＜0 and u，φ：R3→ R.This system has been introduced as a physical model describing a charged wave interacting with its electrostatic field in quantum mechanic. The unknowns u，φ represent the wave functions associated to the particle and electric potential，and the function K is a nonnegative density charge.For a more detailed physical background，we can refer to［4］and the references therein.

To obtain our results，we have to overcome one difficulty in using variational method.The difficulty is that the lack of compactness of the embedding of H1（R3）into Lq（R3），q∈（2，4）. It is well known that the system can be reduced to a single equation with a non-local term.To recover the compactness，we study the behavior of（PS）sequence of its energy functional and establish the compactness lemma for the sequence.

This paper is organized as follows.In Section 2，we give the preliminaries.In Section 3，we consider the problems at infinity.Section 4 is devoted to dealing with the compactness lemma and the proof of main result.

§2.　Preliminaries

Throughout this paper，we use the following notations.

Let H1（R3）be the usual Sobolev space endowed with the standard scalar and norm

D1，2（R3）is the completion ofwith respect to the norm H-1（R3）denotes the dual space of H1（R3）.

Lq（Ω）（1≤q≤+∞），Ω⊂R3，denotes a Lebesgue space；the norm in Lq（Ω）is denoted by‖u‖q，Ω，where Ω is a proper subset of R3，by‖·‖q，when Ω=R3.

S is the best Sobolev constant for the Sobolev embedding H1，2（R3）,→L6（R3），i.e.，

And， is the best Sobolev constant for the Sobolev embedding D1，2（R3）,→L6（R3），i.e.，

Let c，C，Cidenote various positive constants.

We always assume that a（x），K（x）satisfy the following

Under these conditions on a，K and λ＜0，we consider the system（Pλ）.

It is known that（Pλ）can be reduced to a single equation with a non-local term.In fact，for every u∈H1（R3），Riesz-represent Theorem implies that there exists a unique φu∈D1，2（R3）such that-∆φu=K（x）u2with

Moreover，one has that

and

So，we can rewrite（Pλ）as the following equivalent equation

The corresponding functional can be written as for u∈H1（R3），

Let us now define the operators Φ：H1（R3）→D1，2（R3）as Φ（u）=φu.

In the following lemma，we summarize the properties of Φ.

Lemma 2.1

（1）Φ is continuous；

（2）Φ maps bounded sets into bounded sets；

（3）If un*u in H1（R3）then Φ（un）*Φ（u）in D1，2（R3）；

（4）Φ（tu）=t2Φ（u）for all t∈R.

Proof The proof is similar to that of［7，10］and we omit it.

In order to find critical levels of Iλ，we need to look into the geometry of the functional.It is convenient to consider Iλrestricted to a natural constraint，the Nehari manifold，that containsall the critical points of Iλand on which Iλturns out to be bounded from below.We set

where

The manifold Nλsatisfies the following statements and the proof can be made in a similar way as in［7］.

Lemma 2.2（1）Nλis a C1regular manifold diffeomorphic to the sphere of H1（R3）；

（2）Iλis bounded from below on Nλby a positive constant；

（3）u is a free critical point of Iλif and only if u is a critical point of Iλconstrained on Nλ.

§3.　The Problems at Infinity

To establish the compactness lemma and prove our main result in Section 4，we need the problems at infinity.

Since K（x）→ 0，a（x）→a∞as|x|→ ∞，the problem at infinity related to（P′λ）is the following problem

The solutions of（SN1）are the critical points of the functional J∈C2（H1（R3），R）defined as

Let

If K（x）=0，（P′λ）becomes the equation

that has been widely studied when a（x）is a constant.

In particular，when a（x）=1，（NSE）becomes

Proposition 3.1［7］Problem（NSE）∞has a positive ground state solution w∈H1（R3），radially symmetric about the origin，unique up to translations，decaying exponentially，as|x|→+∞.Furthermore，its derivatives have the properties of w.

Moreover，we deal with the problem

The solutions of（SN2）are the critical points of the real functionaldefined on H1（R3）by

Let us define the Nehari manifold related to

and set

By the Mountain Pass Theorem，we can see that

§4.Compactness Lemma and Main Result

First，we deal with the behavior of（PS）sequence of Iλ.

Lemma 4.1Let（un）nbe a（PS）sequence of Iλconstrained on Nλ，i.e.，un∈Nλand

（a）Iλ（un）is bounded；

（b）▽Iλ|Nλ（un）→0 strongly in H1（R3）.

Then replacing（un）n，if necessary，up to a subsequence，there exist a solutionof（P′λ），a number k∈N∪｛0｝，k functions u1，···，ukof H1（R3）and k sequences of points∈ R3，0≤j≤k such that

（iv）ujare nontrivial solutions of（SN1）.

Proof We first observe that for λ＜0，1＜p＜3 and un∈Nλ，

Noticing that Iλ（un）is bounded，we obtain that（un）nis bounded.

We claim that

In fact，we have

for some σn∈R.So，taking the scalar product with un，we obtain

Since un∈Nλ，（▽Iλ（un），un）=0 and by

For u∈Nλ，we have

from which we deduce‖u‖≥C＞0，for all u∈Nλ.

Thus，for u∈Nλ，we deduce

So，σn→0 for n→∞.Moreover，by the boundedness of（un）n，▽Gλ（un）is bounded and this implies that σn▽Gλ（un）→0 and the claim holds.

Since unis bounded in H1（R3），there exists∈H1（R3）such that up to a subsequence，un*in H1（R3）and un*in Lp+1（R3），un（x）→（x）a.e.，on R3.

Furthermore，taking into account（3）of Lemma 2.1，we deduce thatthat isis a weak solution of（P′λ）.

If un→in H1（R3），the proof is done.So we can assume that（un）ndoes not converge strongly toin H1（R3）.Set=un（x）-（x）.Obviously，*0 in H1（R3），but not strongly.A direct computation can show that

Moreover，from the Brezis-Lieb Lemma，we deduce that

And by Lemma 8.1 of［11］，we obtain

and

Therefore，we have

and for all h∈H1（R3），we can see that

so that

Furthermore，one can get that

Setting

we can prove that δ＞0.Then we may assume the existence of y1n⊂R3，such that

it follows that

and then u1≠0.But sinceis unbounded and up to a subsequence， we can assume thatFurthermore，（4.2）implies that▽J（u1）=o（1）.Finally，setThen we have

This implies

Hence from（4.1）and（4.3），we obtain

As before we can prove that

in H1（R3）.Now ifwe complete the proof.Otherwiseand not strongly and we repeat the argument above.By iterating this procedure we obtain sequences of pointssuch thatand a sequence of functionswith j≥2，such that

in H1（R3）.

And

Corollary 4.2Let（un）nbe a（PS）csequence.Then（un）nis relatively compact for allMoreover，ifthen either（un）nis relatively compact or the statement of Lemma 4.1 holds with k=1.

Proof Let us apply Lemma 4.1 to the sequence（un）n.Noticing thatWhenLemma 4.1（iii）gives that k=0 and thenWhenis not compact，then Lemma 4.1（iii）implies that

Proposition 4.3If K（x）≥0 and K≠0，then there existssuch that

Thus，in any case we can complete the proof.

Now we state our main result

Theorem 4.4Let（a）and（K）hold and additionally assume that a satisfies

then problem（Pλ）has at least one ground state solution.

Proof Without loss of generality we can take λ=-1.For simplicity，we set，To prove the existence of a ground state solution for（P′λ），we just need to show that，such thatand let t＞0 such that.We claim t≥1.In fact，since

Hence

and this implies that t≥1.Then

Let us now estimate t.We find

Then，since K（x），a（x）are positive functions，we have

Substituting in（4.4）and by condition

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O175.2Document code:A

date：2014-09-19

Biography：XU Na（1986-），female，native of Baoding，Hebei，a lecturer of Tianjin University of Technology and Education，Ph.D.，engages in differential equations and dynamical systems.