Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 78, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND CONCENTRATION OF POSITIVE GROUND STATES FOR SCHRÖDINGER-POISSON EQUATIONS WITH COMPETING POTENTIAL FUNCTIONS WENBO WANG, QUANQING LI Abstract. This article concerns the Schrödinger-Poisson equation −ε2∆u+ V (x)u+K(x)φu = P (x)|u|p−1u+Q(x)|u|q−1u, x ∈ R3, −ε2∆φ = K(x)u2, x ∈ R3, where 3 < q < p < 5 = 2∗ − 1. We prove that for all ε > 0, the equation has a ground state solution. The methods used here are based on the Nehari manifold and the concentration-compactness principle. Furthermore, for ε > 0 small, these ground states concentrate at a global minimum point of the least energy function. 1. Introduction We study the Schrödinger-Poisson equation −ε2∆u+ V (x)u+ φu = f(x, u), x ∈ R3, −ε2∆φ = u2, x ∈ R3. (1.1) This equation has attracted much attention, and still is a stimulating field of re- search for mathematicians and for physicists. System (1.1) was first introduced in [2] and has been a study object of interest for nonlinear analysis. As a physical model, it describes a charged particle interacting with its own electrostatic field in quantum. And it can be a model to describe semiconductor theory, nonlinear optics and plasma physics. The presence of the nonlinear term f(x, u) simulates the interaction between many particles and external nonlinear perturbations. In fact, it can be described by coupling the nonlinear Schrödinger and Maxwell equations and so it is also known as the Schrödinger-Maxwell system. We refer the readers to [2] and the references therein for the physical aspects of problem (1.1). Especially, the semi-classical state solutions describe the transition from Quantum Mechanics to Newtonian Mechanics from the point of view of physics. In recent years, (1.1) with V (x) ≡ 1 and ε = 1 has been studied under variant assumptions on f . See for example [1, 3, 5, 7, 8, 9], and the references therein. In [12], the authors consider Schrödinger-Poisson equation with a non-constant 2010 Mathematics Subject Classification. 35J15, 35J20, 35J50. Key words and phrases. Schrödinger-Poisson equation; Nehari manifold; ground states; concentration-compactness; concentration. c©2020 Texas State University. Submitted January 11, 2019. Published July 22, 2020. 1 2 W. WANG, Q. LI EJDE-2020/78 potential and double parameters perturbation: −ε2∆u+ V (x)u+ φu = u5 + f(u), x ∈ R3, −ε2∆φ = u2, u(x) > 0, x ∈ R3. (1.2) They use mountain pass to prove that (1.2) has a ground state solution which possesses the concentrating phenomenon, concentrating around global minimum of the potential V in the semi-classical limit. In [28], the authors studied the existence of positive ground state via the Nehari manifold methods. They multiply the nonlinearity by a potential b(x), that is, −ε2∆u+ V (x)u+ φu = u5 + b(x)f(u), x ∈ R3, −ε2∆φ = u2, u(x) > 0, x ∈ R3. (1.3) As for the concentration of ground state solutions, naturally, there is a competition between the linear potential V (x) and the nonlinear potential b(x), i.e., V (x) wants to attract ground state solutions to its minimum points but b(x) wants to attract ground state solutions to its maximum points. For instance in [4], a potential K(x) before non-local term was added, i.e., −∆u+ V (x)u+K(x)φu = a(x)|u|p−1u, x ∈ R3, −∆φ = K(x)u2, x ∈ R3, under suitable assumptions, for p ∈ (3, 5), the authors also obtain a positive ground state solution or positive solution. For other results for this system, see for example [10, 13, 15, 18, 21, 22, 23, 30, 31, 32, 33] and the references therein. We should mention that for the equation − ε2∆u+ V (x)u = |u|p−2u, x ∈ R3, (1.4) if we let ε2 = λ−1, v = λ− 1 p−2u, then (1.4) can be written as −∆v + λV (x)v = |v|p−2v, x ∈ R3. So in [33], the authors consider the system −∆u+ λV (x)u+K(x)φu = a(x)|u|p−2u, x ∈ R3, −∆φ = K(x)u2, x ∈ R3, where 2 < p < 6, the potential V can be sign-changing. Under suitable conditions, they show some concentrations when λ→∞. See [22] for the generalized extensible beam equations. Motivated by above works, and [29], we study the existence and concentration of ground states to (1.1) with competing potentials. More precisely, we are concerned with the Schrödinger-Poisson equation −ε2∆u+ V (x)u+K(x)φu = P (x)|u|p−1u+Q(x)|u|q−1u, x ∈ R3, −ε2∆φ = K(x)u2, x ∈ R3, (1.5) where 3 < q < p < 5 = 2∗ − 1, V , K and P are continuous and bounded positive functions. Q is continuous function and maybe change sign, even be negative. Using the Nehari manifold and the concentration-compactness, we shall prove that the above problem admits a ground state. Furthermore, we want to prove that these ground states concentrate at a point which locates on the middle ground of EJDE-2020/78 SCHRÖDINGER-POISSON EQUATIONS 3 the competing potential functions P (x) and Q(x) as ε → 0+ via a concentration- compactness argument similar to [29]. The previous results of existence and concentration of Schrödinger-Poisson prob- lems (see e.g. [12, Theorem 1.1], [27, Theorems 1.1, 1.2, 1.3], [28, Theorem 1.1]) can not be applied directly to (1.5) when Q 6= 0 is not a constant potential, espe- cially, when Q is sign-changing or negative. To state our main results, we use the following assumptions: (A1) V (x) ∈ C1(R3,R), 0 < V0 := inf x∈R3 V (x) ≤ V (x) ≤ V∞ := lim |x|→∞ V (x) <∞ and V (x) 6≡ V∞. (A2) K(x) ∈ C(R3,R), 0 < K0 := inf x∈R3 K(x) ≤ K(x) ≤ K∞ := lim |x|→∞ K(x) <∞ and K(x) 6≡ K∞. (A3) P (x) ∈ C1(R3,R) and P (x) ≥ P∞ := lim|x|→∞ P (x) > 0 and P (x) 6≡ P∞. (A4) Q(x) ∈ C1(R3,R) and Q(x) ≥ Q∞ := lim|x|→∞Q(x), Q is allowed to change sign or be negative and Q(x) 6≡ Q∞. For each s ∈ R3, we consider the following problem with parameter s ∈ R3, −∆u+ V (s)u+ φu = P (s)|u|p−1u+Q(s)|u|q−1u, x ∈ R3, −∆φ = u2, x ∈ R3. (1.6) Denote the corresponding energy functional by Is and the corresponding least en- ergy by C(s) = c(V (s), P (s), Q(s)) := inf{Is(u) : u is a nontrivial solution of (1.6)}. It is well known that C(s) is well defined. Our main result is as follows. Theorem 1.1. (I) Suppose that (A1)–(A4) are satisfied, and 3 < q < p < 5 = 2∗ − 1. Then for each ε > 0, (1.5) has a positive ground state solution uε. (II) Let the assumptions in (I) be satisfied and let K(x) = 1. Then for ε > 0 small, (1) the positive ground state solution uε obtained in (I) possesses at most one local (hence global) maximum point xε in R3 such that lim ε→0+ C(xε) = inf s∈R3 C(s). (2) xε → x0, wε(x) := uε(εx + xε) converges in H1(R3) to a positive ground state solution of −∆u+ V (x0)u+ φu = P (x0)|u|p−1u+Q(x0)|u|q−1u, x ∈ R3, −∆φ = u2, x ∈ R3. (3) there exist C1, C2 > 0 such that uε(x) ≤ C1e −C2| x−xεε |. Remark 1.2. We point out that the potential K appears both in the first equation and in the second equation of (1.5) which is used in (2.5). In Appendix, we will explain why we let K(x) = 1. 4 W. WANG, Q. LI EJDE-2020/78 This article is organized as follows. In section 2, we verify the existence of ground states. In section 3, we are devoted to prove the properties of ground states including exponential decay and concentration. 2. Existence of ground states Under our assumptions, for the existence of ground states, without loss of gen- erality, we may assume that ε = 1. Then (1.5) becomes −∆u+ V (x)u+K(x)φu = P (x)|u|p−1u+Q(x)|u|q−1u, x ∈ R3, −∆φ = K(x)u2, x ∈ R3. (2.1) Let H1(R3) denote the usual Sobolev space endowed with the standard scalar prod- uct and norm (u, v) = ∫ R3 (∇u∇v + V (x)uv)dx, ‖u‖2 = ∫ R3 (|∇u|2 + V (x)u2)dx. The set D1,2(R3) = {u ∈ L6(R3) : ∇u ∈ L2(R3)} is a Hilbert space endowed with the standard scalar product and norm (u, v) = ∫ R3 ∇u∇vdx, ‖u‖2D1,2 = ∫ R3 |∇u|2dx. For u ∈ H1(R3), we focus on the equation −∆φ = K(x)u2. It is well known that there exists a unique φu ∈ D1,2(R3) such that −∆φu = K(x)u2. Furthermore, we have φu(x) = 1 4π ∫ R3 K(y)u2(y) |x− y| dy. (2.2) Substituting this into (2.1), we can rewrite (2.1) as −∆u+ V (x)u+K(x)φu(x)u = P (x)|u|p−1u+Q(x)|u|q−1u. (2.3) We formally formulate problem (2.3) in a variational way as I(u) = 1 2 ∫ R3 (|∇u|2 + V (x)u2)dx+ 1 4 ∫ R3 K(x)φuu 2dx − 1 p+ 1 ∫ R3 P (x)|u|p+1dx− 1 q + 1 ∫ R3 Q(x)|u|q+1dx, u ∈ H1(R3). For simplicity, define N(u) := ∫ R3 K(x)φuu 2dx, ‖u‖ss := ∫ R3 |u|sdx. Lemma 2.1. (1) If un ⇀ u in H1(R3), then φun ⇀ φu in D1,2(R3) and N(u) ≤ lim inf n→∞ N(un), N(un − u) = N(un)−N(u) + on(1). (2.4) (2) φu ≥ 0, ‖φu‖D1,2 ≤ C‖u‖2, and ∫ R3 K(x)φuu 2dx ≤ C‖u‖412/5 ≤ C‖u‖ 4. (3) φtu = t2φu, for all t ∈ R. EJDE-2020/78 SCHRÖDINGER-POISSON EQUATIONS 5 Proof. (1) It is easy to obtain the conclusions by the method in [7] with slight modification (see also [4]). The last splitting property can be obtained by [32, Lemma 2.1]. (2) Noting that K is positive, it is easy to check the conclusions. (3) It follows form a direct computation, we omit it here. � In view of Lemma 2.1, we can see that I ∈ C1(H1(R3),R). A direct computation, we have 〈N ′(u), ϕ〉 = 1 2 ∫ R3 K(x)φuuϕdx+ 1 2 ∫ R3 K(x)φ√uϕu 2dx = ∫ R3 K(x)φuuϕdx (by Fubini’s theorem). (2.5) Thus, for all ϕ ∈ H1(R3), we have 〈I ′(u), ϕ〉 = ∫ R3 ∇u∇ϕdx+ ∫ R3 V (x)uϕdx+ ∫ R3 K(x)φuuϕdx − ∫ R3 P (x)|u|p−1uϕdx− ∫ R3 Q(x)|u|q−1uϕdx. (2.6) We define the Nehari manifold of I, as N := {u ∈ H1(R3) \ {0} : γ(u) = 0}, (2.7) where γ(u) = 〈I ′(u), u〉. (2.8) The next lemma shows that N 6= ∅. Lemma 2.2. Suppose that u 6= 0 and 3 < q < p < 2∗ − 1. Then there is a unique t = t(u) > 0 such that tu ∈ N and I(ru) < I(tu) if r 6= t. Proof. Set f(t) := I(tu) = At2 2 + Bt4 4 − Ctp+1 p+ 1 − Dtq+1 q + 1 , where A = ∫ R3 (|∇u|2 + V (x)u2)dx, B = ∫ R3 K(x)φuu 2dx, C = ∫ R3 P (x)|u|p+1dx, D = ∫ R3 Q(x)|u|q+1dx. Then f ′(t) = At+ Bt3 − Ctp −Dtq, and hence f ′(t) > 0 for t small and f ′(t) < 0 for t large. Hence there is t = t(u) > 0 such that f ′(t) = 0. Thus γ(tu) = tf ′(t) = 0, which implies tu ∈ N . The uniqueness follows from the fact that the equation A+Bt2 − Ctp−1 −Dtq−1 = 0 has a unique positive solution. � The next lemma is crucial for proving our results. Lemma 2.3. There exists C > 0 such that for any u ∈ N , I(u) ≥ C ∫ R3 (|∇u|2 + V (x)u2)dx. 6 W. WANG, Q. LI EJDE-2020/78 Proof. Since V,K and P are positive, it follows from γ(u) = 0 that I(u) =I(u)− 1 q + 1 〈I ′(u), u〉 = (1 2 − 1 q + 1 )∫ R3 (|∇u|2 + V (x)u2)dx+ (1 4 − 1 q + 1 )∫ R3 K(x)φuu 2dx + ( 1 q + 1 − 1 p+ 1 )∫ R3 P (x)|u|p+1dx ≥C ∫ R3 (|∇u|2 + V (x)u2)dx. The proof is complete. � According to Lemma 2.3, we can define c∗ := inf u∈N I(u). Then c∗ ≥ 0. Furthermore, the following lemma shows that c∗ > 0. Lemma 2.4. There exists r∗ > 0 such that ‖u‖ > r∗, for all u ∈ N . Proof. For any u ∈ N , we have 0 = 〈I ′(u), u〉 ≥ ‖u‖2 − C1‖u‖4 − C2‖u‖p+1 − C3‖u‖q+1, from which we obtain that ‖u‖ > r∗ > 0, which completes the proof. � Let {un} ⊂ N be a minimizing sequence of c∗, i.e. I(un) → c∗ as n → ∞. In the light of Lemma 2.3, {un} is bounded in H1(R3). Extracting a subsequence if necessary, we have un ⇀ u in H1(R3) and un(x) → u(x) a. e. in R3. Up to a subsequence, we have the following lemma. Lemma 2.5. ∫ R3 |un|p+1dx has a positive lower bound with respect to n, that is,∫ R3 |un|p+1dx 6→ 0. Proof. Suppose to the contrary ∫ R3 |un|p+1dx → 0. Invoking the interpolation inequality, we obtain ∫ R3 |un|q+1dx→ 0. So we have c∗ = lim n→∞ I(un) = lim n→∞ ( I(un)− 1 2 〈I ′(un), un〉 ) = lim n→∞ [ − 1 2 ∫ R3 K(x)φunu 2 ndx+ (1 2 − 1 p+ 1 )∫ R3 P (x)|un|p+1dx + (1 2 − 1 q + 1 )∫ R3 Q(x)|un|q+1dx ] ≤ 0. This is a contradiction. � EJDE-2020/78 SCHRÖDINGER-POISSON EQUATIONS 7 Now, we can assume (extracting a subsequence, if necessary) that∫ R3 |un|p+1dx+ ∫ R3 φun |un|2dx n→∞−−−−→ α ∈ (0,∞). We apply the concentration-compactness principle (see [20] or [14]) to ρn := |un|p+1 + φun |un|2 to obtain un → u in Lp+1(R3). By the concentration-compactness lemma, up to a subsequence, there are three possibilities: 1 (compactness). For any ε > 0, there is a R > 0 and {xn} ⊂ RN such that∫ RN\BR(xn) ρndx < ε. 2 (vanishing). For any R > 0, it holds lim n→∞ ( sup x∈RN ∫ BR(x) ρn|dx ) = 0. 3 (dichotomy). There exists a β̃ ∈ (0, α), such that for all ε > 0, there is a R > 0, {xn} ⊂ RN and a sequence R ≤ Rn →∞ satisfied: for n large enough,∣∣ ∫ BRn (xn) ρndx− β̃ ∣∣ < ε, ∣∣ ∫ RN\BRn (xn) ρndx− (α− β̃) ∣∣ < ε. It is sufficient to show that vanishing and dichotomy do not occur. Lemma 2.6. The vanishing does not occur. Proof. The desired result follows from the vanishing Lemma in [14] (see also [24, Lemma 1.21]) and Lemma 2.5. � Lemma 2.7. The dichotomy does not occur. Proof. According to concentration-compactness principle,we can suppose that there exists a subsequence of {ρn}, still denote {ρn}, β ∈ (0, 1] and {xn} ⊂ R3 such that for each ε > 0, there exist rε > 0, rε < rn, rn < rn+1 n→∞−−−−→∞ satisfying lim inf n→∞ ∫ Brn (xn) ρn(x)dx ≥ αβ − ε, (2.9) lim inf n→∞ ∫ Bc2rn (xn) ρn(x)dx ≥ (1− β)α− ε. (2.10) We only need to prove β = 1 to exclude dichotomy. Actually, β = 1 is can be done by using the next lemma. � Let φn be a cut-off function such that φn ≡ 1 in Brn(xn) and φn ≡ 0 in Bc2rn(xn). Write un = φnun + (1− φn)un := vn + wn. Lemma 2.8. limn→∞ ‖wn‖ = 0. 8 W. WANG, Q. LI EJDE-2020/78 Proof. The proof is similar to that in [19] with slight modification. Since it has sign-changing potential Q, here we give the details for completeness. By direct calculations, we obtain∣∣∣ ∫ R3 (|∇un|2 − |∇vn|2 − |∇wn|2)dx ∣∣∣ = 2 ∣∣∣ ∫ R3 ∇vn∇wndx ∣∣∣ = on(1), (2.11)∣∣∣ ∫ R3 V (x)(u2 n − v2 n − w2 n)dx ∣∣∣ = on(1), (2.12)∣∣∣ ∫ R3 P (x) ( |un|p+1 − |vn|p+1 − |wn|p+1 ) dx ∣∣∣ = on(1), (2.13)∣∣∣ ∫ R3 Q(x) ( |un|q+1 − |vn|q+1 − |wn|q+1 ) dx ∣∣∣ = on(1). (2.14) We have the splitting property N(un) = N(vn) +N(wn) + on(1). (2.15) In fact, by a direct computation, we obtain N(un) = N(vn) +N(wn) + on(1) + 1 4 ∫ R3 ∫ R3 K(x)|vn(y)|2|wn(x)|2 4π|x− y| dy dx. Furthermore,∫ {x:|x|≥rn} ∫ {y:|y|≤2rn} |vn(y)|2|wn(x)|2 4π|x− y| dy dx = ∫ {x:|x|≥rn} ∫ {y:rn≤|y|≤2rn} |vn(y)|2|wn(x)|2 4π|x− y| dy dx + ∫ {x:2rn≥|x|≥rn} ∫ {y:|y|2rn} ∫ {y:|y| 0. Take t = [ (p+ 1)M A ] 1 p−3 , where A is from (2.18) and M is large enough such that t > 1 and moreover∫ R3 (|∇un|2 + V (x)u2 n)dx+ 1 2 ∫ R3 K(x)φunu 2 ndx ≤M. (2.20) Thus, for n large enough, we have I(tun) ≤ t 4 2 ( M − 2t p−3 p+ 1 ∫ R3 P (x)|un|p+1dx ) ≤−M t 4 2 + on(1) < 0. (2.21) Case 2. B ≤ 0. Note that an = ∫ R3 ( |∇un|2 + V (x)u2 n ) dx, bn = ∫ R3 K(x)φunu 2 ndx, cn = − ∫ R3 Q(x)|un|q+1dx are bounded. We can choose M1 > 0 independent of n, such that an + 1 2 bn + 2 q + 1 cn ≤M1. Take t̄ = [ (p+ 1)M1 A ] 1 p−q . And let M1 be large enough such that t > 1. For n large enough, we have I(tun) ≤ t q+1 2 ( M1 − 2t p−q p+ 1 ∫ R3 P (x)|un|p+1dx ) ≤−M t q+1 2 + on(1) < 0. (2.22) For the case B = 0, it is easy to obtain a similar result. Thus, by (2.16), for all ε > 0, for n large enough, we obtain I(tun) ≥ I(tvn) + I(twn)− ε. (2.23) Taking into account (2.21), choosing a smaller ε > 0 if necessary, it holds that I(tvn) + I(twn) < 0. It follows that I(tvn) < 0, or I(twn) < 0. By Lemma 2.2 and t(vn) ≤ t(wn), it holds that t(vn) ≤ t. For the lower positive bound, we also need to discuss two cases. Case 1. B > 0. Take t = ( c∗ M )1/2 , 10 W. WANG, Q. LI EJDE-2020/78 where M comes from as in (2.20) and large enough. Note that t < 1, for any t < t, I(tun) ≤ t2 2 (∫ R3 (|∇un|2 + V (x)u2 n)dx+ 1 2 ∫ R3 K(x)φunu 2 ndx ) ≤ c∗ 2 . Case 2. B ≤ 0. Note that there exists a L > 0, such that −L ≤ ∫ R3 Q(x)|un|q+1dx. We take t = min {( c∗ 2M )1/2 , ( c∗ 2L ) 1 q+1 } . So that it holds I(tun) ≤ t 2 2 (∫ R3 (|∇un|2 + V (x)u2 n)dx+ 1 2 ∫ R3 K(x)φunu 2 ndx ) − tq+1 2 ∫ R3 Q(x)|un|q+1dx ≤c ∗ 2 . Similarly to (2.23), jointly with (2.17), I(t(vn)un) ≥ I(t(vn)vn) + I(t(vn)wn) ≥ c∗ − ε. So, by choosing a small ε > 0, for n large enough, I(t(vn)un) > c∗/2. Thus we obtain the lower bound of t(vn). For all t ∈ (0, t(vn)), noting that t(vn) ≤ t(wn), combining with (2.16), we have I(twn) ≤I(t(vn)wn) =I(t(vn)un)− I(t(vn)vn) + on(1) ≤I(un)− c∗ + on(1) = on(1), where {un} ⊂ N and limn→∞ I(un) = c∗ are used in the last inequality. Moreover, it is well-known that there exists a D > 0 independent of n such that 1 p+ 1 ∫ R3 P (x)|wn|p+1dx ≤ D 2 , and 1 q + 1 ∫ R3 Q(x)|wn|q+1dx ≤ D 2 . Observing that 0 < t < t < 1 and q < p, one has on(1) ≥I(twn) ≥ t 2 2 ln − tq+1D 2 − tp+1D 2 ≥ t 2 2 ln −Dtq+1, where ln = ∫ R3 (|∇wn|2 + V (x)w2 n)dx are bounded. Let t = ( ln 4D ) 1 q−1 . Taking D large enough if necessary such that t ∈ (0, t), we obtain t2 2 ln − Etq+1 = t2 4 ln. EJDE-2020/78 SCHRÖDINGER-POISSON EQUATIONS 11 Therefore, on(1) ≥ I(twn) ≥ Cl2q−1 n . In the case t(vn) > t(wn), we can argue analogously to conclude that lim n→∞ ‖vn‖ = 0. This contradicts (2.9). Evidence now allows us to conclude, that limn→∞ ‖wn‖ = 0. � Lemma 2.9. c∗ is achieved. Proof. The proof is divided into two cases. Case (1): {xn} is bounded. In this case, by Lemmas 2.6 and 2.7, un → u 6≡ 0 in Ls(R3) for all s ∈ (2, 6). Combining with Lemma 2.2, there is a unique t > 0 such that γ(tu) = 0 and hence c∗ ≤ I(tu) ≤ lim inf n→∞ I(tun) ≤ lim inf n→∞ I(un) = lim n→∞ I(un) = c∗. Case (2): {xn} is unbounded. Set zn = un(·+xn), and we have zn ⇀ z in H1(R3). It is easy to show zn → z in L2(R3) by following the same method in [19, step 4]. By interpolation inequalities, we obtain zn → z 6≡ 0 in Ls(R3) for all s ∈ [2, 6). Using Lebesgue dominated convergence theorem, one has lim n→∞ ∫ R3 V (x)u2 ndx = lim n→∞ ∫ R3 V (x+ xn)z2 ndx = V∞ ∫ R3 z2dx ≥ ∫ R3 V (x)z2dx = lim n→∞ ∫ R3 V (x)z2 ndx. Similarly, lim n→∞ ∫ R3 P (x)|un|p+1dx ≤ lim n→∞ ∫ R3 P (x)|zn|p+1dx, lim n→∞ ∫ R3 Q(x)|un|q+1dx ≤ lim n→∞ ∫ R3 Q(x)|zn|q+1dx. Especially, for N(un), Lebesgue dominated convergence theorem can be used, so it also holds lim n→∞ N(un) ≥ lim n→∞ N(zn). Thus, similarly to case (1), c∗ ≤ I(t(z)z) ≤ lim inf n→∞ I(t(z)zn) ≤ lim inf n→∞ I(un) = lim n→∞ I(un) = c∗. The proof is complete. � Similar to [16, Lemma 2.5] (see also [19, Theorem 2.2], we have the following result. Lemma 2.10. Suppose that u ∈ N and I(u) = c∗, then u is a weak solution of (2.1). In view of Lemma 2.9 and Lemma 2.10, we can define c∗∗ = inf{I(u) : u is a nontrivial solution of (2.1)}. The next lemma shows that the functional I satisfies the mountain pass geometry. 12 W. WANG, Q. LI EJDE-2020/78 Lemma 2.11. The functional I satisfies (1) there exist α, ρ > 0 such that I(u) ≥ α for all ‖u‖ = ρ. (2) there exists e ∈ H1(R3) \Bρ(0), such that I(e) < 0. Proof. Since K is positive and P , Q are bounded, by the Sobolev embedding and result (2) in Lemma 2.1, we have I(u) ≥ 1 2 ‖u‖2 − C1‖u‖p+1 − C2‖u‖q+1. Hence we can choose some α, ρ > 0 such that I(u) ≥ α for all ‖u‖ = ρ. For u ∈ H1(R3) \ {0}, we have f(t) := I(tu) = At2 2 + Bt4 4 − Ctp+1 p+ 1 − Dtq+1 q + 1 < 0, for t > 0 large enough, where A,B,C,D are similar to Lemma 2.2. Choose e = t0u for some suitable t0. � As a consequence of the Mountain Pass lemma without (PS)c condition, we define the constant c = inf γ∈Γ sup t∈[0,1] I(γ(t)) > 0, where Γ = {γ ∈ C([0, 1], H1(R3)) : γ(0) = 0, I(γ(1)) < 0}. Also we define c∗∗∗ = inf v∈H1(R3){0} max t0 I(tv). Using the mountain pass value c and c∗∗∗ as the connections, we can prove that the minimizer u is a ground state. Lemma 2.12. c = c∗ = c∗∗ = c∗∗∗. Proof. The original research should be attributed to Rabinowitz (see [17, Proposi- tion 3.11]). For the convenience of readers, we sketch the proof. For any u ∈ H1(R3) \ {0}, by Lemma 2.2, the ray Rt = {tu : t ≥ 0} intersects the Nehari manifold N once and only once at t(u)u, where t(u) is given in Lemma 2.2. This implies that c∗ = c∗∗∗. Next, we show c∗ = c∗∗. Obviously, c∗ ≤ c∗∗. On the other hand, since u is a nontrivial solution of (2.1), it holds that I(u) ≤ maxt≥0I(tu). So c∗∗ ≤ c∗∗∗. Now we check c = c∗. Since for all γ ∈ Γ, γ(0) = 0, I(γ(1)) < 0, it follows that γ crosses N . We obtain c ≥ c∗. On the other hand, for fixed u ∈ H1(R3) \ {0}, choosing suitable α = α(u) large enough, let gu(t) = tαu. We have gu ∈ Γ. So c∗∗∗ = inf u∈H1(R3)\{0} max t∈[0,1] I(gu(t)) ≥ c, which completes the proof. � Lemma 2.13. The ground state u can be nonnegative. Furthermore, u > 0. Proof. Since ∫ R3 |∇u|2dx = ∫ R3 |∇|u||2dx, we may assume that u obtained in Lemma 2.10 is nonnegative. Furthermore, a Moser iteration argument implies that u ∈ Ls(R3) for 2 ≤ s ≤ ∞ ([12, Proposition 3.3]). It follows from standard arguments in [6] that u ∈ C1,β loc (R3) for some β > 0. EJDE-2020/78 SCHRÖDINGER-POISSON EQUATIONS 13 Consequently, by Schauder estimate (see [11]), u ∈ C2,β loc (R3). Furthermore, u satisfies the equation −∆u+ c(x)u = P (x)up +Q(x)+uq ≥ 0, where c(x) := V (x) + K(x)φu(x) + Q−(x)uq(x) > 0 and Q± := max{±Q, 0}. Applying the strong maximum principle (see [11]), we have u > 0. � 3. Concentration of positive ground state In this section, we are devoted to the concentration behaviour of the ground state solutions uε of (1.5) as ε → 0+. From now on, we assume K(x) = 1. Introducing the re-scaled transformation x 7→ εx (i.e., vε(x) := u(εx)), we can rewrite (1.5) as −∆vε + Vε(x)vε + φvε = Pε(x)|vε|p−1vε +Qε(x)|vε|q−1vε, x ∈ R3, −∆φ = v2 ε , x ∈ R3, (3.1) where Vε(x) = V (εx), and Pε, Qε defined in a similar way. According to section 2, vε is positive ground state of (3.1). Let Iε be the energy functional associated with (3.1) and Nε be the corresponding Nehari manifold and set least energy cε = infv∈Nε Iε(v). We need the following constant coefficients problem −∆u+ µu+ φuu = ξ|u|p−1u+ τ |u|q−1u, (3.2) where µ > 0, ξ > 0, τ can positive or negative. In the same way, Iµλξτ , Nµξτ and cµξτ correspond to the energy functional, Nehari manifold, least energy associated with (3.2), respectively. Similar to [17, Lemma 3.17] (see also [29, Lemma 2.2] or [26, Lemma 4.1]), we have the following result. Lemma 3.1. Suppose µ1 ≥ µ2, ξ2 ≥ ξ1 and τ2 ≥ τ1. Then cµ1ξ1τ1 ≥ cµ2ξ2τ2 . Furthermore, if one of inequalities is strict, then cµ1ξ1τ1 > cµ2ξ2τ2 . Since assume the potential functions V , P and Q are C1, according to [29, (i) of Lemma 2.3, and Lemma 2.4], we have the following lemma. Lemma 3.2. The ground energy function C(s) is locally Lipschitz continuous in s ∈ R3. If V , K, P and Q are constant functions, then the least energy depends continuously on them. Lemma 3.3. There exists C > 0 independent with ε such that cε ≥ C. Further- more, lim sup ε→0+ cε ≤ inf s∈R3 C(s). (3.3) Proof. By Lemma 3.1, cε ≥ c(inf V, ‖P‖L∞ , ‖Q‖L∞) > 0, we only need to prove (3.3). The original idea is from the proof of [25, Lemma 2.2]. In view of our assumptions on V , P and Q, it holds that infs∈R3 C(s) can be achieved by some s0. Let u0 be a ground state of −∆u+ V (s0)u+ φuu = P (s0)|u|p−1u+Q(s0)|u|q−1u. (3.4) Denote the energy functional Is0 . Take a sequence {yk} such that C(yk)→ C(s0) = infs∈R3 C(s). For any R > 0, take a cut-off function ϕR with ϕR = 1 in BR(0) and ϕR = 0 in BcR+1. Set vR = ϕRu0 and w(x) = vR(x − yk ε ). Then there exists a unique θ > 0 such that θw ∈ Nε and θ → 1 as R→∞, k →∞ and ε→ 0+. Since cε = inf v∈Nε Iε(v) ≤ Iε(θw) 14 W. WANG, Q. LI EJDE-2020/78 =θ2 { Is0(w) + 1 2 ∫ R3 [V (εx)− V (s0)]w2dx + 1− θp−1 p+ 1 ∫ R3 Pε(x)|w|p+1dx+ 1− θq−1 q + 1 ∫ R3 Qε(x)|w|q+1dx + 1 p+ 1 ∫ R3 [P (s0)− Pε(x)]|w|p+1dx+ 1 q + 1 ∫ R3 [Q(s0)−Qε(x)]|w|q+1dx } =θ2 { Is0(vR) + 1 2 ∫ R3 [V (εx+ yk)− V (s0)]v2 Rdx + 1− θp−1 p+ 1 ∫ R3 P (εx+ yk)|vR|p+1dx+ 1− θq−1 q + 1 ∫ R3 Q(εx+ yk)|vR|q+1dx + 1 p+ 1 ∫ R3 [P (s0)− P (εx+ yk)]|vR|p+1dx + 1 q + 1 ∫ R3 [Q(s0)−Q(εx+ yk)]|vR|q+1dx } . Letting R→∞ and k →∞ in the above inequality, we have lim sup ε→0+ cε ≤ inf s∈R3 C(s). � Lemma 3.4. {vε} is bounded in H1(R3). Proof. Since vε is minimizer of Iε on Nε, cε =Iε(vε)− 1 q + 1 〈I ′ε(vε), vε〉 = (1 2 − 1 q + 1 )∫ R3 (|∇vε|2 + Vε(x)v2 ε)dx + (1 4 − 1 q + 1 )∫ R3 φvεv 2 εdx + ( 1 q + 1 − 1 p+ 1 )∫ R3 Pε(x)|vε|p+1dx. In view of Lemma 3.3, we obtain the desired conclusion. � Lemma 3.5. There exists ε∗ > 0 such that, for all ε ∈ (0, ε∗), there exists yε ∈ R3 and R, C > 0 such that ∫ BR(yε) v2 εdx ≥ C. Proof. We assume, for the sake of contradiction, that there is a sequence εk → 0 as k →∞, such that for all R > 0 lim k→∞ sup y∈R3 ∫ BR(y) v2 εk dx = 0. By the vanishing lemma, we have vεk → 0 in Ls(R3) for s ∈ (2, 6). Since cεk =Iεk(vεk) =Iεk(vεk)− 1 2 〈I ′εk(vεk), vεk〉 =− 1 2 ∫ R3 φvεk v 2 εk dx+ (1 2 − 1 p+ 1 ) ∫ R3 Pεk(x)|vεk |p+1dx EJDE-2020/78 SCHRÖDINGER-POISSON EQUATIONS 15 + (1 2 − 1 q + 1 ) ∫ R3 Qεk(x)|vεk |q+1dx, combining this with the result (1) of Lemma 3.3, it follows that 0 < lim inf k→0 cεk = −1 2 lim inf k→0 ∫ R3 φvεk v 2 εk dx ≤ 0. This is a contradiction. � For simplicity, we denote wε(x) := vε(x+ yε) = uε(εx+ εyε), (3.5) so wε is a positive ground state solution to −∆wε + V (εx+ εyε)wε + φwεwε = P (εx+ εyε)w p ε +Q(εx+ εyε)w q ε . By Lemmas 3.4 and 3.5, we obtain wε ⇀ w0 in H1(R3), with w0 ≥ 0, w0 6≡ 0. Furthermore, we will give next lemma which is used in Lemma 3.7. Lemma 3.6. wε → w0 in Lp+1(R3). Proof. We define µε(R3) = (1 2 − 1 q + 1 )∫ R3 (|∇vε|2 + V (εx)v2 ε)dx + (1 4 − 2 q + 1 )∫ R3 φvεv 2 εdx + ( 1 q + 1 − 1 p+ 1 )∫ R3 P (εx)|vε|p+1dx. By following the same methods in [29, Lemma 3.3], we can exclude vanishing and dichotomy. By compactness conditions, for any η > 0, there exists ρ > 0 such that∫ Bcρ(0) (|∇wε|2 + w2 ε)dx < η. By this and Sobolev embedding theorem, we complete the proof. � Lemma 3.7. {εyε} is bounded. Proof. Suppose to the contrary that, if necessary going to a subsequence, εyε →∞. Denote c∞ = c(V∞, P∞, Q∞). Since our assumptions potentials on V , P and Q, we have c∞ > infs∈R3 C(s). Using Lemma 3.2, we can choose ε > 0 small such that Cε := C(V∞ − ε, P∞ + ε,Q∞ + ε) > inf s∈R3 C(s). (3.6) Let N ε be the corresponding Nehari manifold for the Cε. Thus there is a θ > 0 such that θw0 ∈ N ε. Based on [24, Lemma A.1], in view of Lemma 3.6, we can use Lebesgue dominated convergence theorem and Fatou’s Lemma to obtain Cε ≤θ 2 2 ∫ R3 |∇w0|2 + (V∞ − ε)w2 0dx+ θ4 4 ∫ R3 φw0 w2 0dx − θp+1 p+ 1 ∫ R3 (P∞ + ε)|w0|p+1dx− θq+1 q + 1 ∫ R3 (Q∞ + ε)|w0|q+1dx 16 W. WANG, Q. LI EJDE-2020/78 ≤ lim inf ε→0+ {θ2 2 ∫ R3 [|∇wε|2 + V (εx+ εyε)w 2 ε ]dx + θ4 4 ∫ R3 φwεw 2 εdx− θp+1 p+ 1 ∫ R3 P (εx+ εyε)|wε|p+1dx − θq+1 q + 1 ∫ R3 Q(εx+ εyε)|wε|q+1dx } = lim inf ε→0+ (θ2 2 (Iε,1 + Iε,2) + θ4 4 Iε,3 − θp+1 p+ 1 Iε,4 − θq+1 q + 1 Iε,5 ) := lim inf ε→0+ gε(θ). Clearly, gε(θ) < gε(1) for θ ∈ (0.1). Therefore, combining this with Lemma 3.4, it holds Cε ≤ lim inf εk→0+ cεk ≤ inf s∈R3 C(s). It contradicts (3.6). � Without loss of generality, we assume that εyε → x0 as ε → 0+. By a Moser iteration argument, we see that wε → w0 in Ls(R3) for 2 ≤ s ≤ ∞, (see also [12, Proposition 3.3]. It follows that wε → w0 in C1,β loc (R3) for some β > 0. Consequently, by Schauder estimate (see [11]), wε → w0 in C2,β loc (R3), and hence −∆w0 + V (x0)w0 + φw0w0 = P (x0)wp0 +Q(x0)wq0, x ∈ R3. Denote by E the corresponding energy functional. Lemma 3.8. C(x0) = infs∈R3 C(s). Furthermore, wε → w0 in H1(R3). Proof. It is similar to that in [29, Lemma 3.5]. For readers convenience, we sketch the proof. Using Fatou’s lemma and (3.3), we obtain inf s∈R3 C(s) ≤C(x0) ≤ E(w0)− 1 q + 1 〈E′(w0), w0〉 ≤ lim inf ε→0+ {(1 2 − 1 q + 1 )∫ R3 [|∇wε|2 + V (εx+ εyε)w 2 ε ]dx + (1 4 − 2 q + 1 )∫ R3 φwεw 2 εdx + ( 1 q + 1 − 1 p+ 1 )∫ R3 P (εx+ εyε)w p+1 ε dx } = lim inf ε→0+ cε ≤ inf s∈R3 C(s). It follows from the above inequalities and Lemma 3.6 that∫ R3 (|∇wε|2 + V (εx+ εyε)w 2 ε)dx→ ∫ R3 (|∇w0|2 + V (x0)w2 0)dx as ε→ 0+. Which yields wε → w0 in H1(R3). � As in [12, Lemma 3.8 and 3.9], the following lemmas hold. Lemma 3.9. For all ε ∈ (0, ε∗), wε possesses at most one maximum point pε. Moreover lim|x|→∞ wε(x) = 0 uniformly on ε ∈ (0, ε∗). EJDE-2020/78 SCHRÖDINGER-POISSON EQUATIONS 17 Lemma 3.10. There exist constants C1 > 0 and C2 > 0 such that uε(x) ≤ C1e −C2| x−xεε | for all x ∈ R3. Proof. The proof is similar that in [12, Lemma 3.11], but in our case potential Q is allowed to change sign or negative, we check it step by step. Denote fε(x,wε) := P (εx+ εyε)w p ε +Q(εx+ εyε)w q ε . Obviously, lim wε→0 = fε(x,wε) wε = 0. (3.7) Thus by Lemma 3.9, there is R1 > 0, independent of ε ∈ (0, ε∗), such that fε(x,wε) ≤ 1 4 V0wε. Fix ψ(x) = C1e −C2|x| with C2 2 = V0 2 and C1e −C2R1 ≥ wε(x) for all |x| = R1. It is obtained that ∆ψ ≤ C2 2ψ ≤ V0 2 ψ. (3.8) Hence −∆wε + 3 4 V0wε ≤ 1 4 V0wε. (3.9) Define ψε = ψ − wε, using (3.8)) and (3.9), we obtain −∆ψε + V0 2 ψε ≥ 0, if |x| ≥ R1, ψε ≥ 0, if |x| = R1, lim |x|→∞ ψε(x) = 0. The maximum principle implies that ψε ≥ 0 in |x| ≥ R1 and we conclude that wε(x) ≤ C1e −C2|x|, ∀|x| ≥ R1, and all ε ∈ (0, ε∗). By Lemma 3.9, we have wε has a unique maximum point pε. Then vε has a unique maximum point pε+yε and uε has a unique maximum point xε = ε(pε+yε). Thus, we have uε(x) = vε( x ε ) = wε(ε −1x− yε) = wε(ε −1x− ε−1xε + pε) ≤ C1e −C2| x−xεε |. � Proof of Theorem 1.1. The existence result (I) of Theorem 1.1 follows directly from Section 2. The concentration results (1) and (2) of Theorem 1.1 follow from Lemma 3.8, the result (3) of Theorem 1.1 follows from Lemma 3.10. � Remark 3.11. In our paper, we assume that our potentials V , P and Q are C1 functions to ensure that C(s) is continuous. It follows that infs∈R3 C(s) can be achieved. Under our assumptions, the result shows that C(s) is Lipschitz continuous (see Lemma 3.2), so we gesture V , P , Q can be weaker than C1 functions. 18 W. WANG, Q. LI EJDE-2020/78 4. Appendix As mentioned in Remark 1.2, introducing the re-scaled transformation x 7→ εx, the system −ε2∆u+ V (x)u+K(x)φu = P (x)|u|p−1u+Q(x)|u|q−1u, x ∈ R3, −ε2∆φ = K(x)u2, x ∈ R3, can be written as −∆u+ V (εx)u+K(εx)φu = P (εx)|u|p−1u+Q(εx)|u|q−1u, x ∈ R3, −∆φ = K(εx)u2, x ∈ R3. This system is different from (3.1) since in the second equation, φ is dependent on ε. In fact, we can write the corresponding energy functional as Iε(u) = 1 2 ∫ R3 (|∇u|2 + V (εx)u2)dx+ 1 4 ∫ R3 ∫ R3 K(εx) K(εy)u2(y) |x− y| u2(x) dy dx − 1 p+ 1 ∫ R3 P (εx)|u|p+1dx− 1 q + 1 ∫ R3 Q(εx)|u|q+1dx, u ∈ H1(R3). For concentration, it is complicated since it appears terms K(εx) and K(εy). Acknowledgments. This work is supported by the National Natural Science Foundation of China (11901514, 11801153), by the Honghe University Doctoral Research Programs (XJ17B11), by the Yunnan Province Applied Basic Research for Youths (2018FD085), by the Yunnan Province Local University (Part) Basic Research Joint Project (2017FH001-013), and by the Yunnan Province Applied Basic Research for General Project (2019FB001). The authors would like to thank the anonymous referee for constructive sugges- tions and comments, which improved the quality of this paper. References [1] A. Azzollini, P. d’Avenia, A. 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Du; Ground states for asymptotically periodic Schrödinger- Poisson systems with critical growth, Cent. Eur. J. Math., 12(10) (2014), 1484-1499. [32] L. Zhao, F. Zhao; Positive solutions for Schrödinger-Poisson equations with critical exponent, Nonlinear Anal., 70(6) (2009), 2150-2164. [33] L. Zhao, H. Liu, F. Zhao; Existence and concentration of solutions for the Schrödinger- Poisson equations with steep well potential, J. Differential Equations, 255(1) (2013), 1-23. Wenbo Wang School of Mathematics and Statistics, Yunnan University, Kunming,650500, Yunnan, China Email address: wenbowangmath@163.com Quanqing Li (corresponding author) Department of Mathematics, Honghe University, Mengzi, 661100, Yunnan, China Email address: shili06171987@126.com 1. Introduction 2. Existence of ground states 3. Concentration of positive ground state 4. Appendix Acknowledgments References