Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 64, pp. 1–13. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.64 LOCAL WELL-POSEDNESS AND STANDING WAVES WITH PRESCRIBED MASS FOR SCHRÖDINGER-POISSON SYSTEMS WITH A LOGARITHMIC POTENTIAL IN R2 XUECHAO DOU, JUNTAO SUN Abstract. In this article, we consider planar Schrödinger-Poisson systems with a logarithmic external potential W (x) = ln(1 + |x|2) and a general non- linear term f . We obtain conditions for the local well-posedness of the Cauchy problem in the energy space. By introducing some suitable assumptions on f , we prove the existence of the global minimizer. In addition, with the help of the local well-posedness, we show that the set of ground state standing waves is orbitally stable. 1. Introduction We consider the planar Schrödinger-Poisson system iψt −∆ψ +W (x)ψ + γωψ = f(ψ), ∀(t, x) ∈ R1+2, ∆ω = |ψ|2, ψ(0, x) = ψ0(x), (1.1) where ψ : R2 × R → C is the (time-dependent) wave function, x 7→ W (x) is a real external potential and γ ∈ R. The function ω represents an internal potential for a nonlocal self-interaction of the wave function ψ, and the nonlinear term f is used to model the interaction among particles. Such a system arises from quantum mechanics [3, 5, 16] and in semiconductor theory [18, 19]. We refer the reader to [2, 13] for more details on its physical aspects. An important topic is to establish conditions for the well-posedness of Cauchy problem (1.1). From a mathematical point of view, the second equation in the system determines ω : R2 → R up to harmonic functions, it is natural to choose ω as the Newton potential of |ψ|2, i.e. the convolution of |ψ|2 with the fundamental solution Φ(x) = 1 2π ln |x| of the Laplacian. Thus the Newtonian potential ω is given by ω = 1 2π (ln |x| ∗ |ψ|2). We note that the Newtonian potential ω diverges at the spatial infinity no matter how fast ψ decays. In view of this, Masaki [20, 21] proposed a new approach 2020 Mathematics Subject Classification. 35J20, 35J60. Key words and phrases. Nonlinear Schrödinger-Poisson system; normalized solution; logarithmic external potential; local well-posedness. ©2023. This work is licensed under a CC BY 4.0 license. Submitted May 6, 2023. Published September 25, 2023. 1 2 X. DOU, J. SUN EJDE-2023/64 to deal with such a nonlocal term, which can be decomposed into a sum of the linear logarithmic potential and a good remainder. By using the perturbation method, the global well-posedness for the Cauchy problem (1.1) with W (x) ≡ 0 and f(ψ) = |ψ|p−2ψ(p > 2) is established in the space B given by B := { ψ ∈ H1(R2) : ∫ R2 ln (√ 1 + |x|2 ) |ψ(x)|2dx <∞ } . Another interesting topic on (1.1) is to study the standing wave solution of the form ψ(x, t) = eiλtu(x), where λ ∈ R and u : R2 → R. Then (1.1) is reduced to the system −∆u+ (W (x)− λ)u+ γωu = f(u) in R2, −∆ω = u2 in R2, (1.2) which can be further written as the integro-differential equation −∆u+ (W (x)− λ)u+ γ(Φ ∗ |u2|)u = f(u), ∀x ∈ R2. (1.3) At least formally, the energy functional associated with (1.3) is E(u) = 1 2 ∫ R2 ( |∇u|2 + (W (x)− λ)u2 ) dx + γ 8π ∫ R2 ∫ R2 ln(|x− y|2)|u(x)|2|u(y)|2 dx dy − ∫ R2 F (u) dx, where F (t) = ∫ t 0 f(s)ds. Obviously, if u is a critical point of E, then the pair (u,Φ ∗ |u|2) is a weak solution of (1.2). However, the energy functional E is not well-defined on the natural Sobolev space H1(R2), since the logarithm term changes sign and is neither bounded from above nor from below. Inspired by [24], Cingolani and Weth [11] developed a variational framework of ((1.3) with W (x) ≡ 0 in the smaller Hilbert space X := { u ∈ H1(R2) : ∫ R2 ln(1 + |x|)u2dx <∞ } , endowed with the norm ‖u‖2X := ∫ R2 (|∇u|2 + u2(1 + ln(1 + |x|2))) dx. If the frequency λ is a fixed and assigned parameter, then solutions of (1.3) can be obtained as critical points of the functional E in X. Under various types of potentials W and nonlinearities f , there has been much study on this case in recent years, see, for example [1, 8, 9, 14]. For other nonlocal problems, we refer the reader to [22, 25, 26, 27, 29]. If we would like to find solutions of (1.3) with the frequency λ unknown, then λ appears as a Lagrange multiplier, and L2-norms of solutions are prescribed, i.e.∫ R2 |u|2dx = c for a given c > 0, which are usually called normalized solutions. This study seems particularly mean- ingful from the physical point of view, since solutions of (1.1) conserve their mass along time. When W (x) ≡ 0, Cingolani and Jeanjean [10] proved the existence and EJDE-2023/64 SCHRÖDINGER-POISSON SYSTEM WITH A LOGARITHMIC POTENTIAL 3 multiplicity of normalized solutions for 1.3 with f(u) = |u|p−2u(p > 2). When the logarithmic external potential W (x) = ln(1 + |x|2) (1.4) is considered in (1.3), Dolbeault, Frank and Jeanjean [12] studied the existence of normalized solutions for (1.3) with f(u) = ln |u|2u, and recently Guo, Liang and Li [15] proved the existence and uniqueness of L2-critical constraint minimization problem, i.e. f(u) = |u|p−2u with p = 4. Inspired by the analysis mentioned above, in this paper we are concerned with a class of planar Schrödinger-Poisson systems with a logarithmic external potential (1.4) and a general nonlinearity f . First of all, we shall establish conditions of the local well-posedness for the Cauchy problem (1.1). Secondly, we shall focused on the existence of global minimizer when f satisfies some suitable assumptions. In addition, with the help of the local well-posedness of the Cauchy problem (1.1), the orbital stability of the set of ground states is explored as well. To find normalized solutions of (1.3), we consider the associated energy func- tional J(u) := 1 2 ∫ R2 ( |∇u|2 + ln(1 + |x|2)u2 ) dx + γ 8π ∫ R2 ∫ R2 ln(|x− y|)u2(x)u2(y) dx dy − ∫ R2 F (u) dx. (1.5) under the constraint S(c) := { u ∈ H : ∫ R2 u2dx = c } , where H := { u ∈ H1(R2) : ∫ R2 ln(1 + |x|2)u2dx <∞ } , endowed with the norm ‖u‖H := ‖u‖H1 + ‖u‖∗, here ‖u‖2∗ = ∫ R2 ln(1 + |x|2)u2dx. We now summarize our main results. Theorem 1.1. Assume that f satisfies (A1) f ∈ C(R,R) and f(0) = 0, (A2) f(eiθz) = eiθf(z), (A3) there exist z1, z2 and a constant L > 0 such that |f(z1)− f(z2)| ≤ L|z1 − z2|(1 + |z1|+ |z2|)2. Then the Cauchy problem (1.1) is local well-posed in H. That is, for any ψ0 ∈ H, there exists an existence time T = T (‖ψ0‖H) and a unique solution ψ ∈ C((−T, T );H) ∩ Lq0((−T, T );Lr0) ∩ C1((−T, T );H′) of (1.1), where (q0, r0) be an admissible pair with r0 > 2. Theorem 1.2. Assume that condition (A1) holds. In addition, we assume that f satisfies (A4) limt→0 f(t) t = 0; (A5) lim supt→∞ f(t)t |t|4 = 0. 4 X. DOU, J. SUN EJDE-2023/64 Then there exists a constant c∗ > 0 such that for 0 < c < c∗, the infimum Jc := inf u∈S(c) J(u) is achieved by some uc ∈ S(c), i.e. J(uc) = Jc. It is easy to find some examples on the nonlinearity f satisfying conditions (A1), (A4), and (A5), such as f(t) = |t|p−2t+ |t|q−2t with 2 < q < p < 4. By Theorem 1.2, we know that the set of ground states Mc := {eiλtu(x) : u ∈ S(c) and J(u) = Jc} is not empty. Then we have the following stability result. Theorem 1.3. Under the assumptions of Theorems 1.1 and 1.2, the set of ground states Mc is orbitally stable. That is, for any ε > 0, there exists δ > 0 such that for any ψ0 = ψ(0, x) ∈ H satisfying infu∈Mc ‖ψ0 − u‖H < δ, the solution ψ(t, x) of system (1.1) satisfies sup t∈[0,T ) inf u∈Mc ‖ψ(t, x)− u‖H < ε where T is the maximal existence time for ψ(t, x). 2. Preliminary results For sake of convenience, we set A(u) := ∫ R2 |∇u|2dx and V (u) := ∫ R2 ∫ R2 ln(|x− y|)u2(x)u2(y) dx dy. Then the energy functional J defined in (1.5) can be rewritten as J(u) := 1 2 A(u) + 1 2 ∫ R2 ln(1 + |x|2)u2(x) dx+ γ 8π V (u)− ∫ R2 F (u) dx. Definition 2.1. We say that a pair (q, r) is Strichartz admissible if 2 ≤ r < ∞ and 2 q = 1− 2 r . Lemma 2.2 (Strichartz estimates [6]). For any T > 0, the following properties hold: (i) let ϕ ∈ L2(R2). For any admissible pair (q, r), we have ‖eit∆ϕ‖Lq((−T,T );Lr) . ‖ϕ‖L2 ; (ii) let I ⊂ (−T, T ) be an interval and t0 ∈ I. For any admissible pairs (q, r) and (γ, ρ), we have ‖ ∫ t t0 ei(t−s)∆F (s)ds‖Lq(I;Lr) . ‖F‖Lγ′ (I;Lρ′ ) for every F ∈ Lγ′(I;Lρ ′ ). Lemma 2.3 ([21, Lemma 2.2]). Let P be an arbitrary weight function satisfying ∇P,∆P ∈ L∞(R2). Then for all T > 0 and admissible pair (q, r), we have ‖[∇, eitA]ϕ‖Lq((−T,T );Lr) . |T |‖ϕ‖L2 , ‖[P, eitA]ϕ‖Lq((−T,T );Lr) . |T |‖(1 +∇)ϕ‖L2 , EJDE-2023/64 SCHRÖDINGER-POISSON SYSTEM WITH A LOGARITHMIC POTENTIAL 5 where A := −∆ +m ln(1 + |x|2) (2.1) with m := γ 2π‖ψ0‖2L2 + 1. Lemma 2.4 ([21, Lemma 2.3]). Let K(x, y) = ln ( |x−y| 〈x〉 ) 1 + ln〈y〉 for x, y ∈ R2. For any p ∈ [1,∞) and ε > 0, there exist a function H(x, y) ≥ 0 with ‖H‖L∞y Lpx ≤ ε and a constant C0 > 0 such that |K(x, y)| ≤ C0 +H(x, y) for all (x, y) ∈ R2+2. Lemma 2.5 (Gagliardo-Nirenberg Inequality [17]). (i). Let r > 2. Then there exists a sharp constant KGN > 0 such that ‖u‖r ≤ K1/r GN‖∇u‖ r−2 r 2 ‖u‖2/r2 . (ii) (Hardy-Littlewood-Sobolev inequality [30]). Let t, r > 1 and 0 < α < N with 1 t + N−α N + 1 r = 2. For f ∈ Lt(RN ) and h ∈ Lr(RN ), there exists a sharp constant C(t,N, α, r), independent of u and v, such that∫ R2 ∫ R2 f(x)h(x) |x− y|N−α dx dy ≤ C(t,N, α, r)‖f‖t‖h‖r. As in [15], we introduce the symmetric bilinear forms B1(u, v) = ∫ R2 ∫ R2 ln(1 + |x− y|2)u(x)v(y) dx dy, B2(u, v) = ∫ R2 ∫ R2 ln ( 1 + 1 |x− y|2 ) u(x)v(y) dx dy, B0(u, v) = 1 2 [B1(u, v)−B2(u, v)] = ∫ R2 ∫ R2 ln(|x− y|)u(x)v(y) dx dy. Clearly, V (u) = B0(u2, u2) = ∫ R2 ∫ R2 ln(|x−y|)u2(x)u2(y) dx dy. By the continuous embedding from H into Ls(R2) for s ∈ [2,∞), the functionals Bi(u 2, v2) are well- defined on H×H for i = 0, 1, 2. Moreover, we define the associated functionals on H as follows V1(u) = B1(u2, u2) = ∫ R2 ∫ R2 ln(1 + |x− y|2)u2(x)u2(y) dx dy, V2(u) = B2(u2, u2) = ∫ R2 ∫ R2 ln ( 1 + 1 |x− y|2 ) u2(x)u2(y) dx dy. Lemma 2.6 ([15, Lemma 2.1]). The following statements are valid: (i) the space H is compactly embedded in Ls(R2) for all s ∈ [2,∞); (ii) the functionals V, V1, V2 and J are of class C1 on H. Moreover, V ′i (u)v = 4Bi(u 2, uv) for u, v ∈ H and i = 1, 2; (iii) V2 is continuous (in fact continuously differentiable) on L8/3(R2); (iv) V1 is weakly lower semi-continuous on H1(R2); (v) V is weakly lower semi-continuous on H. 6 X. DOU, J. SUN EJDE-2023/64 3. Local well-posedness of the Cauchy problem Following the ideas in [21], the following decomposition holds γωψ = γ 2π ‖ψ‖2L2(ln〈x〉)ψ + γ 2π ψ ∫ R2 ln ( |x− y| 〈x〉 ) |ψ(y)|2dy, where 〈x〉 := (1 + |x|2). According to the conservation of mass ‖ψ‖L2 = ‖ψ0‖L2 , we have m = γ 2π ‖ψ0‖2L2 + 1 > 0. Then, (1.1) is rewritten as i∂tψ + (−∆ +m ln〈x〉)ψ = − γ 2π ψ ∫ R2 ln ( |x− y| 〈x〉 ) |ψ(y)|2dy + f(ψ), ∀(t, x) ∈ R1+2, ψ(0, x) = ψ0(x). We note that A defined as (2.1) is essentially self-adjoint on C∞0 (R2) (see [23]). Since |∂α(ln〈x〉)| → 0 as |x| → ∞ for |α| = 2 and ∂α(ln〈x〉) ∈ L∞(R2) for |α| ≥ 3, the potential is subquadratic. Then for any t ∈ [−T, T ], we have ‖eitAϕ‖L∞ . |t|−1‖ϕ‖L1 , (see [28]). Once we know this type of estimate, the Strichartz estimates follow by interpolation. We are ready to prove Theorem 1.1. We write Lp((−T, T );H) = LpTH for short. We define the Banach space HT,M := {ψ ∈ L∞T H ‖ψ‖HT ≤M} with the norm ‖ψ‖HT := ‖ψ‖L∞T H + ‖ψ‖Lq0T W 1,r0 + ‖ √ ln〈x〉ψ‖Lq0T Lr0 . Now we show that if r0 > 2, then there exist M = M(‖ψ0‖H) and T = T (‖ψ0‖H) such that Q[ψ] := eitAψ0 + i ∫ t 0 ei(t−s)A (γψ 2π ∫ R2 ln ( |x− y| 〈x〉 ) |ψ(y)|2dy − f(ψ) ) ds becomes a contraction map from HT,M to itself. Set K(x, y) := ln ( |x−y| 〈x〉 ) 1 + ln〈y〉 . By Lemma 2.4, there exist a nonnegative function H ∈ L∞y L r′0 y and a constant C0 > 0 such that |K(x, y)| ≤ C0 +H(x, y). Recall that r0 ∈ (2,∞) and so r′0 := r0/(r0 − 1) ∈ (1, 2). We hence see that ωψ = ∫ K(x, y)(1 + ln〈y〉)|ψ(y)|2ψ(x)dy satisfying ‖ωψ‖L2 . (‖ψ‖L2 + ‖ψ‖Lr0 )‖ √ 1 + ln〈x〉ψ‖2L2 . Taking the L1 T -norm one has ‖ωψ‖L1 TL 2 . ( T‖ψ‖L∞T L2 + T r0+2 2r0 ‖ψ‖Lq0T Lr0 ) ‖ √ 1 + ln〈x〉ψ‖2L∞T L2 . (3.1) EJDE-2023/64 SCHRÖDINGER-POISSON SYSTEM WITH A LOGARITHMIC POTENTIAL 7 Note that |f(ψ)| ≤ L(1 + |ψ|)2|ψ|. Then we have ‖f‖L2 . ‖ψ‖L2 + ‖ψ‖L2r0/(r0−2)‖ψ‖Lr0 + ‖ψ2‖L2r0/(r0−2)‖ψ‖Lr0 . ‖ψ‖L2 + (‖ψ‖L2 + ‖∇ψ‖L2 + ‖ψ‖2L2 + ‖∇ψ‖2L2)‖ψ‖Lr0 . Similarly, taking the L1 T -norm yields ‖f‖L1 TL 2 . T r0+2 2r0 ( ‖ψ‖L∞T L2 + ‖∇ψ‖L∞T L2 + ‖ψ‖2L∞T L2 + ‖∇ψ‖2L∞T L2 ) ‖ψ‖Lq0T Lr0 + T‖ψ‖L∞T L2 . (3.2) By Strichartz estimates, we have ‖Q[ψ]‖L∞T L2 + ‖Q[ψ]‖Lq0T Lr0 . ‖ψ0‖L2 + T‖ψ‖HT + T r0+2 2r0 ‖ψ‖2HT + ( T + T r0+2 2r0 ) ‖ψ‖3HT . (3.3) Next, we estimate ∇Q[ψ]. It is easy to see that ∇Q[ψ] = eitA∇ψ0 + [∇, eitA]ψ0 + iγ 2π ∫ t 0 ei(t−s)A(∇(ωψ)−∇f)(s)ds + iγ 2π ∫ t 0 [∇, ei(t−s)A](ωψ − f)(s)ds. From Lemma 2.3 with (q, r) = (∞, 2), we deduce that∫ t 0 ‖[∇, ei(t−s)A](ωψ)(s)‖L2ds ≤ ∫ t 0 (t− s)‖ωψ(s)‖L2ds ≤ |t|‖ωψ‖L1 TL 2 . Similarly, we have∫ t 0 ‖[∇, ei(t−s)A]f(s)‖L2ds ≤ ∫ t 0 (t− s)‖f(s)‖L2ds ≤ |t|‖f‖L1 TL 2 , ‖[∇, eitA]ψ0‖L∞T L2 . |T |‖ψ0‖L2 . Similar to (3.1), we infer that ‖ω∇ψ‖L1 TL 2 . ( T‖∇ψ‖L∞T L2 + T r0+2 2r0 ‖∇ψ‖Lq0T Lr0 ) ‖ √ 1 + ln〈x〉ψ‖2L∞T L2 . (3.4) Now, let us estimate (∇ω)ψ. It can be written as (∇ω(x))ψ(x) = [ ∫ R2 ( x− y |x− y|2 − 2x 1 + x2 ) |ψ(y)|2dy ] ψ(x). It follows from the Hardy-Littlewood-Sobolev and the Sobolev inequalities that ‖(∇ω)ψ‖L2 . ‖(|x|−1 ∗ |ψ|2) + 〈·〉−1‖ψ‖2L2‖L2r0/(r0−2)‖ψ‖Lr0 . ( ‖ψ‖2L2r0/(r0−1) + ‖ψ‖2L2 ) ‖ψ‖Lr0 . ( ‖ψ‖2L2 + ‖∇ψ‖2L2 ) ‖ψ‖Lr0 , which implies that ‖(∇ω)ψ‖L1 TL 2 . T r0+2 2r0 ( ‖ψ‖2L∞T L2 + ‖∇ψ‖2L∞T L2 ) ‖ψ‖Lq0T Lr0 . (3.5) By condition (A3) one has ‖∇f‖L1 TL 2 . T‖∇ψ‖L∞T L2 + T r0+2 2r0 ( ‖ψ‖L∞T L2 + ‖∇ψ‖L∞T L2 ) ‖∇ψ‖Lq0T Lr0 + T r0+2 2r0 ( ‖ψ‖2L∞T L2 + ‖∇ψ‖2L∞T L2 ) ‖∇ψ‖Lq0T Lr0 . 8 X. DOU, J. SUN EJDE-2023/64 We deduce from the Strichartz estimates that ‖∇Q[ψ]‖L∞T L2 + ‖∇Q[ψ]‖Lq0T Lr0 . T‖∇ψ0‖H + T‖ψ‖HT + T r0+2 2r0 ‖ψ‖2HT + ( T + T r0+2 2r0 ) ‖ψ‖3HT . (3.6) Let us proceed to the estimate√ 1 + ln〈x〉Q[ψ] = eitA √ 1 + ln〈x〉ψ0+ iγ 2π ∫ t 0 ei(t−s)A √ 1 + ln〈x〉(ωψ−f)(s)ds+R, where R := [ √ 1 + ln〈x〉, eitA]ψ0 + iγ 2π ∫ t 0 [ √ 1 + ln〈x〉, ei(t−s)A](ωψ − f)(s)ds. (3.7) Let G = √ 1 + ln〈x〉. It follows from Lemma 2.3 and (3.1)–(3.5) that ‖R‖L∞T L2 + ‖R‖Lq0T Lr0 . T‖ψ0‖H + T‖(1 +∇)(ωψ)‖L1 TL 2 + T‖(1 +∇)f‖L1 TL 2 . T‖ψ0‖H + T ( T + T r0+2 2r0 ) ‖ψ‖3HT + T ( T‖ψ‖HT + T r0+2 2r0 ‖ψ‖2HT + T r0+2 2r0 ‖ψ‖3HT ) . As in (3.1), we have ‖ω(Gψ)‖L1 TL 2 . ( T‖Gψ‖L∞T L2 + T r0+2 2r0 ‖Gψ‖Lq0T Lr0 ) ‖Gψ‖2L∞T L2 . ( T + T r0+2 2r0 ) ‖ψ‖3HT , and ‖fG‖L1 TL 2 . T‖Gψ‖L∞T L2 + T r0+2 2r0 ( ‖ψ‖L∞T L2 + ‖∇ψ‖L∞T L2 ) ‖Gψ‖Lq0T Lr0 + T r0+2 2r0 ( ‖ψ‖2L∞T L2 + ‖∇ψ‖2L∞T L2 ) ‖ωψ‖Lq0T Lr0 . T‖ψ‖HT + T r0+2 2r0 ‖ψ‖2HT + T r0+2 2r0 ‖ψ‖3HT . From the Strichartz estimates we have ‖ √ ln〈x〉Q[ψ]‖L∞T L2 + ‖ √ ln〈x〉Q[ψ]‖Lq0T Lr0 . T [ ‖ψ0‖H + (1 + T )‖ψ‖HT + T r0+2 2r0 ‖ψ‖2HT + (T + T r0+2 2r0 )‖ψ‖3HT ] . (3.8) Thus, it follows from (3.3), (3.6), and (3.8) that ‖Q[ψ]‖HT . (1 + T ) [ ‖ψ0‖H + T‖ψ‖HT + T r0+2 2r0 ‖ψ‖2HT + ( T + T r0+2 2r0 ) ‖ψ‖3HT ] . A similar argument shows that ‖Q[ψ1]−Q[ψ2]‖HT . (1 + T ) ( T + T r0+2 2r0 )( (‖ψ1‖HT + ‖ψ2‖HT )2 + ‖ψ1‖HT + ‖ψ2‖HT ) ‖ψ1 − ψ2‖HT . Hence if we take M ≥ 2‖ψ0‖H, then there exists T = T (M) such that Q is a contraction map from HT,M to itself. A similar argument shows that Q has a unique fixed point in this space. EJDE-2023/64 SCHRÖDINGER-POISSON SYSTEM WITH A LOGARITHMIC POTENTIAL 9 4. Existence of a global minimizer Lemma 4.1. Assume that (A1), (A4), (A5) hold. Then there exists c∗ > 0 such that the energy functional J is bounded from below on S(c) for 0 < c < c∗. Proof. Let ε > 0 be arbitrary. By conditions (A4) and (A5), there exists Cε > 0 such that |F (t)| ≤ ε|t|2 + Cε|t|4 for all t ∈ R. For u ∈ S(c), it follows from Lemma 2.5 that∫ R2 |F (u)|dx ≤ ε ∫ R2 |u|2dx+ Cε ∫ R2 |u|4dx ≤ ε ∫ R2 |u|2dx+ CεKGN ∫ R2 |u|2dx ∫ R2 |∇u|2dx = cε+ cKGNCεA(u). (4.1) Since 0 < ln(1 + r) < r holds for all r > 0, by the Hardy-Littlewood-Sobolev inequality, there exists a constant C > 0 such that 1 2 |V2(u)| = 1 2 ∫ R2 ∫ R2 ln ( 1 + 1 |x− y|2 ) u2(x)u2(y) dx dy ≤ ∫ R2 ∫ R2 ln ( 1 + 1 |x− y| ) u2(x)u2(y) dx dy ≤ ∫ R2 ∫ R2 1 |x− y| u2(x)u2(y) dx dy ≤ Cc3/2A(u)1/2. (4.2) From this, (4.1) and (4.2), we obtain J(u) ≥ 1 2 A(u) + ∫ R2 ln(1 + |x|2)u2(x) dx+ γ 16π (V1(u)− V2(u))− ∫ R2 F (u) dx ≥ 1 2 A(u) + 1 2 ∫ R2 ln(1 + |x|2)u2(x) dx− γ 16π V2(u)− cε− cKGNCεA(u) ≥ 1 2 A(u) + 1 2 ∫ R2 ln(1 + |x|2)u2(x) dx− γ 8π Cc3/2A(u)1/2 − cε− cKGNCεA(u) ≥ (1 2 − cCεKGN ) A(u) + 1 2 ∫ R2 ln(1 + |x|2)u2(x) dx − γ 8π Cc3/2A(u)1/2 − cε, (4.3) which implies that J(u) is bounded from below on S(c) when c < c∗ := 1 2KGNCε . The proof is complete. � Proof of Theorem 1.2. By Lemma 4.1, we know that Jc = inf u∈S(c) J(u) > −∞. Then there exists a minimizing sequence {un} ⊂ S(c) such that limn→∞ J(un) = Jc. From (4.3) it follows that A(un) and ∫ R2 ln(1+|x|2)u2 ndx are bounded uniformly 10 X. DOU, J. SUN EJDE-2023/64 in n. Since {un} ∈ S(c), we can deduce that {un} is bounded uniformly in H. According to Lemma 2.6(i), it follows from that uc ∈ S(c), and∫ R2 F (un) dx→ ∫ R2 F (uc) dx as n→∞, (4.4) where we have used the Brezis-Lieb lemma [4]. Moreover, by Lemma 2.6(v), we have ∫ R2 ∫ R2 ln(|x− y|)u2(x)u2(y) dx dy ≤ lim inf n→∞ ∫ R2 ∫ R2 ln(|x− y|)u2 n(x)u2 n(y) dx dy. (4.5) Thus, by (4.4), (4.5) and the weakly lower semi-continuity, we obtain Jc ≤ J(uc) ≤ lim inf n→∞ J(un) = Jc, which indicates that J(uc) = Jc, that is, uc is a minimizer of Jc for c < c∗. Since J(un)→ J(uc) and V2(un)→ V2(uc) as n→∞, together with (4.4) again, we obtain 1 2 [A(un)−A(uc)] + 1 2 ∫ R2 ln(1 + |x|2)(u2 n(x)− u2 c(x)) dx + γ 16π [V1(un)− V1(uc)] = o(1). Note that A(uc) ≤ lim inf n→∞ A(un), V1(uc) ≤ lim inf n→∞ V1(un),∫ R2 ln(1 + |x|2)u2 cdx ≤ lim inf n→∞ ∫ R2 ln(1 + |x|2)u2 ndx. Then we have A(un)→ A(uc) and V1(un)→ V1(uc) as n→∞,∫ R2 ln(1 + |x|2)u2 ndx→ ∫ R2 ln(1 + |x|2)u2 cdx as n→∞. Hence, we deduce that un → uc in H. The proof is complete. � Proof of Theorem 1.3. Following the classical arguments of Cazenave and Lions [7]. we assume that there exist an ε0 > 0, {δn} ⊂ R+ a decreasing sequence converging to 0, and {ψn} ⊂ H satisfying infu∈Mc ‖ψn(0, x)− u‖H < δn such that inf u∈Mc ‖ψn(tn, x)− u‖H ≥ ε0, where ψ(tn, x) is the unique solution of (1.1) with the initial value ψn(0, x). We observe that ‖ψn(0, x)‖2L2 → c as n → ∞ and that J(ψn(0, x)) → Jc by the continuity of J . According to the conservation laws of the energy and mass, we have ‖ψn(t, x)‖2L2 = ‖ψn(0, x)‖2L2 → c as n→∞, (4.6) J(ψn(t, x)) = J(ψn(0, x))→ Jc as n→∞. (4.7) Now, let φn(tn, x) = √ cψn(t,x) ‖ψn(t,x)‖L2 . Then by (4.6) one has ‖φn(tn, x)‖2L2 = c. More- over, it follows from (4.7) that J(φn(tn, x)) EJDE-2023/64 SCHRÖDINGER-POISSON SYSTEM WITH A LOGARITHMIC POTENTIAL 11 = 1 2 ∫ R2 |∇φn(tn, x)|2dx+ 1 2 ∫ R2 ln(1 + |x|2)φ2 n(tn, x) dx + γ 8π ∫ R2 ∫ R2 ln(|x− y|)|φn(tn, x)|2|φn(tn, y)|2 dx dy − ∫ R2 F (φ(tn, x)) dx = c 2‖ψn(t, x)‖2L2 ∫ R2 |∇ψn(t, x)|2dx + c 2‖ψn(t, x)‖2L2 ∫ R2 ln(1 + |x|2)ψ2 n(t, x) dx + γc2 8π‖ψn(t, x)‖4L2 ∫ R2 ∫ R2 ln(|x− y|)|ψn(t, x)|2|ψn(t, y)|2 dx dy − ∫ R2 F ( √ cψn(t, x) ‖ψn(t, x)‖L2 ) dx → Jc as n→∞. So, {φn(tn, x)} is a minimizing sequence to Jc. Thus, there exists ũ ∈ S(c) such that ‖φn(tn, x)− ũ‖H → 0 as n→∞. (4.8) Since ‖φn(tn, x)− ũ‖H = ‖φn(tn, x)− ũ‖H1 + (∫ R2 ln(1 + |x|2)|φn(tn, x)− ũ|2dx )1/2 = ‖ √ cψn(t, x) ‖ψn(t, x)‖L2 − ũ‖H1 + (∫ R2 ln(1 + |x|2)| √ cψn(t, x) ‖ψn(t, x)‖L2 − ũ|2dx )1/2 ≥ inf u∈Mc ‖ψn(tn, x)− u‖H ≥ ε0, which contradicts with (4.8). The proof is complete. � Acknowledgments. J. 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Xuechao Dou School of Mathematics and Statistics, Shandong University of Technology, Shandong, Zibo 255049, China Email address: xcdou178@163.com EJDE-2023/64 SCHRÖDINGER-POISSON SYSTEM WITH A LOGARITHMIC POTENTIAL 13 Juntao Sun (corresponding author) School of Mathematics and Statistics, Shandong University of Technology, Shandong, Zibo 255049, China Email address: jtsun@sdut.edu.cn 1. Introduction 2. Preliminary results 3. Local well-posedness of the Cauchy problem 4. Existence of a global minimizer Acknowledgments References