LOCAL

. In this article, we consider planar Schr¨odinger-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.

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 : R 2 → 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 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−2 ψ(p > 2) is established in the space B given by Another interesting topic on (1.1) is to study the standing wave solution of the form ψ(x, t) = e iλt u(x), where λ ∈ R and u : R 2 → R. Then (1.1) is reduced to the system which can be further written as the integro-differential equation At least formally, the energy functional associated with (1.3) is 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 H 1 (R 2 ), 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 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 L 2 -norms of solutions are prescribed, i.e.R 2 |u| 2 dx = c for a given c > 0, which are usually called normalized solutions.This study seems particularly meaningful 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 multiplicity of normalized solutions for 1.3 with f (u) = |u| p−2 u(p > 2).When the logarithmic external potential is considered in (1.3), Dolbeault, Frank and Jeanjean [12] studied the existence of normalized solutions for (1.3) with f (u) = ln |u| 2 u, and recently Guo, Liang and Li [15] proved the existence and uniqueness of L 2 -critical constraint minimization problem, i.e. f (u) = |u| p−2 u 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 functional under the constraint where endowed with the norm u H := u H 1 + u * , here We now summarize our main results.
Then the Cauchy problem (1.1) is local well-posed in H.That is, for any ψ 0 ∈ H, there exists an existence time 1), where (q 0 , r 0 ) be an admissible pair with r 0 > 2.
Theorem 1.2.Assume that condition (A1) holds.In addition, we assume that f satisfies Then there exists a constant c * > 0 such that for 0 < c < c * , the infimum is achieved by some u c ∈ S(c), i.e.J(u c ) = J c .
It is easy to find some examples on the nonlinearity f satisfying conditions (A1), (A4), and (A5), such as By Theorem 1.2, we know that the set of ground states 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 M c is orbitally stable.That is, for any ε > 0, there exists δ > 0 such that for any where T is the maximal existence time for ψ(t, x).

Preliminary results
For sake of convenience, we set Then the energy functional J defined in (1.5) can be rewritten as 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 ∈ L 2 (R 2 ).For any admissible pair (q, r), we have (ii) let I ⊂ (−T, T ) be an interval and t 0 ∈ I.For any admissible pairs (q, r) and (γ, ρ), we have for every F ∈ L γ (I; L ρ ).
We are ready to prove Theorem 1.1.We write L p ((−T, T ); H) = L p T H for short.We define the Banach space T H + ψ L q 0 T W 1,r 0 + ln x ψ L q 0 T L r 0 .Now we show that if r 0 > 2, then there exist M = M ( ψ 0 H ) and T = T ( ψ 0 H ) such that  x, y).Recall that r 0 ∈ (2, ∞) and so r 0 := r 0 /(r 0 − 1) ∈ (1, 2).We hence see that By Strichartz estimates, we have It is easy to see that From Lemma 2.3 with (q, r) = (∞, 2), we deduce that Similar to (3.1), we infer that (3.4) Now, let us estimate (∇ω)ψ.It can be written as It follows from the Hardy-Littlewood-Sobolev and the Sobolev inequalities that By condition (A3) one has We deduce from the Strichartz estimates that Let us proceed to the estimate where As in (3.1), we have Thus, it follows from (3.3), (3.6), and (3.8) that A similar argument shows that Hence if we take M ≥ 2 ψ 0 H , then there exists T = T (M ) such that Q is a contraction map from H T,M to itself.A similar argument shows that Q has a unique fixed point in this space.

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 For u ∈ S(c), it follows from Lemma 2.5 that 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 From this, (4.1) and (4.2), we obtain ) which implies that J(u) is bounded from below on S(c) when c < c * := 1 2K GN Cε .The proof is complete.
Proof of Theorem 1.2.By Lemma 4.1, we know that Then there exists a minimizing sequence {u n } ⊂ S(c) such that lim n→∞ J(u n ) = J c .From (4.3) it follows that A(u n ) and R 2 ln(1+|x| 2 )u 2 n dx are bounded uniformly in n.Since {u n } ∈ S(c), we can deduce that {u n } is bounded uniformly in H.According to Lemma 2.6(i), it follows from that u c ∈ S(c), and where we have used the Brezis-Lieb lemma [4].Moreover, by Lemma 2.6(v), we have Thus, by (4.4), (4.5) and the weakly lower semi-continuity, we obtain Then we have Hence, we deduce that u n → u c 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 ε 0 > 0, {δ n } ⊂ R + a decreasing sequence converging to 0, and where ψ(t n , x) is the unique solution of (1.1) with the initial value ψ n (0, x).We observe that ψ n (0, x) 2 L 2 → c as n → ∞ and that J(ψ n (0, x)) → J c by the continuity of J.According to the conservation laws of the energy and mass, we have J(ψ n (t, x)) = J(ψ n (0, x)) → J c as n → ∞.