ENERGY SOLUTION TO SCHRO¨DINGER-POISSON SYSTEM IN THE TWO-DIMENSIONAL WHOLE SPACE SATOSHIMASAKI Abstract. We consider the Cauchy problem of the two-dimensional Schrödinger-Poisson system in the energy class. Though the Newto- 0 1 nian potential diverges at the spatial infinity in the logarithmic order, 0 global well-posedness is proven in both defocusing and focusing cases. 2 The key is a decomposition of the nonlinearity into a sum of the linear logarithmic potential and a good remainder, which enables us to n apply the perturbation method. Our argument can be adapted to the a J one-dimensional problem. 5 2 ] 1. Introduction P A This paper is devoted to the study of the Scho¨dinger-Poisson system . h 1 i∂ u+ ∆u= λPu, (t,x) R1+2, t t a 2 ∈ m (1.1)  ∆P = u2,  − | | [  u(0,x) = u0(x), 1 where λ is a real constant. We suppose P is the Newtonian potential v  8 1 0 (1.2) P = (log x u2) −2π | |∗| | 3 4 where denotes the convolution. For a suitable u, this is the unique strong ∗ . solution of ∆P = u2 under the condition 1 − | | 0 0 P 0 as x , P L∞(R2), P(0) = (log y )u(y)2dy 1 |∇ | → | | → ∞ ∇ ∈ R2 | | | | Z : v (see [11]). When the dimensions are larger than two, the Schro¨dinger- i PoissonsystemisaspecialcaseoftheHartreeequationandoneofthetypical X example of the nonlinear Schro¨dinger equation with a nonlocal nonlinearity, r a and there is large amount of literature (see [6] and references therein). On the other hand, the two-dimensional case is less studied. In [1, 18], (1.1) is considered with some restrictive assumptions such as a neutrality condition which confirms that the Newtonian potential (1.2) does not diverge at the spatial infinity and in particular belongs to L2 space. The Poisson equation is sometimes posed with a background (or doping profile): ∆P = u 2 b, − | | − wherebisagivenpositivefunction. Then,theneutralityconditionis u2 | | − bdx = 0 or equivalently (u2 b)(0) = 0. When we consider the problem F | | − R in dimensions less than three, this condition is useful to control P. Notice that this condition excludes all nontrivial solutions when b 0, and that ≡ we need to remove this condition for the study of (1.1). In [11], the above 1 2 S.MASAKI assumptions are removed and the existence of a unique local solution is proven for data in the usual Sobolev space Hs(R2) (s > 2) despite the fact that the nonlinear potential diverges at the spatial infinity. Since (1.2) is not necessarily defined for u Hs (s >2) we introduced a new formula ∈ 1 x y P = log | − | u(y)2dy −2π R2 y | | Z (cid:18) | | (cid:19) which makes sense merely if u2 Lp(R2) (p (1,2)). We underline that | | ∈ ∈ the local solutions given there do not have finite energy (the energy is given in (1.5) below). Our aim in this paper is to prove that there exists a time- global solution if initial data has finite energy. For our analysis, the following reduction is crucial: We guess that the Newtonian potential (1.2) may behave like 1 u 2 log x at the spatial −2π k kL2 | | infinity, which will be the bad part of the nonlinearity, and decompose the nonlinearity as λ λ x y λPu= u 2 (log x )u u log | − | u(y)2dy, −2π k kL2 h i − 2π R2 x | | Z (cid:18) h i (cid:19) where x = (1+ x 2)1/2. We then obtain h i | | 1 λ λ x y i∂ u+ ∆u+ u 2 (log x )u= u log | − | u(y)2dy. t 2 2π k kL2 h i −2π R2 x | | Z (cid:18) h i (cid:19) ItwillturnoutthatthebadpartofP iscorrectlyextractedfromtheoriginal nonlinearity and therefore the behavior of the “new nonlinearity” becomes better. Notice that one can also expect that u is conserved because λ k kL2 is a real number. Hence, putting λ m := u 2 , −2π k 0kL2 we reach to the equation 1 λ x y i∂ u+ ∆ mlog x u = u log | − | u(y)2dy, t (1.3) 2 − h i −2π R2 x | |  (cid:18) (cid:19) Z (cid:18) h i (cid:19) u(0,x) = u0(x). Noticethat mlog x isnowcompletely independentofuandthatitthere- − h i fore can be regarded as a linear potential. In what follows, we work with this equation. Observe that if there exists a solution to (1.3) conserving u , then it is also a solution of (1.1). k kL2 Now, thelinear partoftheequation isnoti∂ +(1/2)∆ buti∂ +(1/2)∆ t t − mlog x . Thus, a natural choice of the function space on which we shall work ihs niot the Sobolev space H1(R2) any more, but the following one: := u H1(R2); log x u L2 , H { ∈ h i ∈ } (1.4) u := u p+ log u . k kH k kH1(R2) h·i L2(R2) (cid:13)p (cid:13) If m > 0, that is, if λ < 0, then the a(cid:13)bove space(cid:13)coincides with the form (cid:13) (cid:13) domain of the positive operator 1∆ +mlog x . Our main result is the −2 h i following: ENERGY SOLUTION TO 2D-SP SYSTEM 3 Theorem 1.1. The problem (1.3) is globally well-posed in . Moreover, H the solution conserves u(t) and the energy k kL2 1 λ (1.5) E(t) = u(t) 2 (log x y )u(t,x)2 u(t,y)2dxdy. 2k∇ kL2 − 4π R2 | − | | | | | Z Corollary 1.2. The Problem (1.1) is globally well-posed in . H Remark 1.3. Let u C(R; ) be a solution of (1.3) (and of (1.1)) given in ∈ H Theorem 1.1. Then, v := uexp( i λ t log u(s) 2 ds) solves − 2π 0 k |·| kL2 1 λ R p x y i∂ v+ ∆v = v log | − | v(y)2dy, t (1.6) 2 −2π R2 y | |  Z (cid:18) | | (cid:19) v(0,x) = u0(x). Notice that the nonlinearity of (1.6) makes sense without the momentum condition log v L2. This observation explains why existence of a time-local soluti|o·n|ca∈n be proven by assuming only u Hs(R2) (s > 1) in p 0 ∈ [11]. 1.1. Consequent results. Our argument is also applicable to (1.1) involv- ing a power type nonlinearity: 1 i∂ u+ ∆u= λPu+η up−1u, (t,x) R1+2, t 2 | | ∈ (1.7)  ∆P = u2,  − | |  u(0,x) = u (x), 0  where η is a real number and p >2. Theorem 1.4. The problem (1.7) is globally well-posed in if either one H of the following conditions is satisfied: (1) η > 0, λ R and p > 2; (2) η < 0, λ ∈ R, and 2 6 p < 3; ∈ (3) η < 0, λ > 0, p = 3, and u is small; k 0kH (4) η < 0, λ < 0, p > 3, and u is small. k 0kH Moreover, the solution conserves u(t) and the energy k kL2 1 λ E (t):= u(t) 2 (log x y )u(t,x)2 u(t,y)2dxdy (1.8) p 2k∇ kL2 − 4π ZR2 | − | | | | | η p+1 + u(t) . p+1k kLp+1 The proof is done with a straight-forward modification (see Section 4). The case where p = 3 is known as the L2-critical case. Since the -norm H contains derivative, it seems difficult to treat the case 1 < p < 2. Never- theless, we can show global well-posed in a slightly smaller function space 1,2 := u H1(R2);ulog x L2 . H { ∈ h i∈ } Theorem 1.5. Suppose 1 < p < 2. For η,λ R The problem (1.7) is glob- ∈ ally well-posed in the space 1,2. Moreover, the solution conserves u(t) H k kL2 and the energy E (t) given in (1.8). p 4 S.MASAKI We can also handle the one-dimensional problem 1 λ i∂ u+ ∂ u= (x u2)u+η up−1u, (t,x) R1+1, t xx (1.9) 2 −2 | |∗| | | | ∈  u(0,x) = u (x),  0 where λ,η R and p > 2. The one dimensional problem was studied in ∈ [7, 14, 15]. The global well-posedness of (1.9) was shown in the space f { ∈ H1(R); x f L2(R) in [15], and in the space f H1(R); x f L2(R) | | ∈ } { ∈ | | ∈ } with a presence of background in [7], provided λ > 0 and data is small p relative to the background. We can prove the global well-posedness result of (1.9) including these results. Theorem1.6. Theproblem (1.9)isgloballywell-posedin f H1(R); x f L2(R) if λ R and either one of the following condition{s is∈satisfied: | | ∈ } ∈ p (1) η > 0, λ R, and p > 2; (2) η < 0, λ ∈ R, and 2 6 p < 5; ∈ (3) η < 0, λ > 0, p = 5, and u + u is small; k 0kH1 k |·| 0kL2 (4) η < 0, λ < 0, p > 5, and u + u is small. k 0kH1 kp|·| 0kL2 The solution conserves u and the energy k kL2 p (1.10) 1 λ η E(t) := ∂ u 2 x y u(x)2 u(y)2dxdy+ u p+1 . 2 k x kL2(R)− 2 R2| − || | | | p+1k kLp+1(R) ZZ e Theone-dimensionalversionofTheorem1.5isasfollows,whichreproduce the same result in [15, Theorem 2.1] when η < 0 and λ > 0. Theorem 1.7. Suppose 1 < p < 2. For η,λ R The problem (1.9) is globally well-posed in the space Σ := u H1(R∈2); x u L2 . Moreover, { ∈ | | ∈ } the solution conserves u(t) and the energy E(t) given in (1.10). k kL2 As in the two dimensional case, the key is a “reduction” of (1.9) to e 1 λ u 2 λ i∂ u+ ∂ u+ k 0kL2 x u = u (x y x ) u(y)2dy+η up−1u, t xx  2 2 | | −2 R | − |−| | | | | | Z u(0,x) = u0(x). We briefly mention about other related works. Oh considered in [12]  the Cauchy problem of the nonlinear Schro¨dinger equation with general potentialandL2-subcriticalpower-typenonlinearity, andprovedglobalwell- posedness in the form domain of 1∆+V, provided the potential V > 0 −2 satisfies ∂αV L∞ for α > 2 (see also [6]). In particular, the case where ∈ | | thepotentialV isaquadraticpolynomialisextensivelystudied. Inthiscase, we have several special properties such as explicit representations of linear solutions,calledMehler’sformula,and/oroftheHeisenbergobservables. We refer the reader to [2, 3, 4, 10, 19] for H1-subcritical and H1-critical power- type nonlinearity and to [5] for H1-subcritical Hartree type nonlinearity. In [16], the ground states of (1.1) is treated. The rest of the paper is organized as follows: We collect some basic estimates in Section 2, and, in Section 3 we prove Theorem 1.1. Section 4 is devoted to the study of (1.7). ENERGY SOLUTION TO 2D-SP SYSTEM 5 2. Preliminaries 2.1. Strichartz estimate. We first summarize the properties on the operator 1 (2.1) A:= ∆ mlog x , 2 − h i where m = 0 is a real constant. For any m, A is essentially self-adjoint on C∞(R26 ) (see [13]). Since our potential is sub-quadratic, that is, since 0 ∂αlog x 0 as x for α = 2 and ∂αlog x L∞ for α > 3, the | h i|→ | | → ∞ | | h i ∈ | | following estimate is established in [17]: For any T > 0, eitAϕ 6C t −1 ϕ L∞ | | k kL1 for t [ T,T], where C(cid:13)depen(cid:13)ds on T (see also [8]). Once we know this ∈ − (cid:13) (cid:13) type of estimate, the Strichartz estimate follows by interpolation. We say that a pair (q,r) is admissible if 26 r < and 2/q = δ(r) := 1 2/r. ∞ − Lemma 2.1 (Strichartz’sestimate). For any T > 0, the following properties hold: Suppose ϕ L2(R2). For any admissible pair (q,r), there exists a • ∈ constant C = C(T,q,r) such that eitAϕ 6 C ϕ . Lq((−T,T);Lr) k kL2 (cid:13) (cid:13) Let I ( T,T(cid:13)) be a(cid:13)n interval and t0 I. For any admissible pairs • ⊂ − ∈ (q,r) and (γ,ρ), there exists a constant C = C(t,q,r,γ,ρ) such that t ei(t−s)AF(s)ds 6 CkFkLγ′(I;Lρ′) (cid:13)Zt0 (cid:13)Lq(I;Lr) (cid:13) (cid:13) for ever(cid:13)(cid:13)y F Lγ′(I;Lρ′). (cid:13)(cid:13) ∈ 2.2. Some estiamtes. Lemma 2.2. Let W be an arbitrary weight function such that W, ∆W L∞(R2). It holds for all T > 0, admissible pair (q,r), and ϕ ∇ that ∈ ∈ H [ ,eitA]ϕ 6 C T ϕ , ∇ Lq((−T,T);Lr) | |k k2 (cid:13)[W,eitA]ϕ(cid:13) 6 C T (1+ )ϕ . (cid:13) (cid:13)Lq((−T,T);Lr) | |k ∇ k2 (cid:13) (cid:13) (cid:13) (cid:13) Proof. Since v = eitAϕ solves i∂ v+Av = 0, an explicit calculation shows t t mx [ ,eitA]ϕ = i ei(t−s)A eisAϕds ∇ − 1+x2 Z0 and t 1 [W,eitA]ϕ = i ei(t−s)A W + ∆W eisAϕds. ∇ ·∇ 2 Z0 (cid:18) (cid:19) The Strichartz estimate therefore gives the desired estimates. (cid:3) The following is useful for estimates of the nonlinearity in (1.3). 6 S.MASAKI Lemma 2.3. Set a function |x−y| log hxi K(x,y)= 1+log y h i of x,y R2. For any p [1, ) and ε > 0, there exist a function W(x,y) > ∈ ∈ ∞ 0 with kWkL∞y Lpx 6 ε and a constant C0 such that K(x,y) 6 C +W(x,y) 0 | | holds for all (x,y) R2+2. ∈ Proof. Take η (0,1] and set W(x,y) = K(x,y)1 (x,y). If η is |x−y|6η ∈ | | sufficiently small then log x +log y +η 1 W(,y) 6 k | |kLp(|x|6η) h| | ik kLp(|x|6η) 6 ε k · kLp 1+log y h i since log x belongs to Lp (R2) for all p < . Moreover, by (2.12) of [11], | | loc ∞ √3 sup K(x,y) 6 1+log η |x−y|>η for any η 6 1, which completes the proof. (cid:3) Remark 2.4. In 1D case, the corresponding estimate is x y x | − |−| | 6 1. 1+ y (cid:13) | | (cid:13)L∞x,y(R2) (cid:13) (cid:13) (cid:13) (cid:13) 3.(cid:13)Proof of t(cid:13)he theorem 3.1. Local well-posedness. Lemma 3.1. Let (q ,r ) be an admissible pair with r > 2. For any u , 0 0 0 0 ∈ H there exist an existence time T = T( u ) and a unique solution u k 0kH ∈ C(( T,T); ) Lq0(( T,T);Lr0) C1(( T,T); ∗). The solution conserves − H ∩ − ∩ − H u(t) andtheenergy (1.5). Moreover, themapu uiscontinuousfrom k kL2 0 7→ to C(( T,T); ). H − H Proof. We write Lp(( T,T);X) = LpX, for short. Define a Banach space − T := f L∞(( T,T); ); f 6 M HT,M { ∈ − H k kHT } with norm kfkHT := kfkL∞T H+kfkLqT0W1,r0 + loghxif Lq0Lr0 . T (cid:13)p (cid:13) We show that if r > 2 then there exist M = M(cid:13) ( u ) a(cid:13)nd T = T( u ) 0 (cid:13) k 0kH (cid:13) k 0kH such that Q[u](t,x) := (eitAu )(x) 0 i t y + ei(t−s)A log |·− | u(s,y)2dy u(s, )ds (x) 2π (cid:18)Z0 (cid:18)ZR2 h·i | | (cid:19) · (cid:19) becomes a contraction map from to itself, where A is defined in (2.1). T,M H ENERGY SOLUTION TO 2D-SP SYSTEM 7 Set |x−y| log hxi K(x,y) = . 1+log y h i Then, by Lemma 2.3, there exist a nonnegative function W L∞Lr0′ and a ∈ y x constant C such that 0 K(x,y) 6 C +W(x,y). 0 | | Recall that r (2, ) and so r′ := r /(r 1) (1,2). We hence see that 0 ∈ ∞ 0 0 0− ∈ Pu= K(x,y)(1+log y )u(y)2u(x)dydx h i | | ZZ satisfies 2 Pu 6 C( u + u ) 1+log x u . k kL2 k kL2 k kLr0 h i L2 Take L1 norm to yield (cid:13)(cid:13)p (cid:13)(cid:13) T (cid:13) (cid:13) (3.1) kPukL1TL2 6 C(T kukL∞T L2 +T21+r10 kukLqT0Lr0) 1+loghxiu 2L∞L2. T (cid:13)p (cid:13) By the Strichartz estimate, we end up with (cid:13) (cid:13) (cid:13) (cid:13) (3.2) kQ[u]kL∞T L2 +kQ[u]kLqT0Lr0 6 Cku0kL2 +C(T +T21+r10)kuk3HT . We next estimate Q[u]. One easily sees that ∇ t Q[u] = eitA u i ei(t−s)A (Pu)(s)ds 0 ∇ ∇ − ∇ Z0 t +[ ,eitA]u i [ ,ei(t−s)A](Pu)(s)ds. 0 ∇ − ∇ Z0 We deduce from Lemma 2.2 with (q,r) = ( ,2) that ∞ t t [ ,ei(t−s)A](Pu)(s) ds 6 (t s) Pu(s) ds 6 t Pu . Z0 (cid:13)∇ (cid:13)L2 Z0 − k kL2 | |k kL1TL2 The r(cid:13)ight hand side is bo(cid:13)unded as in (3.1). [ ,eitA]u is handled similarly. (cid:13) (cid:13) 0 ∇ Mimicking (3.1), we infer that (3.3) kP∇ukL1TL2 6 C(T k∇ukL∞T L2 +T12+r10 k∇ukLqT0Lr0) 1+loghxiu 2L∞L2 T (cid:13)p (cid:13) Now, let us estimate ( P)u. It writes (cid:13) (cid:13) (cid:13) (cid:13) ∇ x y x ( P)(x)u(x) = − u(y)2dy u(x), ∇ R2 x y 2 − 1+x2 | | (cid:18)Z (cid:18)| − | (cid:19) (cid:19) and so ( P)u 6 C (x −1 u2)+ −1 u 2 u k ∇ kL2 | | ∗| | h·i k kL2 Lr20r−02 k kLr0 (cid:13) (cid:13) 6 C((cid:13)u 2 + u 2 ) u (cid:13) (cid:13)k kLr20r−01 k kL2 k kLr0 (cid:13) 6 C( u 2 + u 2 ) u k kL2 k∇ kL2 k kLr0 8 S.MASAKI by the Hardy-Littlewood-Sobolev and the Sobolev inequalities. We see that (3.4) k(∇P)ukL1TL2 6 C(T kuk2L∞T L2 +T21+r10 kuk2LqT0Lr0)kukL∞T L2. We deduce from the Strichartz estimate that (3.5) k∇Q[u]kL∞T L2+k∇Q[u]kLqT0Lr0 6 Ck∇u0kH+C(T+T12+r10)kuk3HT . Let us proceed to the estimate of log x Q[u]. It holds that h i p t 1+log x Q[u]= eitA 1+log x u i ei(t−s)A 1+log x Pu(s)ds 0 h i h i − h i Z0 p +Rp, p where t R = [ 1+log x ,eitA]u i [ 1+log x ,ei(t−s)A]Pu(s)ds. 0 h i − h i Z0 p p A use of Lemma 2.2 with W = 1+log x yields h i kRkL∞T L2 +kRkLqT0Lr0 6pCT ku0kH+CT k(1+∇)(Pu)kL1TL2 6 CT ku0kH+CT(T +T21+r10)kuk3HT where we have used (3.1), (3.3), and (3.4). As in (3.1), it holds that kP(Wu)kL1TL2 6 C(T kWukL∞T L2 +T21+r10 kWukLqT0Lr0)kWuk2L∞T L2 6 C(T +T21+r10) u 3 , k kHT where W = 1+log x . We conclude from the Strichartz estimate, (3.2), h i and (3.5) that p kQ[u]kHT 6 C1ku0kH+C2(T +T21+r10)kuk3HT . A similar argument shows kQ[u1]−Q[u2]kHT 6 C3(T +T12+r10)(ku1kHT +ku2kHT)2ku1−u2kHT . Thus, if we take M > 2C u then there exists T = T(M) such that Q 1k 0kH is a contraction map from to itself. T,M H The conservations of u(t) is shown by multiplying (1.3) by u and k kL2 integrating the imaginary part. To prove the energy conservation, we need a regularizing argument. Note that (1.3) can be solved also in the space f H2(R2) : log x f L2 , which is one of dense subsets of , in an { ∈ h i ∈ } H essentially same way. We omit details. (cid:3) 3.2. Global existence. We first give a useful blow-up criteria. Lemma 3.2. Suppose u . Let u C(( T ,T ); ) be a unique 0 min max ∈ H ∈ − H maximal solution given by Lemma 3.1. If T < (resp. T < ), then max min ∞ ∞ u(t) as t T (resp. t T ). k∇ kL2 → ∞ ↑ max ↓ − min ENERGY SOLUTION TO 2D-SP SYSTEM 9 Proof. We only consider positive time. Suppose T < . Then, u(t) max ∞ k kH has to diverge as t T . Otherwise, we can extend the solution beyond max ↑ T by Lemma 3.1. Recall that u(t) = u . Since max k kL2 k 0kL2 d 2 log x u(t) = 2Re (log x )∂ u(t)u(t)dx t dt h i L2 h i (cid:13)p (cid:13) Z (cid:13) (cid:13) (cid:13) (cid:13) = Im (log x )∆u(t)u(t)dx − h i Z x = Im u(t)u(t)dx, 1+x2 ·∇ Z it holds that 2 2 log x u(t ) 6 log x u(t ) + t t u u h i 2 L2 h i 1 L2 | 2− 1|k∇ kL∞((t1,t2);L2)k 0kL2 f(cid:13)(cid:13)opr all Tmin <(cid:13)(cid:13)t1 < t(cid:13)(cid:13)2p< Tmax. This(cid:13)(cid:13)implies that if we assume (cid:13) − (cid:13) (cid:13) (cid:13) limsup u(t) < k∇ kL2 ∞ t↑Tmax then u(t) never blows up. We hence obtain the lemma. (cid:3) k kH Remark 3.3. As in [9], the solution breaks down with concentration at a point if log x u(t) = 0. However, this does not occur when u(t) L2 k h i k k∇ k is bounded above. Indeed, since p 1 1 kukL2(|x| 0 and since log x u u = u u > u k h i kL2, k kL2(|x|r) k 0kL2 − (log r )1/2 p h i by letting r = log x u , we obtain L2 k h i k 1 p 2 log x u −12 6 C k loghxiukL2 +C u 12 , k h i kL2   k∇ kL2 log p log x u L2 p k h i k  D p E which implies log x u is strictly positive if u < . k h i kL2 k∇ kL2 ∞ Proof of Theorepm 1.1. Let us establish a priori estimate of u(t) . k∇ kL2 We first consider the case λ < 0. Since log x > 0 for x > 1, | | | | λ log x y u(x)2 u(y)2dxdy − 4π R2+2 | − || | | | ZZ λ > | | log x y u(x)2 u(y)2dxdy −4π | | − ||| | | | ZZ|x−y|<1 λ > | | log x 2 u 2 u 2 −4π k | |kL2(|x|61)k kL4k kL2 By the L2-conservation and the Sobolev embedding, we have (3.6) u(t) 2 6 2E +C u(t) . k∇ kL2 0 k∇ kL2 Therefore,thereexistsaconstantM independentoftsuchthat u(t) 6 k∇ kL2 M. 10 S.MASAKI We now suppose λ > 0. By Lemma 2.3, for any ε > 0 there exists a constant C such that the following estimate holds: 0 λ log x y u(x)2 u(y)2dxdy 4π R2+2 | − || | | | ZZ λ x y λ 2 6 log | − | u(x)2 u(y)2dxdy+ u 2 log x u 4π R2+2 x | | | | 4π k 0kL2 h i L2 λ ZZ h i 2 λ (cid:13)(cid:13)p (cid:13)(cid:13) 2 6 (C u 2 +ε u 2 ) 1+log x u + u(cid:13)2 log(cid:13)x u 4π 0k 0kL2 k kL4 h i L2 4π k 0kL2 h i L2 λC λ(C +1) (cid:13)p (cid:13) 2 (cid:13)p (cid:13) 6 4π0 ku0k4L2 + 40π (cid:13)(cid:13)ku0k2L2 loghx(cid:13)(cid:13)iu L2 +Cεku0k(cid:13)(cid:13)3L2k∇ukL2 (cid:13)(cid:13) (cid:13)2p (cid:13) +Cε u u log x u(cid:13) (cid:13) k 0kL2k∇ kL2 h i (cid:13)L2 (cid:13) 6 C1+C2(ε+|t|) sup(cid:13)(cid:13)(cid:13)kp∇u(s)kL2(cid:13)(cid:13)(cid:13)+C3ε|t| sup k∇u(s)k2L2, s∈[0,t] s∈[0,t] where C (i = 1,2,3) depends only on λ, C , u , and ε. Fix T > 0. i 0 k 0kH Taking ε < (8C T)−1, we deduce from the conservation of E(t) that 3 2 (3.7) sup u(s) 6 4E(0)+4C +4C (ε+2T) sup u(s) k∇ kL2 1 2 k∇ kL2 s∈[0,t] ! s∈[0,t] for 0 6 t 6 2T. This implies that sup u(t) 6 C( u ,T) < . k∇ kL2 k 0kH ∞ t∈[0,2T] Since T is arbitrary, we obtain the global existence. (cid:3) 4. Remarks on the problem with power nonlinearity WegivearoughsketchoftheproofsofTheorem1.4and1.5inthissection. Proof of Theorem 1.4. Thelocal well-posedness partholds if p > 2 as in the proof of Lemma 3.1. The restriction p > 2 is required when we estimate (u p−1u u p−1u ) 6 C (u p−2+ u p−2)( u + u )u u 1 1 2 2 p 1 2 1 2 1 2 |∇ | | −| | | | | | | |∇ | |∇ | | − | +C (u p−1+ u p−1) (u u ). p 1 2 1 2 | | | | |∇ − | By exactly the same argument as in Lemma 3.2, the problem of global existence boils down to obtaining an a priori bound of u(t) . Recall k∇ kL2 that the conserved energy is 1 λ E (t):= u(t) 2 (log x y )u(t,x)2 u(t,y)2dxdy p 2k∇ kL2 − 4π R2 | − | | | | | Z η p+1 + u(t) . p+1k kLp+1 The case η > 0. We have λ u(t) 2 6 E (t)+ (log x y )u(t,x)2 u(t,y)2dxdy. k∇ kL2 p 4π R2 | − | | | | | Z By the same argument as in the case η = 0, we prove global existence.

Energy solution to Schrödinger-Poisson system in the two-dimensional whole space

Similar Energy solution to Schrödinger-Poisson system in the two-dimensional whole space