Positive energy static solutions for the Chern-Simons-Schr\"odinger system under a large-distance fall-off requirement on the gauge potentials

In this paper we prove the existence of a positive energy static solution for the Chern-Simons-Schr\"odinger system under a large-distance fall-off requirement on the gauge potentials. We are also interested in existence of ground state solutions.


INTRODUCTION
The following Chern-Simons-Schrödinger system (CSS) 2 |φ| 2 , has been object of interest for many authors, physicists and mathematicians, in the last thirty years.
For p = 3, it corresponds to the model proposed by Jackiw-Pi [16], and studied also in [10,11,15,17,18], to describe the dynamics of a nonrelativistic solitary wave that behaves like a particle, in the three dimensional gauge Chern-Simons theory.
Here t ∈ R, x = (x 1 , x 2 ) ∈ R 2 , φ : R × R 2 → C is the scalar field, A µ : R × R 2 → R are the components of the gauge potential and D µ = ∂ µ + iA µ is the covariant derivative (µ = 0, 1, 2). The initial value problem, wellposedness, global existence and blow-up, scattering, etc. have been considered in [4,12,14,[23][24][25] for the case p = 3. In particular Jackiw and Pi were able to find self-dual solitons deduced by static solutions of (CSS) transfomed by means of Galilean boost or conformal invariance. Since, as usual in Chern-Simons theory, problem (CSS) is invariant under the gauge transformation (1) φ → φe iχ , A µ → A µ − ∂ µ χ for any arbitrary C ∞ function χ : R × R 2 → R, we easily see that the definition of static solution, that is time-indipendent solution, makes sense once we have removed the gauge freedom. In [16] it has be done assuming the Coulomb gauge choice ∇ · A = 0 (here A = (A 1 , A 2 )), supplemented by large-distance fall-off requirements on the differential equations satisfied by A 0 , A 1 and A 2 (see [18]). In particular, we require that being this asymptotic behaviour physically relevant, as it is the reflection of the possible presence of, respectively, electric charges and magnetic monopoles.
In order to find standing waves, we introduce the following ansatz (2) φ(t, x) = u(|x|)e iωt , A 0 (t, x) = A 0 (|x|), where ω ∈ R is a given frequency and u is a radial real valued function that, with an abuse of notation, has to be meant as a one or two variables function according to the situation. In [6] the authors proved that (φ, A 0 , A 1 , A 2 ) solves (CSS) if we set h(r) = h u (r) = 1 2 r 0 su 2 (s) ds, r > 0, in the previous ansatz (2), with ξ ∈ R arbitrary, and u is a solution of the equation Therefore, given a standing wave solution we can consider, for any c ∈ R, the function χ(t) = c t and use the gauge invariance (1) to obtain the family of standing wave solutions u(x)e i(ω+c)t , ξ − c + +∞ |x| h u (s) s u 2 (s) ds, − x 2 |x| 2 h(|x|), x 1 |x| 2 h(|x|) c∈R which is characterized by the constant ω + ξ that results to be a gauge invariant. In order to differentiate and classify the solutions, as in [18] we fix the gauge freedom imposing the following decay at infinity condition on the potential A 0 (4) lim |x|→+∞ A 0 (x) = 0.
We point out that, assuming the square integrability of u (which, as we are going to show, means that the solution has a finite total charge), our ansatz, together with (4), is consistent with the Coulomb gauge choice ∇ · A = 0, supplemented by largedistance fall-off requirements (FO).
Static solutions of (CSS) deduced from (5) have been found only when p = 3 in [6]. In detail, in [6] the authors proved that when p = 3 solutions to (CSS) satisfying the ansatz (2) and which have a field of matter that is nowhere zero (in the sense that u > 0 everywhere) must be static and belong to a one-parameter family which can be explicitly described. In particular, it is quite interesting to observe that such solutions are real valued, differently from the complex valued static field of matter found in [16]. Both solutions found in [6] and those found in [16] have zero energy (see [6, sec.5] and [18, sec.4]).
When p > 1, p = 3, equation (5) has been approached by variational methods looking for non-static solutions of (CSS) with ω > 0. Indeed as showed in [6], the equation (5) is nonlocal and it corresponds to the Euler-Lagrange equation of the functional I ω : : u is radially symmetric}. Observe that I ω presents a competition between the nonlocal term and the local nonlinearity of power-type.
When p > 3, in [6] the authors showed that I ω is unbounded from below and exhibits a mountain-pass geometry. However the existence of non-static solutions is not so direct, since for p ∈ (3, 5) the Palais-Smale condition is not known to hold. This problem is bypassed by using a constrained minimization taking into account the Nehari and Pohozaev identities. Up to our knowledge, there is no information about the sign of the energy of these solutions.
Finally, non-static solutions of (CSS) deduced from (5) are found for p ∈ (1, 3) in [6] as minimizers on a L 2 -sphere: here the gauge freedom is exploited to combine the value ω with a Lagrange multiplier, generating a family of non-static, not gauge equivalent solutions which do not in general satify the large-distance falling-off condition. Later, the result for p ∈ (1, 3) has been extended in [27] by investigating the geometry of I ω . Through a careful analysis for a limit equation, the authors showed that there exist 0 < ω 0 <ω <ω such that if ω >ω, the unique solutions to (5) are the trivial ones; if ω 0 < ω <ω, there are at least two positive solutions to (5); if 0 < ω < ω 0 , there is a positive solution to (5) for almost every ω. In particular, in [27] the authors proved that one of the two solutions found in the interval (ω 0 ,ω) has negative energy. We mention, moreover, [8,13] where multiplicity results are provided.
Inspired by the original paper by Jackiw and Pi [16] and the following literature, the aim of this paper is to study (CSS) looking for positive energy solutions. We recall the following result that can be easily deduced by the definition of energy and charge and direct computations is a solution of (CSS) satifying the ansatz (2). Then the energy and the charge of the solution are, respectively, Q(u) = 1 2 u 2 2 . By a comparison between (6) and (7), we see that E = I 0 , that is (5) corresponds to the Euler-Lagrange equation of the functional of the energy, when we are looking for static solutions.
From a mathematical point of view, the equation falls in that class which is usually called zero mass equations. A variational approach to it immediately presents several difficulties, starting with the definition of a suitable functional setting. Indeed, at least formally, solutions of (8) can be found as critical points of the functional E for which, differently from the case ω > 0, the space H 1 r (R 2 ) seems to be "too small" to apply the techniques of the calculus of variations in a usual way. On the other hand, the idea of introducing the functional framework as a specific Sobolev space endowed with a norm containing an expression of the nonlocal term (see for example Ruiz' approach in [29]) does not seem to be immediately applicable. In order to overcome this difficulty, we will make use of a perturbation argument as that presented inside [2], where the problem of defining the functional setting is due to the dimension N = 2, and recovered in [1] where another type of nonlocal equation is considered in the zero mass case.
Combining equation (8) with a condition at infinity, the problem reads as follows where u : R 2 → R is radially symmetric and p > 3.
As a first step, we have to clarify what we mean as solution of (P). We start with the solutions in the sense of distribution. Definition 1.2. We say that a measurable function u : Even if solutions in the sense of distribution have of course mathematical relevance, it is absolutely clear that they are in general too weak for having any phisical significance. Indeed observe that, without any global integrability information, we are not able to prevent the infinite energy phenomenon arising, as it is well known, in classical electrodynamics models.

the operators
Then we introduce a new setting and proceed with the definition of solution in a stronger sense. Definition 1.3. We define the sets H 2,4 (R 2 ) and H 2,4 r (R 2 ) as the completion respectively of C ∞ 0 (R 2 ) and of the set of radial functions in C ∞ 0 (R 2 ) with respect to the norm · 2,4 = ∇ · 2 + · 4 . Moreover, we denote by We will discuss the properties of H 2,4 (R 2 ) and H 2,4 r (R 2 ) in Section 2.
. We say that u is a weak solution of (P), if it satisfies (8) in a weak sense, namely there holds the following equality Finally we give the definition of classical solution.
are well defined and continuous in R 2 , u satisfies (8) pointwise and goes to 0 as x goes to ∞.
In Proposition 3.9, we will show that Definition 1.4 and Definition 1.5 coincide when the energy of the solution is finite, namely every u ∈ H is weak solution of (P) if and only if u is a classical solution of (P).
In the Appendix A, we will study sufficient integrability conditions on u for U u and V u to be well defined on R 2 .
We can state now our first result, which guarantees the existence of a static finite energy solution of system (CSS), satisfying (2) and (4). Theorem 1.6. For any p > 3, there exists u ∈ H classical positive solution of (P). As a consequence the quadruplet (φ, A 0 , A 1 , A 2 ) defined as in (2) for ω = 0 is in C 2 (R 2 ) × (C 1 (R 2 )) 3 and it is a static positive energy solution of (CSS) satisfying the following weak formulation of the large-distance fall-off requirement In the previous result, the positiveness of the energy is a consequence of Nehari and Pohozaev identities (see Proposition 4.3). We underline that the failure to use variational methods to find solutions causes non-trivial difficuties in deducing these identities. In particular, the fundamental Nehari and Pohozaev identities are not immediately available by means of direct computations based on standard arguments as in [6], but they both require quite tricky ad-hoc strategies.
These identities also play a key role in view of an analysis of the energy levels and in particular in order to estimate the zero-point energy of our system. The crucial question of establishing whether a ground state (at least limiting to static waves satisfying our ansatz) exists, translates into a minimum problem consisting in minimizing the functional of the energy in the set of solutions in H. Observe that, since by Theorem 1.6 the set (10) S := {u ∈ H \ {0} : u is a classical solution of (P) } is not empty, and by positiveness of energy the set {E(u) : u ∈ S} is bounded below, the minimizing problem makes sense. Actually, we will prove that the infimum is attained.
Theorem 1.7. For any p > 3, there exists a non-trivial radial ground state, namely there existsū ∈ S such that E(ū) = inf u∈S E(u).
As for the energy, the estimate of the total charge of our static wave presents analogous difficulties due to the particular zero mass structure of equation (8). In addition to evident problems related with the possibility that the total charge may be infinite, by (2) this fact is reflected in (FO) which is, in general, hard to verify. However, a priori considerations, based on a comparison argument, lead to the following (quite surprising) result Theorem 1.8. Assume that p > 9 and let u be the solution found in Theorem 1.6. Then u has finite total charge (that is u is in L 2 (R 2 )) and the corresponding quadruplet This paper is organized as follows.
In Section 2, we present the functional framework introducing some useful properties of the spaces H 2,4 (R 2 ) and H 2,4 r (R 2 ). Section 3 is devoted to the most of the proof of Theorem 1.6 (positive energy of our static solution is a consequence of Proposition 4.3 in Section 4). Following [1,2], as first step, roughly speaking we add a positive mass to the functional E; more precisely, for any ε > 0, we consider the following perturbed functional . By [6], it is easy to see that there exists a critical point u ε of I ε , for any ε > 0. The second step consists in studying the behaviour of the family {u ε } ε>0 , as ε ց 0. By concentration-compactness arguments, we show that, up to a subsequence, there exists u 0 ∈ H such that the family converges weakly to such u 0 in H 2,4 r (R 2 ), as ε ց 0. This will be enough to prove that, actually, u 0 is the desired solution.
In Section 4, we perform a deep analysis of the properties related with the energy of our static wave, and prove Theorem 1.7. An interesting consequence of this study and the result in [6] is the existence of a continuum of positive energy non-static standing waves stated in the Corollary 4.4. Moreover, the existence of a ground state will be obtained, again by a concentration-compactness argument, by means of Nehari and Pohozaev identities holding for (P). Finally, in Section 5 we show that, when p > 9, our static wave has finite total charge and Theorem 1.8 holds. The proof is based on a contradiction argument and a precise estimate of the decay at infinity of the solution will play a crucial role.
We conclude this introduction fixing some notations. For any τ 1, we denote by L τ (R 2 ) the usual Lebesgue spaces equipped by the standard norm · τ . In our estimates, we will frequently denote by C > 0, c > 0 fixed constants, that may change from line to line, but are always independent of the variable under consideration. Moreover, for any R > 0, we denote by B R the ball of R 2 centred in the origin with radius R. Finally the letters x, y indicate two-dimensional variables and r, s denote one-dimensional variables.

FUNCTIONAL FRAMEWORK
In this section we introduce the functional framework presenting some useful properties of the spaces H 2,4 (R 2 ) and H 2,4 r (R 2 ). The following inequality will play an essential role in our arguments. It is essentially already contained in [6], where it is proved for H 1 r (R 2 ) functions (see [6,Proposition 2.4]), but actually it holds also in H 2,4 r (R 2 ). The proof is based on the same density argument used in [6] after having showed its validity in C ∞ 0 (R 2 ) and therefore we omit it.
, the following inequality holds:

Remark 2.2.
We observe that the right hand side in inequality (11) could be also infinity, while it is surely finite if u ∈ H 2,4 r (R 2 ) with finite energy.
Proof. To prove that the normed space is reflexive it is sufficient to observe that · 2,4 is equivalent to . We want to prove that ∇u = U in the distributions sense, i.e. that for every Obviously, for every ϕ ∈ C ∞ 0 (R 2 ) and for every n ∈ N So it is sufficient to prove that Proposition 2.4. The space H 2,4 (R 2 ) corresponds to the set Moreover, if we define Proof. We have just to show that the functions in W 2,4 (R 2 ) can be approximate in the norm · 2,4 by functions in the same space, with compact support. The rest of the proof proceeds following standard arguments (see [20,Theorem 7.6]). Indeed, consider u ∈ W 2,4 (R 2 ) and let k : Certainly v M has a compact support and it is in where o M (1) denotes a vanishing function as M → +∞. Moreover and then we conclude.
In the following proposition we study the embedding's properties of H 2,4 (R 2 ).
Proof. Going back the proof of the Sobolev inequality, if u ∈ C ∞ 0 (R 2 ), one has See [5, (19), P. 280]. Let m 2. Applying (12) to |u| m−1 u, we get By the Young inequality, it follows that In (13), we first choose 2(m − 1) = 4, that is, m = 3. Thus from (13), we obtain Iterating this procedure with m = 3 + j for j ∈ N, and applying the interpolation inequality, one gets u q C u 2,4 for all u ∈ C ∞ 0 (R 2 ) and q ∈ [4, +∞). This completes the proof by a density argument.

Proposition 2.7.
For any τ ∈ 0, 1 4 , there exists C τ > 0 and R τ > 0 such that, for all u ∈ H 2,4 r (R 2 ), we have Proof. Let k ∈ 0, 1 2 and consider u a radial function in C ∞ 0 (R 2 ). For any r 0, we have that d dr The conclusion follows easily by density arguments.
The following compact embedding result holds.
Proof. Taking into account Proposition 2.5 and Proposition 2.7 the proof follows the same arguments as in [30, Compactness Lemma 2].

EXISTENCE OF A STATIC SOLUTION
First, we will study the following perturbed equation adding a positive small mass term to (P). More precisely, for any ε > 0 we consider Solutions of (P ε ) can be found as critical points of the functional which is well defined in classical Sobolev space : u is radially symmetric}. Following [6], we define a Pohozaev-Nehari type manifold and we have fixed α > 1 and such that 2 p−1 < α < 2 5−p , for p ∈ (3, 5) and α > 1 arbitrary, for p 5.
Moreover these minimum's levels are uniformly bounded by positive constants both from above and from below. Indeed we have Proposition 3.2. There exists C > 0 such that for any ε ∈ (0, 1) we have C m ε m 1 .
Proof. In the following, for every w ∈ H 1 r (R 2 ), we set Consider u ∈ M 1 and for any t > 0 assume the following notation u t := t α u(t·), where α is choosen as in the definition of J ε . If we denote by t ε > 0 the unique value for which J ε (u tε ) = 0 (see [6]), by simple computations we see that t ε < 1 for ε ∈ (0, 1). Now, we have that Passing to the infimum, we have m ε m 1 . Now suppose by contradiction that, for a suitable ε n → 0, it results that m εn → 0. For any n ∈ N, let u n ∈ M εn such that I εn (u n ) = m εn . Then we have that (14) a(u n ) → 0 and c(u n ) → 0.
Since u n ∈ M εn , by Proposition 2.5 we have that, for suitable positive constants C 1 and C 2 , On the other hand, by (11) and taking into account that a(u n ) → 0, for a suitable constant C > 0, we obtain u n 2,4 = (a(u n )) 1 2 + u n 4 (a(u n ))   Inequalities (15) and (16) contradict (14).
As an immediate consequence of Proposition 3.2, we have In the following we fix a decreasing sequence {ε n } n which tends to zero as n → +∞.
Lemma 3.6. Dichotomy does not hold.
Proof. As usual, we perform a proof by contradiction assuming that, on the contrary, dichotomy holds. Define ρ n ∈ C 1 0 (R 2 , [0, 1]) radial such that, for any n 1, ρ n ≡ 1 in B Rn , ρ n ≡ 0 in B c 2Rn and sup x∈R 2 |∇ρ n (x)| 2 Rn . Moreover set v n = ρ n u n and w n = (1 − ρ n )u n , observing that v n , w n ∈ H 1 r (R 2 ). Now we proceed by steps.
1st step: we prove that, defined Ω n = {x ∈ R 2 : R n |x| 2R n }, we have and then we have proved (18) also for v n . The proof for w n is analogous. 2nd step: lim inf n G n (v n ) =m. Observe, indeed, that since h un = h vn in B Rn , we have (19) G n (v n ) ν n (B Rn ) ν 1 n (B Rn ) →m, Now, observe that, by the first step and considering that ν n ν 2 n , m = lim Since lim n ν 2 n (R 2 ) = m −m and Supp ν 2 n ⊂ B c 2Rn , we conclude that lim inf n G n (v n ) =m.
3rd step: conclusion. First of all observe that, since u n = v n + w n and both v n and w n are nonnegative, then by the first step (20) G n (u n ) G n (v n ) + G n (w n ) + o n (1).
Observe that, by step 1, For any n ∈ N, let t n , s n > 0 be the numbers, respectively, such that (v n ) tn ∈ M εn and (w n ) sn ∈ M εn . There are three possibilities. Case 1: up to a subsequence, J εn (v n ) 0. By simple computations we see that t n 1 and then we have which, for a large n 1, leads to a contradiction due to the fact that, by (17) and step 2, lim n m εn = m >m = lim inf n G n (v n ).
Case 2: up to a subsequence, J εn (w n ) 0. Then, proceeding as in the first case, by (19) and using (20), we have, for n sufficiently large, which, by (17), implies m = lim n G n (w n ). Then, passing to the limit in (20), we have which contradicts the result obtained in step 2.
Case 3: there exists n 0 1 such that for all n n 0 both J εn (v n ) > 0 and J εn (w n ) > 0. Then lim inf n t n 1 and, by (21), we also have that J εn (v n ) = o n (1). If lim inf n t n = 1, we can repeat the computations performed in the first case and get the contradiction. If lim inf n t n > 1, from and, as a consequence, also v n p+1 → 0 by Propositions 2.1 and 2.5. Of course, we get a contradiction since lim inf n G n (v n ) > 0 by step 2.
Proof of Proposition 3.4. By the previous two lemmas we conclude that concentration holds. Moreover, the symmetry property of the functions u n guarantees the boundedness of {ξ n } n . The next two propositions provide fundamental integrability properties related to the nonlocal terms. Proposition 3.7. There exists u 0 ∈ H 2,4 r (R 2 ) such that, up to a subsequence, u n ⇀ u 0 in H 2,4 (R 2 ) and moreover Proof. The existence of u 0 ∈ H 2,4 (R 2 ) is guaranteed by the fact that, since {G n (u n )} n is bounded, {u n } n is bounded in H 2,4 r (R 2 ) and then it possesses a weakly convergent subsequence by Proposition 2.3. We can assume that such a sequence, relabeled {u n } n , is such that u n → u 0 a.e. in R 2 (and then u 0 is radial and nonnegative) u n → u 0 in L q (B), for all B ⊂ R 2 bounded and q 1.
To prove (i), observe that, for any u ∈ L 4 (R 2 ) and for any x ∈ R 2 \ {0}, we have that Therefore, since u 0 ∈ L 4 (R 2 ) and {u n } n is bounded in L 4 (R 2 ), we have We prove (ii). First of all we show that, for all B ⊂ R 2 bounded, we have Indeed, since u n → u 0 in L 2 (B) for every B ⊂ R 2 bounded, we have that By (23), (25) and the dominated convergence theorem we obtain Hence we deduce that and we obtain (24). By contradiction, suppose now that hu 0 |x| u 0 / ∈ L 2 (R 2 ). Then, for every M 0, there exists R > 0 such that In particular, there exists R m > 0 such that where m is defined in (17). By (24) and (26) which leads to a contradiction comparing with (17).
Let us now prove that (22) holds. By Proposition 3.4, we know that for any δ > 0 there exists R δ > 0 such that uniformly for n 1 Of course we can assume R δ large enough to have also Then, by (24), we have and we conclude. The proof of (iii), follows immediately by (i) and (ii). Finally we prove (iv) showing that which implies also the continuity of V u 0 . Observe that This, together with (i), implies that Observe, moreover, that u 0 |x| ∈ L 2 (B c 1 ). Indeed, we have This, together with (ii), implies that Now (29) is a direct consequence of (30) and (31).
By (iii) of Proposition 3.7 we deduce that Indeed, let B a bounded domain in R 2 , then by (23), (25) and the dominated convergence theorem, we get Hence we deduce that Moreover, by (23), (27) and (28), we have that, for any δ > 0 there exists R δ > 0 such that, uniformly for n 1, and we conclude the proof of (32). Now we prove (ii). Observe that For R > 0, we have while, taking into account the inequality a 3 2 1 + a 2 that holds true for any a 0, due to (ii) of Proposition 3.7. We deduce, therefore, that hu 0 Moreover, observe that, for any R > 0, Now, B 1 n → 0 by compact embedding in bounded domain and a proper application of Hölder inequality, whereas B 2 n and B 3 n go to zero by dominated convergence, again using properly the Hölder inequality (the scheme of the proof is similar to that used to obtain (22)). As to B 4 n , observe that by Proposition 3.4, for δ > 0 we can take R > 0 such that uniformly for n 1. Since for every a 0 we know that a 3 2 1 + a 2 , by Holder and (33), Finally we prove that, for R large enough, B 5 is less then δ arguing as for B 4 n and taking into account that hu 0 As to (iii), observe that we only have to prove that we can apply Fubini-Tonelli Theorem to the function f : It is easy to see that f is measurable in R 4 endowed with the product measure of R 2 -Lebesgue measures. Moreover, denoted by g(x) := R 2 f (x, y) dy and byg(x) : u 0 (y)|v(y)| dy dx < +∞ by (ii). Then, by Fubini-Tonelli Theorem, for almost every y ∈ R 2 there exists k(y) := It is easy to check that this corresponds exactly to what we claimed in (iii).
Now we can prove Theorem 1.6, except the positivity of the energy of the solution, which will be a direct consequence of Proposition 4.3.
Proof of Theorem 1.6. By Proposition 3.1, for any n ∈ N, there exists u n ∈ H 1 r (R 2 ) such that u n > 0 and I ′ εn (u n ) = 0 in H −1 . Hence, for every v ∈ H 1 (R 2 ), we have that By Proposition 3.7 there exists u 0 ∈ H 2,4 r (R 2 ) such that, up to a subsequence, u n ⇀ u 0 in H 2,4 (R 2 ). It is immediate that u 0 0. Moreover R 2 ∇u n · ∇v dx → R 2 ∇u 0 · ∇v dx and, by boundedness of √ ε n u n in L 2 (R 2 ), we also deduce that ε n u n v dx √ ε n √ ε n u n 2 v 2 → 0.
By compact embedding of H 2,4 r (R 2 ) into L q (R 2 ) for q > 4 (see Proposition 2.8), we also have u p n → u p 0 in L p+1 p (R 2 ) and then By Proposition 3.8, we conclude that (9) holds, namely u 0 is a weak solution of (P). By (i) and (iv) of Proposition 3.7 and by [9, Theorem 8.8] we infer that u 0 ∈ W 2,2 loc (R 2 ) and so u 0 ∈ C(R 2 ). Observing that the conclusions of [6, Proposition 2.1] hold for u 0 , by bootstraps arguments, following again [9], we conclude that u ∈ C 2 (R 2 ) and u > 0 by the maximum principle. Keeping in mind that A 0 ∈ L ∞ (R 2 ) by Proposition 3.7, we can show that A i ∈ C 1 (R 2 ), for i = 0, 1, 2, arguing as in [6, Proposition 2.1]. Finally the potentials verify the weak formulation of the large-distance fall-off requirement by (i) and (iv) in Proposition 3.7.
We conclude this section showing that the definitions of weak solutions and classical solutions coincide for finite energy functions. More precisely the following holds. Proposition 3.9. Let u ∈ H. Then u is weak solution of (P) if and only if u is a classical solution of (P).
Proof. Observing that all the integrability conditions of Propositions 3.7 and 3.8 hold for functions belonging to H, then, arguing as in the last part of proof of Theorem 1.6, we conclude.

ENERGY OF STATIC SOLUTIONS
We now prove that any weak solution with finite energy in the sense of Definition 1.4 satisfies a Nehari type identity. We would like to remark that this fact cannot be deduced as a trivial consequence of (9) since, in general, we do not know if a weak solution is in H 1 (R 2 ). Moreover, while, in general, the Nehari identity is given by E ′ (u)[u] = 0, in our case, not only the weak solution is not found as a critical point of the functional but also the functional could be not well defined on the weak solution.
Proposition 4.1. Let u ∈ H be a weak solution of (P), then it satisfies the following Nehari type identity Proof. For any n ∈ N, let ψ n : Being ψ n u ∈ H 1 (R 2 ), for any n ∈ N, we have that Observe that, being u ∈ H 2,4 r (R 2 ), = o n (1), Analogously, being u with finite energy and u ∈ L p+1 (R 2 ), we have easily that Finally observe that, due to the fact that u has finite energy, arguing as in Proposition 3.8, we have that Therefore, using again the fact that u has finite energy, we have Now the conclusion follows by (35) together with (36), (37), and (38).
We now prove that each classical solution of (P) with finite energy satisfies a Pohozaev type identity. We point out that even if a similar identity is present also in [6], we have to provide a different proof since their arguments need the essential information that the solution belongs to L 2 (R 2 ). Hence a new and different strategy is necessary.

Proposition 4.2.
Let u ∈ H be a classical solution of (P), then u satisfies the following Pohozaev type identity Proof. Let u ∈ H be a classical solution of (P) and fix R > 0. Multiplying by ∇u · x and integrating by parts on B R we have Arguing as in [6], we infer that where o R (1) denotes a vanishing function as R → +∞. Observe that we cannot repeat the arguments of [6] to study also the remaining terms, because in their arguments it is essential the fact that u belongs to L 2 (R 2 ). Therefore, we use another approach which seems, actually, less involved than that of [6]. Integrating by parts, we have Being u with finite energy, as observed in [3], we have and so, by radial symmetry, Using again the fact that u has finite energy, by Fubini-Tonelli Theorem we deduce Hence, arguing as before, we have Finally, another immediate consequence of the fact that +∞ |x| hu(s) s u 2 (s) ds u 2 is in L 1 (R 2 ), we have that By this, considering a suitable diverging sequence {R n } n , we conclude taking into account (40), (41), (42), and (43).

Proposition 4.3. Every static finite energy solution of the form
Proof. By Theorem 1.6, we know that S is not empty. Now, if we compute E on S, we have and then, by the choice of α, for any p > 3, we have that inf u∈S E(u) 0. Assume by contradiction that, for a suitable sequence {u n } n in S, we have E(u n ) → 0, then, by (11), we deduce also that u n → 0 in H 2,4 (R 2 ).
Using again (11), we have, moreover, that C u n p+1 p+1 . Therefore, taking into account that u n 2,4 → 0 and by the continuous embedding H 2,4 (R 2 ) ֒→ L p+1 (R 2 ), we have that, for any n ∈ N large enough, As by-product of our results, we now prove the existence of positive energy nonstatic solution of (CSS) satisfying the ansatz (2) with sufficiently small frequency.

STATIC SOLUTIONS WITH FINITE CHARGE
In all this section we assume that p > 9 and we prove that, in this case, the solution found in Theorem 1.6 belongs to L 2 (R 2 ).
We fix a decreasing sequence {ε n } n which tends to zero as n → +∞ and, for any n 1, we define u n := u εn , where u εn is as in Proposition 3.1. By Proposition 3.3 we know that {u n } n is bounded and, up to a subsequence, weakly convergent in H 2,4 (R 2 ). Finally let u 0 ∈ H be the solution found in Theorem 1.6 as the weak limit of {u n } n in H 2,4 (R 2 ).

Remark 5.2.
Arguing as in the proof of Theorem 1.8, if u 0 2 > 16π 2 , then {u n } n is bounded in L 2 (R 2 ).

APPENDIX A.
By Hölder inequality it is easy to see that if u ∈ L 4 loc (R 2 ) radially symmetric, then the function In the following for a measurable function u : R 2 → R, we want to understand under which assumptions on u we have that Lemma A.2. If u ∈ L q (R 2 ) and is radially symmetric with q ∈ (2, 4) and u ∈ L τ loc (R 2 ) with τ ∈ (4, +∞), then V u is well defined in R 2 .