Abstract
In this paper we prove the existence of a positive energy static solution for the Chern–Simons–Schrödinger system under a large-distance fall-off requirement on the gauge potentials. We are also interested in existence of ground state solutions.
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1 Introduction
The following Chern–Simons–Schrödinger system
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 \(t \in {\mathbb {R}}\), \(x=(x_1, x_2) \in {\mathbb {R}}^2\), \(\phi : {\mathbb {R}}\times {\mathbb {R}}^2 \rightarrow {\mathbb {C}}\) is the scalar field, \(A_\mu : {\mathbb {R}}\times {\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) are the components of the gauge potential and \(D_\mu = \partial _\mu + i A_\mu \) is the covariant derivative (\(\mu = 0,\ 1,\ 2\)).
The initial value problem, well-posedness, 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 (\(\mathcal {CSS}\)) transformed by means of Galilean boost or conformal invariance.
Since, as usual in Chern–Simons theory, problem (\(\mathcal {CSS}\)) is invariant under the gauge transformation
for any arbitrary \(C^\infty \) function \(\chi :{\mathbb {R}}\times {{\mathbb {R}}^2}\rightarrow {\mathbb {R}}\), we easily see that the definition of static solution, that is time-independent solution, makes sense once we have removed the gauge freedom. In [16] it has be done assuming the Coulomb gauge choice \(\nabla \cdot \mathbf{A}=0\) (here \(\mathbf{A}=(A_1,A_2)\)), 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.
The existence of standing waves for (\(\mathcal {CSS}\)) and general \(p>1\) has been studied in [6, 8, 13, 27, 28, 31, 32], whereas standing waves with a vortex point have been studied in [7, 19] (see also the review paper [26]).
In order to find standing waves, we introduce the following ansatz
where \(\omega \in {\mathbb {R}}\) 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 \((\phi , A_0,A_1,A_2)\) solves (\(\mathcal {CSS}\)) if we set
in the previous ansatz (2),
with \(\xi \in {\mathbb {R}}\) arbitrary, and u is a solution of the equation
Therefore, given a standing wave solution
we can consider, for any \(c\in {\mathbb {R}}\), the function \(\chi (t) = c\, t\) and use the gauge invariance (1) to obtain the family of standing wave solutions
which is characterized by the constant \(\omega + \xi \) 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 \(A_0\)
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 \(\nabla \cdot \mathbf{A} =0\), supplemented by large-distance fall-off requirements (\(\mathcal {FO}\)).
According to the above discussion, in what follows we will take \(\xi =0\) which is a necessary condition for (\(\mathcal {FO}\)) as it is assumed for example in [4, 18].
Equation (3), therefore, becomes
Observe that static solutions of (\(\mathcal {CSS}\)) having the form (2) are deduced from (5) for \(\omega =0\).
Static solutions of (\(\mathcal {CSS}\)) 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 (\(\mathcal {CSS}\)) 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 \(p>1\), \(p\ne 3\), Eq. (5) has been approached by variational methods looking for non-static solutions of (\(\mathcal {CSS}\)) with \(\omega >0\). Indeed as showed in [6], the Eq. (5) is nonlocal and it corresponds to the Euler–Lagrange equation of the functional \( I_{\omega }: H^1_r({{\mathbb {R}}^2})\rightarrow {\mathbb {R}}\),
where
Observe that \(I_{\omega }\) presents a competition between the nonlocal term and the local nonlinearity of power-type.
When \(p>3\), in [6] the authors showed that \(I_\omega \) is unbounded from below and exhibits a mountain-pass geometry. However the existence of non-static solutions is not so direct, since for \(p \in (3,5)\) 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 (\(\mathcal {CSS}\)) deduced from (5) are found for \(p\in (1,3)\) in [6] as minimizers on a \(L^2\)-sphere: here the gauge freedom is exploited to combine the value \(\omega \) 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 \(p \in (1,3)\) has been extended in [27] by investigating the geometry of \(I_\omega \). Through a careful analysis for a limit equation, the authors showed that there exist \(0< \omega _0< {\tilde{\omega }} < {{\bar{\omega }}}\) such that if \(\omega > {{\bar{\omega }}}\), the unique solutions to (5) are the trivial ones; if \(\omega _0< \omega < {\tilde{\omega }}\), there are at least two positive solutions to (5); if \(0< \omega < \omega _0\), there is a positive solution to (5) for almost every \(\omega \).
In particular, in [27] the authors proved that one of the two solutions found in the interval \((\omega _0, {\tilde{\omega }})\) 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 (\(\mathcal {CSS}\)) 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
Proposition 1.1
Assume that \((\phi , A_0, A_1, A_2)\) is a solution of (\(\mathcal {CSS}\)) satisfying the ansatz (2). Then the energy and the charge of the solution are, respectively,
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 \(\omega >0\), the space \(H^1_r({{\mathbb {R}}^2})\) 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 Eq. (8) with a condition at infinity, the problem reads as follows
where \(u:{\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) is radially symmetric and \(p>3\).
As a first step, we have to clarify what we mean as solution of (\(\mathcal {P}\)). We start with the solutions in the sense of distribution.
Definition 1.2
We say that a measurable function \(u:{{\mathbb {R}}^2}\rightarrow {\mathbb {R}}\) is a solution of (\(\mathcal {P}\)) in the sense of distribution if
-
1.
u is in \(L^p_{\mathrm{loc}}({{\mathbb {R}}^2})\),
-
2.
for every \(\varphi \in C_0^{\infty }({{\mathbb {R}}^2})\)
$$\begin{aligned} \frac{u(x)\varphi (x)}{|x|^2}\left( \int _{B_{|x|}}u^2 dy\right) ^2\in L^1({{\mathbb {R}}^2})\hbox { and }\frac{u^2}{|x|^2}\left( \int _{B_{|x|}}u^2 dy\right) \left( \int _{B_{|x|}}u\varphi \, dy\right) \in L^1({{\mathbb {R}}^2}), \end{aligned}$$ -
3.
the operators
$$\begin{aligned}&\varphi \in C_0^{\infty }({{\mathbb {R}}^2})\mapsto \int _{{{\mathbb {R}}^2}}\frac{u(x)\varphi (x)}{|x|^2}\left( \int _{B_{|x|}}u^2 dy\right) ^2\,dx\\&\varphi \in C_0^{\infty }({{\mathbb {R}}^2})\mapsto \int _{{{\mathbb {R}}^2}}\frac{u^2}{|x|^2}\left( \int _{B_{|x|}}u^2 dy\right) \left( \int _{B_{|x|}}u\varphi \, dy\right) \,dx \end{aligned}$$are in \({\mathcal {D}} '\),
-
4.
for every \(\varphi \in C_0^{\infty }({{\mathbb {R}}^2})\)
$$\begin{aligned} \int _{{{\mathbb {R}}^2}}-u\Delta \varphi \, dx + \int _{{{\mathbb {R}}^2}}\frac{u(x)\varphi (x)}{|x|^2}\left( \int _{B_{|x|}}u^2 dy\right) ^2\,dx\\ + \int _{{{\mathbb {R}}^2}}\frac{u^2}{|x|^2}\left( \int _{B_{|x|}}u^2 dy\right) \left( \int _{B_{|x|}}u\varphi \, dy\right) \,dx =\int _{{{\mathbb {R}}^2}}|u|^{p-1}u\varphi \,dx, \end{aligned}$$ -
5.
for every \(\delta >0\) the Lebesgue measure of the set \(\{x\in {{\mathbb {R}}^2}: |u(x)|\geqslant \delta \}\) is finite.
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 physical 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.
Then we introduce a new setting and proceed with the definition of solution in a stronger sense.
Definition 1.3
We define the sets \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\) and \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\) as the completion respectively of \(C_0^\infty ({{\mathbb {R}}^2})\) and of the set of radial functions in \(C_0^\infty ({{\mathbb {R}}^2})\) with respect to the norm \(\Vert \cdot \Vert _{2,4}=\Vert \nabla \cdot \Vert _2+\Vert \cdot \Vert _4\).
Moreover, we denote by
We will discuss the properties of \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\) and \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\) in Sect. 2.
Definition 1.4
Let \(u\in \mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\). We say that u is a weak solution of (\(\mathcal {P}\)), if it satisfies (8) in a weak sense, namely there holds the following equality
for all v in \(H^1({{\mathbb {R}}^2})\).
Finally we give the definition of classical solution.
Definition 1.5
A classical solution of (\(\mathcal {P}\)) is a radial function \(u\in C^2({{\mathbb {R}}^2})\) such that
and
are well defined and continuous in \({{\mathbb {R}}^2}\), u satisfies (8) pointwise and goes to 0 as x goes to \(\infty \).
In Proposition 3.9, we will show that Definitions 1.4 and 1.5 coincide when the energy of the solution is finite, namely every \(u\in \mathcal {H}\) is weak solution of (\(\mathcal {P}\)) if and only if u is a classical solution of (\(\mathcal {P}\)).
In the Appendix 1, we will study sufficient integrability conditions on u for \(U_u\) and \(V_u\) to be well defined on \({{\mathbb {R}}^2}\).
We can state now our first result, which guarantees the existence of a static finite energy solution of system (\(\mathcal {CSS}\)), satisfying (2) and (4).
Theorem 1.6
For any \(p>3\), there exists \(u\in \mathcal {H}\) classical positive solution of (\(\mathcal {P}\)).
As a consequence the quadruplet \((\phi , A_0, A_1,A_2)\) defined as in (2) for \(\omega =0\) is in \(C^2({{\mathbb {R}}^2})\times (C^1({{\mathbb {R}}^2}))^3\) and it is a static positive energy solution of (\(\mathcal {CSS}\)) 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 difficulties 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 \(\mathcal {H}\). Observe that, since by Theorem 1.6 the set
is not empty, and by positiveness of energy the set \(\{E(u): u\in \mathcal {S}\}\) 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 exists \({\bar{u}}\in \mathcal {S}\) such that
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 Eq. (8). In addition to evident problems related with the possibility that the total charge may be infinite, by (2) this fact is reflected in (\(\mathcal {FO}\)) 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 \(L^2({{\mathbb {R}}^2})\)) and the corresponding quadruplet \((\phi , A_0, A_1,A_2)\) is a positive energy static solution of (\(\mathcal {CSS}\)) satisfying (\(\mathcal {FO}\)) .
This paper is organized as follows.
In Sect. 2, we present the functional framework introducing some useful properties of the spaces \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\) and \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\).
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 Sect. 4). Following [1, 2], as first step, roughly speaking we add a positive mass to the functional E; more precisely, for any \(\varepsilon >0\), we consider the following perturbed functional
defined in \(H^1_r({{\mathbb {R}}^2})\). By [6], it is easy to see that there exists a critical point \(u_\varepsilon \) of \(I_\varepsilon \), for any \(\varepsilon >0\). The second step consists in studying the behaviour of the family \(\{u_\varepsilon \}_{\varepsilon >0}\), as \(\varepsilon \searrow 0\). By concentration-compactness arguments, we show that, up to a subsequence, there exists \(u_0\in \mathcal {H}\) such that the family converges weakly to such \(u_0\) in \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\), as \(\varepsilon \searrow 0\). This will be enough to prove that, actually, \(u_0\) is the desired solution.
In Sect. 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 (\(\mathcal {P}\)).
Finally, in Sect. 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 \(\tau \geqslant 1\), we denote by \(L^\tau ({\mathbb {R}}^2)\) the usual Lebesgue spaces equipped by the standard norm \(\Vert \cdot \Vert _{\tau }\). 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 \({{\mathbb {R}}^2}\) centred in the origin with radius R. Finally the letters x, y indicate two-dimensional variables and r, s denote one-dimensional variables.
2 Functional framework
In this section we introduce the functional framework presenting some useful properties of the spaces \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\) and \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\).
The following inequality will play an essential role in our arguments. It is essentially already contained in [6], where it is proved for \(H^1_r({{\mathbb {R}}^2})\) functions (see [6, Proposition 2.4]), but actually it holds also in \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\). The proof is based on the same density argument used in [6] after having showed its validity in \(C_0^\infty ({{\mathbb {R}}^2})\) and therefore we omit it.
Proposition 2.1
For any \(u \in \mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\), 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 \(u\in \mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\) with finite energy.
Proposition 2.3
\((\mathcal {H}^{2,4}({{\mathbb {R}}^2}),\Vert \cdot \Vert _{2,4})\) is a reflexive Banach space.
Proof
To prove that the normed space is reflexive it is sufficient to observe that \(\Vert \cdot \Vert _{2,4}\) is equivalent to \(\Vert \cdot \Vert _*=\sqrt{\Vert \nabla \cdot \Vert _2^2+\Vert \cdot \Vert _4^2}\) and \((\mathcal {H}^{2,4}({{\mathbb {R}}^2}),\Vert \cdot \Vert _*)\) is an uniformly convex normed space.
Now we prove it is complete. Let \(\{ u_n\}_n\) be a Cauchy sequence in \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\). Then \(\{ u_n\}_n\) is a Cauchy sequence in \(L^4({{\mathbb {R}}^2})\) and \(\{ \nabla u_n\}_n\) is a Cauchy sequence in \(L^2({{\mathbb {R}}^2})\). Since \(L^4({{\mathbb {R}}^2})\) is complete, there exists \(u \in L^4({{\mathbb {R}}^2})\) such that \(\lim _n u_n = u\) in \(L^4({{\mathbb {R}}^2})\). Since \(L^2({{\mathbb {R}}^2})\) is complete, then there exists \(\mathbf{U} \in L^2({{\mathbb {R}}^2})\) such that \(\lim _n \nabla u_n = \mathbf{U}\) in \(L^2({{\mathbb {R}}^2})\). We want to prove that \(\nabla u = \mathbf{U}\) in the distributions sense, i.e. that for every \(\varphi \in C_0^\infty ({{\mathbb {R}}^2})\)
Obviously, for every \(\varphi \in C_0^\infty ({{\mathbb {R}}^2})\) and for every \(n\in {\mathbb {N}}\)
So it is sufficient to prove that
Indeed, since \(\lim _n u_n = u\) in \(L^4({{\mathbb {R}}^2})\), then
while, since \(\lim _n \nabla u_n = \mathbf{U}\) in \(L^2({{\mathbb {R}}^2})\) then
\(\square \)
Proposition 2.4
The space \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\) corresponds to the set
Moreover, if we define
then \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})={{\mathcal {W}}}_r^{2,4}({{\mathbb {R}}^2})\).
Proof
We have just to show that the functions in \({{\mathcal {W}}}^{2,4}({{\mathbb {R}}^2})\) can be approximate in the norm \(\Vert \cdot \Vert _{2,4}\) 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 \(u\in {{\mathcal {W}}}^{2,4}({{\mathbb {R}}^2})\) and let \(k:{{\mathbb {R}}^2}\rightarrow [0,1]\) be a cut off smooth function such that \(k\equiv 1\) in \(|x|\leqslant 1\) and \(k\equiv 0\) in \(|x|\geqslant 2.\) For any \(M>0\), define \(v_M= k_M u \), where \(k_M(x)=k(x/M),\) and set \(A_M=\{x\in {{\mathbb {R}}^2}: M\leqslant |x|\leqslant 2M\}\). Certainly \(v_M\) has a compact support and it is in \(L^4({{\mathbb {R}}^2})\).
Moreover, since \(\nabla v_M= k_M\nabla u+u\nabla k_M\), of course \(\nabla v_M \in L^2({{\mathbb {R}}^2}).\) We easily have that
where \(o_M(1)\) denotes a vanishing function as \(M\rightarrow +\infty .\)
Moreover
and then we conclude. \(\square \)
In the following proposition we study the embedding’s properties of \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\).
Proposition 2.5
The space \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\) is continuously embedded into \(L^q({{\mathbb {R}}^2})\), for any \(q\in [4,+\infty )\).
Proof
Going back the proof of the Sobolev inequality, if \(u\in C_0^{\infty }({{\mathbb {R}}^2})\), one has
See [5, (19), P. 280]. Let \(m \geqslant 2\). 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 \(j \in {\mathbb {N}}\), and applying the interpolation inequality, one gets
This completes the proof by a density argument.
\(\square \)
Remark 2.6
It is easy to see that \(\mathcal {H}^{2,4}_{\mathrm{loc}}({{\mathbb {R}}^2})=H^{1,2}_{\mathrm{loc}}({{\mathbb {R}}^2})\) and so \(\mathcal {H}^{2,4}_{\mathrm{loc}}({{\mathbb {R}}^2})\) is compactly embedded into \(L^q_{\mathrm{loc}}({{\mathbb {R}}^2})\), for any \(q\in [1,+\infty )\).
We now introduce a new Strauss Radial Lemma (see [30]) in \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\).
Proposition 2.7
For any \(\tau \in \left( 0,\frac{1}{4}\right) \), there exists \(C_\tau >0\) and \(R_\tau >0\) such that, for all \(u\in \mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\), we have
Proof
Let \(k\in \left( 0,\frac{1}{2}\right) \) and consider u a radial function in \(C_0^\infty ({{\mathbb {R}}^2})\). For any \(r\geqslant 0\), we have that
Now, fix \(r\geqslant 1\) and integrate \(-\frac{d}{ds}\left( s^ku^2(s)\right) \) in the interval \([r,+\infty )\). We have
The conclusion follows easily by density arguments. \(\square \)
The following compact embedding result holds.
Proposition 2.8
The space \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\) is compactly embedded into \(L^q({{\mathbb {R}}^2})\), for any \(q\in (4,+\infty )\).
Proof
Taking into account Propositions 2.5 and 2.7 the proof follows the same arguments as in [30, Compactness Lemma 2]. \(\square \)
3 Existence of a static solution
First, we will study the following perturbed equation adding a positive small mass term to (\(\mathcal {P}\)). More precisely, for any \(\varepsilon >0\) we consider
Solutions of (\(\mathcal {P}_\varepsilon \)) can be found as critical points of the functional
which is well defined in classical Sobolev space
Following [6], we define a Pohozaev-Nehari type manifold
where
and we have fixed \(\alpha >1\) and such that \(\frac{2}{p-1}<\alpha <\frac{2}{5-p}\), for \(p\in (3,5)\) and \(\alpha >1\) arbitrary, for \(p\geqslant 5\).
We have the following
Proposition 3.1
( [6]) For any \(\varepsilon >0\), there exists \(u_\varepsilon \in H^1_r({{\mathbb {R}}^2})\) which is a positive solution of (\(\mathcal {P}_\varepsilon \)) and such that
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 \(\varepsilon \in (0,1)\) we have \(C\leqslant m_\varepsilon \leqslant m_1\).
Proof
In the following, for every \(w\in H^1_r({{\mathbb {R}}^2})\), we set
Consider \(u\in \mathcal {M}_1\) and for any \(t>0\) assume the following notation \(u_t:=t^\alpha u(t\cdot )\), where \(\alpha \) is chosen as in the definition of \(J_\varepsilon \). If we denote by \(t_\varepsilon >0\) the unique value for which \(J_\varepsilon (u_{t_\varepsilon })=0\) (see [6]), by simple computations we see that \(t_\varepsilon <1\) for \(\varepsilon \in (0,1)\). Now, we have that
Passing to the infimum, we have \(m_\varepsilon \leqslant m_1\).
Now suppose by contradiction that, for a suitable \(\varepsilon _n\rightarrow 0\), it results that \(m_{\varepsilon _n}\rightarrow 0\). For any \(n\in {\mathbb {N}}\), let \(u_n \in \mathcal {M}_{\varepsilon _n}\) such that \(I_{\varepsilon _n}(u_n)=m_{\varepsilon _n}\). Then we have that
Since \(u_n\in \mathcal {M}_{\varepsilon _n}\), 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 \(a(u_n)\rightarrow 0\), for a suitable constant \(C>0\), we obtain
Inequalities (15) and (16) contradict (14). \(\square \)
As an immediate consequence of Proposition 3.2, we have
Proposition 3.3
The family \(\{u_\varepsilon \}_{\varepsilon >0}\) is bounded in \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\).
In the following we fix a decreasing sequence \(\{\varepsilon _n\}_n\) which tends to zero as \(n \rightarrow +\infty \).
We define
observing that, for \(\delta >0\) small enough and
\(a_i>0\) for any \(i=1,\ldots ,4\).
For any \(n\geqslant 1\) define \(u_n:=u_{\varepsilon _n}\), where \(u_{\varepsilon _n}\) is as in Proposition 3.1,
for any measurable \(\Omega \subset {\mathbb {R}}^2\), and
for any \(u\in H^1_r({{\mathbb {R}}^2})\). Of course \(\nu _n({{\mathbb {R}}^2})=G_n(u_n)=I_{\varepsilon _n}(u_n)=m_{\varepsilon _n}=\inf _{u\in \mathcal {M}_{\varepsilon _n}}I_{\varepsilon _n}(u)\).
By Proposition 3.2, we assume that, up to a subsequence,
By [21, 22] there are three possibilities:
-
1.
concentration: there exists a sequence \(\{\xi _n\}_n\) in \({{\mathbb {R}}^2}\) with the following property: for any \(\epsilon > 0\), there exists \(r = r(\epsilon ) > 0\) such that
$$\begin{aligned} \nu _n(B_r(\xi _n))\geqslant c-\epsilon ; \end{aligned}$$ -
2.
vanishing: for all \(r > 0\) we have that
$$\begin{aligned} \lim _n \sup _{\xi \in {{\mathbb {R}}^2}} \nu _n(B_r(\xi ))=0; \end{aligned}$$ -
3.
dichotomy: there exist two sequences of positive measures \(\{\nu _n^1\}_n\) and \(\{\nu _n^2\}_n\), a positively diverging sequence of numbers \(\{R_n\}_n,\) and \({\tilde{m}} \in (0,m)\) such that
$$\begin{aligned}&0\leqslant \nu _n^1 + \nu _n^2\leqslant \nu _n,\quad \nu _n^1({{\mathbb {R}}^2})\rightarrow {\tilde{m}},\quad \nu _n^2({{\mathbb {R}}^2})\rightarrow m-{\tilde{m}} \\&\mathrm{Supp}\, \nu _n^1\subset B_{R_n},\quad \mathrm{Supp}\, \nu _n^2\subset B_{2R_n}^c. \end{aligned}$$
Proposition 3.4
Concentration holds and, moreover, the sequence \(\{\xi _n\}_n\) is bounded.
We preliminary prove the following two lemmas.
Lemma 3.5
Vanishing does not hold.
Proof
If vanishing held, then we would have that
Since \(p>3\), we have also that
Therefore, since by Proposition 3.3, the sequence \(\{u_n\}_n\) is bounded in \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\), by [22, Lemma I.1], we deduce that \(u_n\rightarrow 0 \) in \(L^{p+1}({{\mathbb {R}}^2})\), as \(n \rightarrow +\infty \), and so, being \(J_{\varepsilon _n}(u_n)=0\), also \(m_{\varepsilon _n}\rightarrow 0\), contradicting Proposition 3.2. \(\square \)
Lemma 3.6
Dichotomy does not hold.
Proof
As usual, we perform a proof by contradiction assuming that, on the contrary, dichotomy holds.
Define \(\rho _n\in C^1_0({{\mathbb {R}}^2},[0,1])\) radial such that, for any \(n\geqslant 1\), \(\rho _n\equiv 1\) in \(B_{R_n}\), \(\rho _n\equiv 0\) in \( B_{2R_n}^c\) and \(\sup _{x\in {{\mathbb {R}}^2}}|\nabla \rho _n(x)|\leqslant \frac{2}{R_n}\). Moreover set \(v_n=\rho _nu_n\) and \(w_n=(1-\rho _n)u_n\), observing that \(v_n, w_n\in H^1_r({{\mathbb {R}}^2})\).
Now we proceed by steps.
1st step: we prove that, defined \(\Omega _n=\{x\in {{\mathbb {R}}^2}: R_n\leqslant |x|\leqslant 2R_n\}\), we have
for \(z_n\) equal to \(u_n\), \(v_n\) and \(w_n\).
Indeed observe that
and then we deduce (18) for \(u_n\).
By simple computations
and then we have proved (18) also for \(v_n\). The proof for \(w_n\) is analogous.
2nd step: \(\liminf _n G_n(v_n)=\tilde{m}\).
Observe, indeed, that since \(h_{u_n}=h_{v_n}\) in \(B_{R_n}\), we have
Now, observe that, by the first step and considering that \(\nu _n\geqslant \nu _n^2\),
Since \(\lim _n\nu _n^2({{\mathbb {R}}^2})=m-{\tilde{m}}\) and \(\mathrm{Supp}\, \nu _n^2\subset B_{2R_n}^c\), we conclude that
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
Observe that, by step 1,
For any \(n\in {\mathbb {N}}\), let \(t_n, s_n>0\) be the numbers, respectively, such that \((v_n)_{t_n}\in \mathcal {M}_{\varepsilon _n}\) and \((w_n)_{s_n}\in \mathcal {M}_{\varepsilon _n}\).
There are three possibilities.
Case 1: up to a subsequence, \(J_{\varepsilon _n}(v_n)\leqslant 0\).
By simple computations we see that \(t_n\leqslant 1\) and then we have
which, for a large \(n\geqslant 1\), leads to a contradiction due to the fact that, by (17) and step 2,
Case 2: up to a subsequence, \(J_{\varepsilon _n}(w_n)\leqslant 0.\)
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 \(n_0\geqslant 1\) such that for all \(n\geqslant n_0\) both \(J_{\varepsilon _n}(v_n)>0\) and \(J_{\varepsilon _n}(w_n)>0\).
Then \(\liminf _nt_n\geqslant 1\) and, by (21), we also have that \(J_{\varepsilon _n}(v_n)=o_n(1)\).
If \( \liminf _n t_n = 1\), we can repeat the computations performed in the first case and get the contradiction. If \(\liminf _n t_n >1\), from
we deduce that
and, as a consequence, also \(\Vert v_n\Vert _{p+1}\rightarrow 0\) by Propositions 2.1 and 2.5 . Of course, we get a contradiction since \(\liminf _n G_n(v_n)>0\) by step 2. \(\square \)
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 \(\{\xi _n\}_n\). \(\square \)
The next two propositions provide fundamental integrability properties related to the nonlocal terms.
Proposition 3.7
There exists \(u_0\in \mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\) such that, up to a subsequence, \(u_n \rightharpoonup u_0\) in \( \mathcal {H}^{2,4}({{\mathbb {R}}^2})\) and moreover
-
(i)
\(\frac{h_{u_0}}{|x|}\in L^\infty ({{\mathbb {R}}^2})\);
-
(ii)
\(\frac{h_{u_0}}{|x|}u_0\in L^2({{\mathbb {R}}^2}),\) and
$$\begin{aligned} \frac{h_{u_n}}{|x|}u_n\rightarrow \frac{h_{u_0}}{|x|}u_0\qquad \hbox { in }L^2({{\mathbb {R}}^2}); \end{aligned}$$(22) -
(iii)
\(\frac{h^2_{u_0}}{|x|^2}u_0\in L^2({{\mathbb {R}}^2})\);
-
(iv)
\(V_{u_0}(x)=\displaystyle \int _{|x|}^{+\infty }\frac{h_{u_0}(s)}{s}u_0^2(s)\,ds\) is well defined and continuous in \({{\mathbb {R}}^2}\).
Proof
The existence of \(u_0\in \mathcal {H}^{2,4}({{\mathbb {R}}^2})\) is guaranteed by the fact that, since \(\{G_n(u_n)\}_n\) is bounded, \(\{u_n\}_n\) is bounded in \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\) and then it possesses a weakly convergent subsequence by Proposition 2.3.
We can assume that such a sequence, relabelled \(\{u_n\}_n\), is such that
To prove (i), observe that, for any \(u\in L^4({{\mathbb {R}}^2})\) and for any \(x\in {{\mathbb {R}}^2}{\setminus } \{0\}\), we have that
Therefore, since \(u_0\in L^4({{\mathbb {R}}^2})\) and \(\{u_n\}_n\) is bounded in \(L^4({{\mathbb {R}}^2})\), we have
We prove (ii). First of all we show that, for all \(B\subset {{\mathbb {R}}^2}\) bounded, we have
Indeed, since \(u_n\rightarrow u_0\) in \(L^2(B)\) for every \(B\subset {{\mathbb {R}}^2}\) 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 \(\frac{h_{u_0}}{|x|}u_0\notin L^2({{\mathbb {R}}^2})\). Then, for every \(M\geqslant 0\), 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), we get
which leads to a contradiction comparing with (17).
Let us now prove that (22) holds.
By Proposition 3.4, we know that for any \(\delta >0\) there exists \(R_\delta >0\) such that uniformly for \(n\geqslant 1\)
Of course we can assume \(R_\delta \) 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 \(\frac{u_0^2}{|x|}\in L^1(B_1)\). Indeed, we have
This, together with (i), implies that
Observe, moreover, that \(\frac{u_0}{|x|}\in L^2(B_1^c)\). Indeed, we have
This, together with (ii), implies that
Now (29) is a direct consequence of (30) and (31). \(\square \)
Proposition 3.8
For every \(v\in L^2({{\mathbb {R}}^2})\) we have
-
(i)
\(\displaystyle \int _{{{\mathbb {R}}^2}}\frac{h_{u_n}^2}{|x|^2}u_nv\,dx\rightarrow \int _{{{\mathbb {R}}^2}}\frac{h_{u_0}^2}{|x|^2}u_0v\,dx,\)
-
(ii)
\( \displaystyle \frac{h_{u_0}}{|x|^2}u_0^2\left( \int _{B_{|x|}}u_0 v \, dy \right) \in L^1({{\mathbb {R}}^2})\) and
$$\begin{aligned} \int _{{{\mathbb {R}}^2}}\frac{h_{u_n}}{|x|^2}u_n^2\left( \int _{B_{|x|}}u_n v \, dy \right) dx\rightarrow \int _{{{\mathbb {R}}^2}}\frac{h_{u_0}}{|x|^2}u_0^2\left( \int _{B_{|x|}}u_0 v \, dy \right) dx, \end{aligned}$$ -
(iii)
\(\displaystyle \left( \int _{|x|}^{+\infty }\frac{h_{u_0}(s)}{s}u_0^2(s)\,ds\right) u_0 \in L^2({{\mathbb {R}}^2})\) and
$$\begin{aligned} 2\pi \int _{{{\mathbb {R}}^2}}\left( \int _{|x|}^{+\infty }\frac{h_{u_0}(s)}{s}u_0^2(s)\,ds\right) u_0v\, dx=\int _{{{\mathbb {R}}^2}}\frac{h_{u_0}}{|x|^2}u_0^2\left( \int _{B_{|x|}}u_0 v \, dy\right) dx. \end{aligned}$$
Proof
Let \(v\in L^2({{\mathbb {R}}^2})\).
By (iii) of Proposition 3.7 we deduce that \( \frac{h_{u_0}^2}{|x|^2}u_0v\in L^1({{\mathbb {R}}^2})\). Moreover, we prove easily (i) if we show that
Indeed, let B a bounded domain in \({{\mathbb {R}}^2}\), 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 \(\delta >0\) there exists \(R_\delta >0\) such that, uniformly for \(n\geqslant 1\),
Therefore
and we conclude the proof of (32).
Now we prove (ii). Observe that
For \(R>0\), we have
while, taking into account the inequality \(a^{\frac{3}{2}}\leqslant 1 +a^2\) that holds true for any \(a\geqslant 0\),
due to (ii) of Proposition 3.7. We deduce, therefore, that \(\frac{h_{u_0}}{|x|^2}u_0^2\left( \int _{B_{|x|}}u_0 v \, dy\right) \in L^1({{\mathbb {R}}^2})\).
Moreover, observe that, for any \(R>0\),
Now, \(B_n^1\rightarrow 0\) by compact embedding in bounded domain and a proper application of Hölder inequality, whereas \(B_n^2\) and \(B_n^3\) 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 \(B_n^4\), observe that by Proposition 3.4, for \(\delta >0\) we can take \(R>0\) such that
uniformly for \(n\geqslant 1\). Since for every \(a\geqslant 0\) we know that \(a^{\frac{3}{2}}\leqslant 1 +a^2\), by Holder and (33),
Finally we prove that, for R large enough, \(B^5\) is less then \(\delta \) arguing as for \(B_n^4\) and taking into account that \(\frac{h_{u_0}}{|x|^2}u_0^2\left( \int _{B_{|x|}}u_0 |v|dy\right) \in L^1({{\mathbb {R}}^2})\).
As to (iii), observe that we only have to prove that we can apply Fubini-Tonelli Theorem to the function \(f:{{\mathbb {R}}^2}\times {{\mathbb {R}}^2}\rightarrow {\mathbb {R}}\), where for almost every \((x,y)\in {{\mathbb {R}}^2}\times {{\mathbb {R}}^2},\)
It is easy to see that f is measurable in \({\mathbb {R}}^4\) endowed with the product measure of \({{\mathbb {R}}^2}\)-Lebesgue measures.
Moreover, denoted by \(g(x):=\int _{{{\mathbb {R}}^2}}f(x,y)\,dy\) and by \({\tilde{g}}(x):=\int _{{{\mathbb {R}}^2}}|f(x,y)|\,dy\) we have
by (ii). Then, by Fubini-Tonelli Theorem, for almost every \(y\in {{\mathbb {R}}^2}\) there exists \(k(y):=\int _{{{\mathbb {R}}^2}}f(x,y)\, dx\). Moreover \(k(y)\in L^1({{\mathbb {R}}^2})\) and
It is easy to check that this corresponds exactly to what we claimed in (iii). \(\square \)
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 \(n\in {\mathbb {N}}\), there exists \(u_n \in H^1_r({{\mathbb {R}}^2})\) such that \(u_n>0\) and \(I_{\varepsilon _n}'(u_n)=0\) in \(H^{-1}\). Hence, for every \(v\in H^1({{\mathbb {R}}^2})\), we have that \(I_{\varepsilon _n}'(u_n)[v]=0\), namely
By Proposition 3.7 there exists \(u_0\in \mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\) such that, up to a subsequence, \(u_n\rightharpoonup u_0\) in \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\). Moreover, by Proposition 3.4 we know that \(u_0\) is nontrivial.
It is immediate that \(u_0\geqslant 0\). Moreover \(\int _{{{\mathbb {R}}^2}}\nabla u_n\cdot \nabla v\,dx\rightarrow \int _{{{\mathbb {R}}^2}}\nabla u_0\cdot \nabla v\,dx\) and, by boundedness of \(\sqrt{\varepsilon _n} u_n\) in \(L^2({{\mathbb {R}}^2})\), we also deduce that
By compact embedding of \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\) into \(L^q({{\mathbb {R}}^2})\) for \(q>4\) (see Proposition 2.8), we also have \(u_n^p\rightarrow u_0^p\) in \(L^{\frac{p+1}{p}}({{\mathbb {R}}^2})\) and then
By Proposition 3.8, we conclude that (9) holds, namely \(u_0\) is a weak solution of (\(\mathcal {P}\)). By (i) and (iv) of Proposition 3.7 and by [9, Theorem 8.8] we infer that \(u_0\in W^{2,2}_{\mathrm{loc}}({{\mathbb {R}}^2})\) and so \(u_0\in C({{\mathbb {R}}^2})\). Observing that the conclusions of [6, Proposition 2.1] hold for \(u_0\), by bootstraps arguments, following again [9], we conclude that \(u_0\in C^2({{\mathbb {R}}^2})\) and \(u_0>0\) by the maximum principle.
Keeping in mind that \(A_0\in L^\infty ({{\mathbb {R}}^2})\) by Proposition 3.7, we can show that \(A_i\in C^1({{\mathbb {R}}^2})\), 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. \(\square \)
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 \(u\in \mathcal {H}\). Then u is weak solution of (\(\mathcal {P}\)) if and only if u is a classical solution of (\(\mathcal {P}\)).
Proof
Observing that all the integrability conditions of Propositions 3.7 and 3.8 hold for functions belonging to \(\mathcal {H}\), then, arguing as in the last part of proof of Theorem 1.6, we conclude. \(\square \)
4 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 \(H^1({{\mathbb {R}}^2})\). 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 \(u\in \mathcal {H}\) be a weak solution of (\(\mathcal {P}\)), then it satisfies the following Nehari type identity
Proof
For any \(n\in {\mathbb {N}}\), let \(\psi _n :{{\mathbb {R}}^2}\rightarrow {\mathbb {R}}\), where
Being \(\psi _n u\in H^1({{\mathbb {R}}^2})\), for any \(n\in {\mathbb {N}}\), we have that
Observe that, being \(u\in \mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\),
where \(A_n:= B_{2n}{\setminus } B_n\).
Analogously, being u with finite energy and \(u\in L^{p+1}({{\mathbb {R}}^2})\), 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). \(\square \)
We now prove that each classical solution of (\(\mathcal {P}\)) 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 \(L^2({{\mathbb {R}}^2})\). Hence a new and different strategy is necessary.
Proposition 4.2
Let \(u\in \mathcal {H}\) be a classical solution of (\(\mathcal {P}\)), then u satisfies the following Pohozaev type identity
Proof
Let \(u\in \mathcal {H}\) be a classical solution of (\(\mathcal {P}\)) and fix \(R>0\). Multiplying by \(\nabla u \cdot x\) 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 \(R\rightarrow +\infty .\)
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 \(L^2({{\mathbb {R}}^2})\). 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 that \(\left( \int _{|x|}^{+\infty }\frac{h_u(s)}{s}u^2(s)\,ds\right) u^2\) is in \(L^1({{\mathbb {R}}^2})\), since
Hence, arguing as before, we have
Finally, another immediate consequence of the fact that \(\left( \int _{|x|}^{+\infty }\frac{h_u(s)}{s}u^2(s)\,ds\right) u^2\) is in \(L^1({{\mathbb {R}}^2})\), we have that
By this, considering a suitable diverging sequence \(\{R_n\}_n\), we conclude taking into account (40), (41), (42), and (43). \(\square \)
Recalling the definition of \(\mathcal {S}\) given in (10), observe that, by (34) and (39), any \(u\in \mathcal {S}\) satisfies
where we have fixed \(\alpha >1\) and such that \(\frac{2}{p-1}<\alpha <\frac{2}{5-p}\), for \(p\in (3,5)\) and \(\alpha >1\) arbitrary, for \(p\geqslant 5\). Moreover we have that the functional E is well defined in \(\mathcal {S}\).
Proposition 4.3
Every static finite energy solution of the form (2) generated by \(u\in \mathcal {S}\) has positive energy. Moreover we have that \(\inf _{u\in \mathcal {S}}E(u)>0\).
Proof
By Theorem 1.6, we know that \(\mathcal {S}\) is not empty.
Now, if we compute E on \(\mathcal {S}\), we have
and then, by the choice of \(\alpha \), for any \(p>3\), we have that \(\inf _{u\in \mathcal {S}}E(u)\geqslant 0\).
Assume by contradiction that, for a suitable sequence \(\{u_n\}_n\) in \(\mathcal {S}\), we have \(E(u_n)\rightarrow 0\), then, by (11), we deduce also that \(u_n\rightarrow 0\) in \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\).
Using again (11), we have, moreover, that
and then, since \(u_n\) satisfies (44), we have
Therefore, taking into account that \(\Vert u_n\Vert _{2,4}\rightarrow 0\) and by the continuous embedding \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\hookrightarrow L^{p+1}({{\mathbb {R}}^2})\), we have that, for any \(n\in {\mathbb {N}}\) large enough,
which contradicts the fact that \(u_n\rightarrow 0\) in \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\). \(\square \)
As by-product of our results, we now prove the existence of positive energy non-static solution of (\(\mathcal {CSS}\)) satisfying the ansatz (2) with sufficiently small frequency.
Corollary 4.4
There exists \(\omega _0>0\) such that, for all \(\omega \in (0,\omega _0)\), there exists \((\phi ,A_0,A_1,A_2)\), a positive energy non-static solution of (\(\mathcal {CSS}\)) satisfying the ansatz (2).
Proof
Suppose by contradiction that that there exists a decreasing sequence \(\{\omega _n\}_n\) which tends to zero as \(n \rightarrow +\infty \) and, for any \(n\geqslant 1\), we define \(u_n:=u_{\omega _n}\), where \(u_{\omega _n}\) is as in Proposition 3.1 and with \(E(u_n)\leqslant 0\). By Proposition 3.3 we infer that \(\{u_n\}_n\) is bounded in \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\) and there exists \(u_0\in \mathcal {H}\) the weak limit of \(\{u_n\}_n\) in \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\). Arguing as in the previous section we deduce that \(u_0\) is a solution of (\(\mathcal {P}\)) which has positive energy by Proposition 4.3 and such that the conclusions of Proposition 3.7 hold. Then, by the weak lower semicontinuity of the norm, by the compact embedding of \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\) into \(L^{p+1}({{\mathbb {R}}^2})\) and by (22), we have
reaching a contradiction. \(\square \)
Now we have all the tools to conclude the prove Theorem 1.7.
Proof of Theorem 1.7
Consider \(\{u_n\}_n\) a sequence in \(\mathcal {S}\) such that \(E(u_n)\rightarrow \inf _{u\in \mathcal {S}}E(u)\). By (45), the sequence is bounded in \(\mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\) and then there exists \({\bar{u}}\in \mathcal {H}^{2,4}({{\mathbb {R}}^2})\) such that, up to a subsequence, \(u_n\rightharpoonup {\bar{u}}\) in \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\) and
Of course \({\bar{u}}\in \mathcal {H}_r^{2,4}({{\mathbb {R}}^2})\).
Arguing as in Sect. 3, we can see that also the minimizing sequence \(\{u_n\}_n\) concentrates in the sense of [21, 22] and, arguing as in Propositions 3.7 and 3.8 , this implies that \({\bar{u}}\) is a classical solution of (\(\mathcal {P}\)) with finite energy and so it satisfies (44).
By (44) and (46), therefore, we have that
Since \(\int _{{{\mathbb {R}}^2}}\frac{h_{u_n}^2u_n^2}{|x|^2} dx\) is bounded, we can assume that, up to a subsequence, it is convergent.
We prove that
By (47) we have that \(u_n \rightarrow {\bar{u}}\) in \(L^2(B_{|x|})\), for all \(x\in {{\mathbb {R}}^2}\). This implies that
By (48), (51) and Fatou Lemma, we prove our claim (50).
Using the weak lower semicontinuity property of the norms, inequality (50), and formula (49), we obtain
By (50) we deduce that
and, again by (49), \(\lim _n\Vert \nabla u_n\Vert _2=\Vert \nabla {\bar{u}}\Vert _2\). Taking into account also (46), \(E(u_n)\rightarrow E({\bar{u}})\) and we conclude. \(\square \)
5 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 \(L^2({{\mathbb {R}}^2})\).
We fix a decreasing sequence \(\{\varepsilon _n\}_n\) which tends to zero as \(n \rightarrow +\infty \) and, for any \(n\geqslant 1\), we define \(u_n:=u_{\varepsilon _n}\), where \(u_{\varepsilon _n}\) 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 \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\). Finally let \(u_0\in \mathcal {H}\) be the solution found in Theorem 1.6 as the weak limit of \(\{u_n\}_n\) in \(\mathcal {H}^{2,4}({{\mathbb {R}}^2})\).
Proof of Theorem 1.8
We need only to prove that \(u_0\in L^2({{\mathbb {R}}^2})\): this and the Strauss radial Lemma [30] imply that \((\phi ,A_0,A_1,A_2)\) is a positive energy static solution of (\(\mathcal {CSS}\)) satisfying (\(\mathcal {FO}\)).
By contradiction, assume that \(u_0\notin L^2({{\mathbb {R}}^2})\). Then there exists \(R_\sigma >0\) such that \(\Vert u_0\Vert ^4_{L^2(B_{R_\sigma })}= \sigma > 16\pi ^2\).
Fix \(\sigma '\in (16\pi ^2,\sigma )\). Since \(u_n\rightarrow u_0\) in \(L^2_{\mathrm{loc}}({{\mathbb {R}}^2})\) up to a subsequence, we can assume that there exists \(n_0\in {\mathbb {N}}\) such that
By Proposition 2.7, there exists \(\tau \in \left( \frac{2}{p-1},\frac{1}{4}\right) \), \(C_\tau >0\) and \(R_\tau >0\) such that
for \(r\geqslant R_\tau \) and any \(n\geqslant 1\). In particular, since \(\tau (p-1)>2\), taken \(\delta >0\) such that \(\gamma :=\frac{\sigma '}{16\pi ^2}-\delta >1\), there exists \(R_\tau '\) such that
for \(r\geqslant R'_h\) and any \(n\geqslant 1\). Up to replace \(R_\sigma \) with \(R'_\tau \) and \(\sigma \) with a larger number, we can assume \(R_\sigma =R'_\tau \).
Observe that, by (52) and (53), we have that
for \(|x|\geqslant R_\sigma \). Now consider the problem
which is solved by \(w_n(x)=u_n(R_\sigma )R_\sigma ^{\sqrt{\gamma }}|x|^{-\sqrt{\gamma }}\). Observe that
in \( H^{-1}({{\mathbb {R}}^2}{\setminus } \overline{B_{R_\sigma }})\) and, since \(u_n-w_n=0\) in \(\partial B_{R_\sigma }\) and \(u_n - w_n\rightarrow 0\) as \(|x| \rightarrow +\infty \), we have that \((u_n-w_n)^+\in H^{1}_0({{\mathbb {R}}^2}{\setminus } \overline{B_{R_\sigma }})\).
So, multiplying in (55) by \((u_n-w_n)^+\) and integrating, by (54) and the fact that \(u_n>0\) we have
and then, for \(|x|\geqslant R_\sigma \) and any \(n\geqslant n_0\), \(0\leqslant u_n\leqslant w_n\).
In conclusion, by Proposition 2.7,
By this and (52) we deduce that \(\{u_n\}_n\) is (up to a subsequence) bounded in \(L^2({{\mathbb {R}}^2})\) and so also in \(H^1({{\mathbb {R}}^2})\). Then, there exists \(u\in H^1({{\mathbb {R}}^2})\) and a subsequence of \(\{u_n\}_n\) such that \(u_n\rightharpoonup u\) in \(H^1({{\mathbb {R}}^2})\). Since we can assume that the same subsequence is such that \(u_n\rightarrow u_0\) a.e., we have \(u_0=u\in L^2({{\mathbb {R}}^2})\), and we obtain the contradiction.
\(\square \)
Remark 5.1
Using similar arguments as before and taking into account the Strauss Lemma [30], we have that for any \(\tau \in (0,a)\) there exists \(C_\tau >0\) and \(R_\tau >0\) such that
where \(a=\lim _{r\rightarrow +\infty }h_{u_0}^2(r)\).
Remark 5.2
Arguing as in the proof of Theorem 1.8, if \(\Vert u_0\Vert _2>16 \pi ^2\), then \(\{u_n\}_n\) is bounded in \(L^2({{\mathbb {R}}^2})\).
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Appendix A
Appendix A
By Hölder inequality it is easy to see that if \(u\in L^4_{\mathrm{loc}}({{\mathbb {R}}^2})\) radially symmetric, then the function
is well defined in \({{\mathbb {R}}^2}\).
In the following for a measurable function \(u:{{\mathbb {R}}^2}\rightarrow {\mathbb {R}}\), we want to understand under which assumptions on u we have that
is well defined.
Lemma 5.3
If \(u\in L^q({{\mathbb {R}}^2})\) and is radially symmetric with \(q\in (2,4)\), then \(V_u\) is well defined in \({{\mathbb {R}}^2}{\setminus }\{0\}\).
Proof
Fix \(x\ne 0\). Observe that for any \(s>|x|\), by Hölder inequality we have
Therefore, being \(q<4\), we have
\(\square \)
Lemma 5.4
If \(u\in L^q({{\mathbb {R}}^2})\) and is radially symmetric with \(q\in (2,4)\) and \(u\in L^\tau _{\mathrm{loc}}({{\mathbb {R}}^2})\) with \(\tau \in (4,+\infty )\), then \(V_u\) is well defined in \({{\mathbb {R}}^2}\).
Proof
By Lemma 5.3, we have to prove only that \(V_u(0)<+\infty \).
Observe that
By Lemma 5.3, we need to estimate only \(A_u\). Since
being \(\tau >4\), we have
\(\square \)
Remark 5.5
By [6], we already know that, if \(u\in L^2({{\mathbb {R}}^2})\cap L^\infty _ \mathrm{loc}({{\mathbb {R}}^2})\), then \(V_u\in L^\infty ({{\mathbb {R}}^2})\).
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Azzollini, A., Pomponio, A. Positive energy static solutions for the Chern–Simons–Schrödinger system under a large-distance fall-off requirement on the gauge potentials. Calc. Var. 60, 165 (2021). https://doi.org/10.1007/s00526-021-02031-4
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DOI: https://doi.org/10.1007/s00526-021-02031-4