NONLOCAL PLANAR SCHRÖDINGER-POISSON SYSTEMS IN THE FRACTIONAL SOBOLEV LIMITING CASE

. We study the nonlinear Schrödinger equation for the s − fractional p − Laplacian strongly coupled with the Poisson equation in dimension two and with p = 2 s , which is the limiting case for the embedding of the fractional Sobolev space W s,p ( R 2 ). We prove existence of solutions by means of a variational approximating procedure for an auxiliary Choquard equation in which the uniformly approximated sign-changing logarithmic kernel competes with the exponential nonlinearity. Qualitative properties of solutions such as symmetry and decay are also established by exploiting a suitable moving planes technique.


Introduction
In this paper we investigate existence and symmetry of positive solutions for the following strong coupling of the (s, 2 s )−fractional Schrödinger equation with the Poisson equation in the whole plane where s ∈ (0, 1), f is a nonnegative real function and F its primitive vanishing at zero.The nonlocal operator in the Schrödinger equation is the s−fractional p−Laplacian with p = 2 s , which is the so-called limiting case for the Sobolev embedding of the fractional Sobolev space (see Section 2) and it is well-known that this implies the nonlinearity may grow exponentially at infinity.
In the local case s = 1, (SP s ) reduces to the deeply studied Schrödinger-Poisson system (SP) which emerges in several fields of Physics: in electrostatics, it models the interaction of two identically charged particles; in quantum mechanics yields a model for the self-interaction of the wave function with its own gravitational field; it is also related with the Hartree model for crystals, see [BF] and references therein.In the higher-dimensional case N ≥ 3, and when f has a polynomial growth, there is an extensive literature about (SP), see the survey [MV] and references therein.A strategy to investigate (SP) is to move the attention to the corresponding Choquard equation where C N is an explicit positive constant depending on the dimension N, and then formally (see the discussion carried out in Section 3) inserting φ = φ u := I N * F (u) in the Schrödinger equation.Besides the effect of variables reduction, the advantage of this approach is that the Choquard equation (Ch) can be studied by variational techniques.Moreover, new interesting phenomena arise, such as the appearance of a lower-critical exponent in addition to the usual upper-critical exponent, as in the Sobolev case, see [CZ, CVZ] and references therein.
In dimension N = 2 just a few results are available.In this case the Riesz kernel is logarithmic (1.1) and therefore unbounded both from below and from above, which together with the fact that it is sign changing introduces a major difficulty with respect to the higher dimensional case.In fact, the functional associated to the corresponding Choquard equation (Ch) may be not well-defined in the natural space H 1 (R 2 ).When f (u) = u, the approach developed in [CW, DW, BCV,CW2], originating from the unpublished work of Stubbe [St], allows one to study the equation in a constrained subspace of H 1 (R 2 ), in which the log-convolution term is well-defined.However, in dimension two it is well-known that H 1 (R 2 ) ֒→ L q (Ω) for all q ≥ 1 but not in L ∞ (Ω) and that the maximal degree of summability for functions with membership in H 1 (R 2 ) is exponential, namely (e αu 2 − 1) dx stays bounded for all α > 0, see [R].Several extensions and refinements to this result have been proposed, among which the extension to the higher-order Sobolev spaces W m,p with m ∈ N, in the limiting case N = mp, see [RS].Exponential nonlinearities in (Ch) when N = 2, still mantaining the polynomial Riesz kernel and so loosing the connection to the Schrödinger-Poisson system, have been first considered in [ACTY].On the other hand, a special Schrödinger-Poisson system with logarithmic kernel and exponential nonlinearity, but not in gradient form as (SP), has been studied in [AF].The tuning of the two phenomena: logarithmic growth of the Riesz kernel and maximal exponential growth for system (SP) was first tackled in [CT].Here, the authors establish a proper functional setting by means of a log-weighted version of the Pohožaev-Trudinger inequality, so that the functional associated to (Ch) turns out to be well-defined.
An extension of these techniques to quasilinear Schrödinger-Poisson systems was established in [BCT].A different approach has been recently proposed in [LRZ, LRTZ, CDL].Instead of working directly on the logarithmic Choquard equation, the authors consider a family of approximating problems, each of them involving only a polynomial Riesz kernel, and prove that the limit of a sequence of approximating solutions converges to a solution to the original problem.This approach has the advantage of working in the H 1 (R 2 ) space context and one finds a posteriori that the logarithmic convolution term turns out to be well-defined at least on the solution.Let us finally mention that this method has been recently used in [Ro] to cover also the zero-mass case.
As a consequence of [PR, Z], it is meaningful to study system (SP s ) in presence of exponential nonlinearities.Fractional Schrödinger-Poisson systems and fractional Choquard equations have been recently studied in the subcritical regime, precisely in higher dimension N > sp and with polynomial nonlinearities, see [dASS, LM, HW, GRW].To the very best of our knowledge, the only results for fractional Choquard equations with exponential nonlinearities are obtained in [CDB] in the one dimensional case, in [BM] where the logarithmic kernel and the exponential nonlinearity are not combined, and in [YRTZ] where the Riesz kernel is polynomial; see also [YRCW] for related results.
Here we address both the problem of proving existence and establishing qualitative properties of solutions to the fractional planar system (SP s ), in the limiting case of logarithmic kernel and exponential nonlinearity.The first nontrivial step is proving that solutions of the system can be obtained by solving the related Choquard equation Before stating our main results, let us briefly introduce the functional setting and make precise the definition of solution we deal with for (SP s ) and (Ch s ).The (s, 2 s )−fractional Laplace operator is pointwisely defined as where PV stands for the Cauchy Principal Value which is well-defined for all x ∈ R 2 , for functions in C 1,1 loc (R 2 ) which enjoy suitable integrability conditions at infinity, see Section 3 and [CL,Lemma 5.2].The fractional Sobolev space W s, 2 s (R 2 ) is defined by .
The fractional Sobolev-Slobodeckij space W s, 2 s (R 2 ) is a uniformly convex Banach space with norm For γ > 0 the weighted Lebesgue space L γ (R 2 ) is defined as Definition 1.2 (Solution of (SP s )).We say that (u, φ) is a weak solution of the Schrödinger- for all ϕ ∈ W s, 2 s (R 2 ).Assumptions.Throughout the paper, we consider the following assumptions on the nonlinearity f : for all t > T (s), for some β > β 0 .The values of β 0 and T ( • ) are explicitly given in Lemma 5.5.

Main results.
We are now in the position to state our main results.The first one, proved in Section 3, concerns the relationship between the solutions of the Choquard equation (Ch s ) and the corresponding Schrödinger-Poisson system (SP s ): this is the fractional counterpart of [BCT, Theorem 2.1].In particular we make rigorous the fact that from a weak solution u of (Ch s ) one obtains a weak solution to (SP s ).
for some ν ∈ (0, 1), and Note that we do not prove that the two problems are equivalent, namely that they have the same set of solutions.Finding a proper functional setting in which this holds is still open, even for the local case (SP)-(Ch), see [BCT, Section 2].
The system (SP s ) is autonomous, thus we expect that positive solutions enjoy radial symmetry.In the next result, by exploiting the connection between (Ch s ) and (SP s ) established in Theorem 1.4, we show that this is indeed the case for solutions which are regular enough.The regularity will be needed to have a pointwisely defined (s, 2 s )-fractional Laplace operator and to be able to exploit a moving plane technique.This method has been firstly employed to (s, p)-fractional operators by Chen and Li in [CL] for the single Schrödinger equation; here, we further develop it to cover the case of systems.
Theorem 1.5 (Symmetry for (Ch s )).Suppose (f 1 )-(f 2 ) are satisfied, and let ) be a positive solution of (Ch s ).Then u is radially symmetric about the origin and monotone decreasing.
Observe that in Theorems 1.4 and 1.5 we only need that f is subcritical or critical in the sense of Proposition 2.1.In order to establish the existence of solutions for (Ch s ) -and therefore for (SP s ) by Theorem 1.4 -we require the full set of assumptions (f 1 )-(f 5 ).
Overview.In the next section we give motivations and discuss consequences of our assumptions together with some preliminaries.In Section 3 we study the relationship between solutions of the Choquard equation (Ch s ) and the related Schrödinger-Poisson system (SP s ) which is quite delicate as it depends on the notion of solution and regularity issues.Apparently there is no equivalence in general, even in the local case, see [BCT]; Theorem 1.4 is a step forward towards a complete understanding of this phenomenon.In Section 4 we prove the symmetry result of Theorem 1.5 by extending the moving-plane technique of [CL].Finally, in Section 5 we exploit all previous results to prove Theorem 1.6 by means of a variational approach together with a uniform asymptotic approximation technique.

Notation.
For R > 0 and x 0 ∈ R 2 we denote by B R (x 0 ) the ball of radius R and center x 0 .Given a set Ω ⊂ R 2 , we denote Ω c := R 2 \ Ω, and its characteristic function as χ Ω .The space of the infinitely differentiable functions which are compactly supported is denoted by for ν ∈ (0, 1) are usual spaces of Hölder continuous functions.The space S is the Schwartz space of rapidly decreasing functions and S ′ the dual space of tempered distributions.For q > 0 we define q! := q(q − 1) • • • (q − ⌊q⌋), where ⌊q⌋ denotes the largest integer strictly less than q; if q > 1 its conjugate Hölder exponent is q ′ := q q−1 .Finally, o n (1) denotes a vanishing real sequence as n → +∞.Hereafter, the letter C will be used to denote positive constants which are independent of relevant quantities and whose value may change from line to line.

Preliminaries
The Pohožaev-Trudinger inequality has been extended to the Sobolev fractional setting by Parini and Ruf [PR] for bounded domains and by Zhang [Z] for the whole space, results which we resume below.Let for t ≥ 0, where j2 Moreover, there exists α * ∈ (0, +∞) such that for 0 ≤ α < α * , Let us point out some immediate consequences of the assumptions (f 1 )-(f 5 ): Remark 2.3.
(i) From (f 1 ) and (f 2 ) it is easy to infer that the exists a constant C such that for all t > 0 ; (ii) Assumption (f 3 ) implies that f is monotone increasing.Moreover, d dt from which one infers Then (2.2) follows, by restricting to t > M ε ≥ N ε , where M ε is chosen such that which is bigger than or equal to 1 − s + τ , e.g. with the choice τ = τ (s) := s(1+s) 2+s .Hence the lower bound in (f 3 ) is satisfied on (0, t * ).Assumption (f 4 ) is also easily verified, since By the same computation we also note that the lower bound in (f 3 ) is trivially satisfied on (t . In order for f = F ′ to be continuous on R, one needs to impose a linear dependence between the constants A and B. Finally, (f 5 ) is also satisfied, provided one chooses A and B large enough.This is a working example, however, let us point out that these restrictions on constants can be relaxed.
For s ∈ (0, 1) and Ω ⊂ R 2 we define The subspace of radial functions in W s, 2 s (R 2 ) is defined as The following compactness result due to Lions [L] will be crucial in our analysis.
Let us also recall the well-known Hardy-Littlewood-Sobolev inequality, see [LL, Theorem 4.3], which will be frequently used throughout the paper.Lemma 2.5.(Hardy-Littlewood-Sobolev inequality) Let N ≥ 1, q, r > 1, and α ∈ (0, N) There exists a constant C = C(N, α, q, r) such that for all f ∈ L q (R N ) and h ∈ L r (R N ) one has The following technical lemma will be useful in the sequel, Lemma 2.6.Assume u ∈ W s, 2 s (R 2 ) and (f 1 )-(f 2 ) hold.Then for any 0 < κ ≤ 1 we have Observe that g(x, y) → 0 as |x| → +∞ for every y ∈ R 2 .Moreover, Hence, Lebesgue's theorem implies that  , MTV].This fact, together with a suitable polynomial estimate of the decay at infinity, yields a log-weighted L 1 -estimate for F (u), which is the crucial ingredient to prove Theorem 1.4.Then, we end up with C 0,ν loc (R 2 ) for some ν ∈ (0, 1), by standard regularity results.Throughout this section we assume which implies that u − ≡ 0. Hence, using the strong maximum principle for the p−fractional Laplacian [DQ1,Theorem 1.4] where in the last inequality we have used the fact that 1 |•| * F (u) is bounded, which follows arguing as in [ACTY,Lemma 4.1].For any L > 0 and γ > 1 define for t > 0. By Jensen's inequality it is possible to show that where (3.4) Recall that from (f 1 )-(f 2 ) we have that for any ε > 0, there exists Hence, from (3.4) we infer On the other hand, if u(x) > L, one has Hence, taking ε = 1 2C in (3.5) and using the Sobolev embedding , where S q is the best constant of the embedding inequality (see [CD] for related results).
Next, in order to prove a crucial decay estimate, let us recall a standard lemma from [CL, Lemma 5.1].
Then there exists a constant c 0 > 0 independent of t 1 and t 2 such that ) be a positive solution of the Choquard equation (Ch s ).Then there exist C, R > 0 for which where C 0 is a constant independent of x.Indeed, denoting by with C depending on u ∞ , from which (3.7) follows.Hence, there exists ) as a test function in (3.8), and noting that by homogeneity the inequality (3.9) holds also for w := C 1 w, we deduce that (3.10) Note that by Lemma 3.3 one has As a consequence we get ) be a positive solution of (Ch s ).Then ln(e + |x|)F (u) dx < +∞ .
Proof.Lemma 3.4 and (f 2 ) imply that there exists R > 0 for which We are now in the position to prove Theorem 1.4.
Proof of Theorem 1.4.We already proved that u ∈ L ∞ (R 2 ) in Lemma 3.2.To show that and Lemma 3.5.In order to show that (u, φ u ) solves (SP s ), we follow the approach of [BCT, Theorem 2.1].By [H, Lemma 2.3] the function Lemma 2.4] we know that all such solutions of −∆φ = F (u) in R 2 are of the form φ = v u + p with p polynomial of degree at most one.We aim to prove that φ u solves the Poisson equation in (SP s ) by showing that v u − φ u is constant.Indeed, by Lemma 3.5.Finally, we prove the logarithmic behaviour (1.3) of φ u at ∞ in the spirit of [CW, Proposition 2.3 (ii)].One has where l(x, y) := ln |x−y| |x| .Note that for |x| → +∞ one has l(x, y) → 0 for any fixed y This yields ), and we have that as |x| → +∞.Eventually we conclude that also |x−y|≤ 1 2 l(x, y)F (u(y)) dy → 0 as |x| → +∞, which together with (3.12) implies (1.3).
Finally, we prove that solutions of (Ch s ) are Hölder continuous.Following [DKP, IMS], for all measurable u : R 2 → R, we define its s, 2 s −non-local tail centered at x ∈ R 2 with radius R > 0 as for some ν ∈ (0, 1).
Proof.First, as in the proof of Lemma 3.2, one has (−∆) s 2 s u ≤ C weakly in R 2 for some constant C > 0. To prove the weak bound from below, take a test function ϕ ≥ 0 and estimate ) and by the decay estimate of Lemma 3.4 we have and analogously Then, by [IMS, Corollary 5.5] there exists a universal constant C and ν ∈ (0, 1) such that for all 4. Symmetry of positive solutions: proof of Theorem 1.5 As a consequence of Theorem 1.4, we have that solutions u of (Ch s ) (which are positive, see Remark 3.1) correspond to solutions (u, φ) of (SP s ) which enjoy the following Next we are going to prove Theorem 1.5 and for this purpose we rely on the method of moving planes, which has been adapted to the p-fractional context by Chen and Li in [CL], provided the operator (−∆) is pointwisley defined.For this reason we require Moreover, for any x ∈ R 2 , denote by x β the reflection of x with respect to ∂Σ β , that is Note that φ is a continuous function on R 2 by (f 1 ) and Lemma 3.4.Defining where where In what follows, for a function v we denote by v − := min{v, 0} its negative part.Notice that φ − is a nonpositive function with this convention.Actually we have the following lemma which is inspired by Proof.From the expression of φ β in (4.4) we know that u β ≥ 0 in Σ β implies φ β ≥ 0 in Σ β .In particular, φ β > 0 in Σ β if u β ≡ 0. Conversely, if φ β ≡ 0, then also u β ≡ 0 again using (4.4).We claim: Indeed, if not, there exists On the one hand, a direct computation using the definition of the (s, 2 s )-fractional operator yields (−∆) where ξ(y) lies between u β (x 0 ) − u β (y) and u(x 0 ) − u β (y), η(y) between u β (x 0 ) − u(y) and u(x 0 ) − u(y), and θ(y) between u β (x 0 ) − u β (y) and u(x 0 ) − u(y).Here we have used the mean value theorem in the third and fourth identities, and the fact that On the other hand, by assumption (f 1 ) and since w β (x 0 ) > 0, observe that which contradicts (4.6).
Let us next recall a maximum principle for the p−fractional laplacian and a key boundary estimate lemma for anti-symmetric functions, both established in [CL].These results play an important role in carrying out the method of moving planes.

Lemma 4.2. (A maximum principle for anti-symmetric functions).
Let Ω be a bounded domain in x ∈ R 2 .These conclusions hold for an unbounded region Ω if we further assume that Lemma 4.4.There exists β < 0 such that (4.9) Proof.Note that u satisfies (4.2).Suppose on the contrary that (4.9) is violated.Then, there exists β 0 < 0 such that for all β < β 0 one is always able to find x * ∈ Σ β for which Recalling that, by assumption (f 1 ), K β is non-negative function.Based on the above facts, we deduce from the expression of φ β in (4.4) that It is easy to check that c β is finite and by Lemma 3.4, and using (f 1 ) we obtain Moreover, by assumption (f 1 ), for any fixed ε > 0 and β sufficiently negative, (4.12) Thanks to (4.1) and the fact that h β is a non-negative function, for sufficiently negative β, Combining (4.11), (4.12), and (4.13) yields It follows from Lemma 3.3 that there exists c o > 0 such that (4.15) By using (4.14) and (4.15) in (4.2) and β sufficiently negative so that c β u(x * ) < c o , we get Next, similar computations as in (4.6) yield (4.17) is a strictly increasing function, and Proof of Theorem 1.5.So far, Lemma 4.4 provides a starting point to move the plane ∂Σ β .Now, let us move the plane to the right, as long as (4.9) holds, up to some limiting position.In particular, define (4.18) Let us divide the proof into two steps: Step 1.Let us prove that u is symmetric about the limiting plane Σ β 0 , that is, By contradiction, assume that (4.19) is false, that is to say, u β 0 (x) ≥ 0 and u β 0 (x) = 0 for some x ∈ Σ β 0 .Moreover, it follows from (4.2) that So, by the strong maximum principle (see Lemma 4.2) together with the fact that the map t → |t| 2 s −2 t is increasing and t → φh β 0 t is linear, we have u β 0 > 0 and hence by (4.4) also φ β 0 (x) > 0, ∀x ∈ Σ β 0 by Lemma 4.1.Now, according to the definition of β 0 , there exists a sequence β n ց β 0 , and Σ βn u βn < 0, and ∇u βn (x n ) = 0 .
Remind that u ∈ C 1 (R 2 ).Up to subsequence, we claim: where c o has appeared in (4.15).Assume by contradiction that there exists N > 0 such that Then by (4.23) and the definition of h β , one has for n > N Moreover, by assumption (f 1 ), for any fixed ε > 0, taking N large enough, Set M βn := {u ∈ Σ βn : u βn < 0} .We deduce from (4.4), Lemma 3.4, (4.10), and (4.24) that for n > N with N large enough On the other hand, as for (4.15), it follows from Lemma 3.3 that for n > N with N large enough ) .Combining (4.25), (4.26) and (4.27), we have Take ε > 0 sufficiently small such that εu(x n ) < 2, when n is large enough.Hence, from (4.28) and (4.2) we immediately deduce that for n > N with N large enough Moreover, arguing as in Lemma 4.4, one may deduce This is a contradiction.Therefore, (x n ) n must be bounded.The claim (4.22) holds true.Thus, by (4.21) we have (4.29) By the definitions of δ n , h βn and recalling that φ is a continuous function, it follows that where δ n := |β n − x n 1 |.By the definitions of K β and M β , Lemma 3.4, (4.10), and (f 1 )-(f 2 ), there exists C > 0 such that for R > 1 which implies, together with (4.31) and (4.2) for β = β n and x = x n , that lim inf This contradicts Lemma 4.3 and therefore (4.19) holds true.
Step 2. We next complete the proof showing that u is radially symmetric.Recalling Lemma 3.4 and the definition of β 0 in (4.18), we first have β 0 < ∞.It follows from Lemma 4.3 that β 0 > −∞.According to Lemma 4.1 and Step 1, we get u β 0 ≡ 0 and φ β 0 ≡ 0. By using the same argument for the second coordinate direction x 2 , we can find β 2 ∈ R such that u β 2 ≡ 0 and φ β 2 ≡ 0. Consider β = (β 0 , β 2 ), then ũ(x) := u(x−β) and φ(x) := φ(x−β) is a solution of equation (4.2).By invariance under translation, we may assume that ũ(x) = ũ(−x) and φ(x) := φ(−x) for x ∈ R 2 .So it is not hard to check that each symmetry hyperplane of ũ(x) and φ(x) contains the origin.Thus, repeating the above arguments for an arbitrary direction replacing the x 1 -coordinate direction, we deduce that ũ(x) and φ(x) are symmetric with respect to any hyperplane containing the origin, thus radially symmetric.Moreover, as a byproduct of the method, ũ(x) and φ(x) are also strictly decreasing in the distance from the symmetry center.

5.
Existence results for (Ch s ) by asymptotic approximation: proof of Theorem 1.6 As we mentioned in the Introduction, the applicability of variational methods to the planar Choquard equation (Ch s ) is not straightforward.Indeed (Ch s ) has, at least formally, a variational structure related to the energy functional (5.1) However, this energy functional is not well-defined on the natural Sobolev space W s, 2 s (R 2 ) because of the presence of the convolution term and the fact that the logarithm is unbounded both from below and from above.To overcome this difficulty, inspired by [LRTZ, CDL] we will use an approximation technique as follows.Set x ∈ R 2 , and consider the modified approximating problem with corresponding functional Unlike the original functional I, the power-type singularity in G α can be handled by the Hardy-Littlewood-Sobolev inequality (Lemma 2.5), and it is standard to prove that I α is well-defined and C 1 on W s, 2 s (R 2 ) with ).In order to retrieve compactness in light of Lemma 2.4, we will restrict the functional I α to the subspace W s, 2 s r (R 2 ) of radially symmetric functions.Note that in the previous section we established symmetry of all strong positive solution for (Ch s ) belonging to C 1,1 loc (R 2 ).Because of the singular behaviour of the (s, 2 s )-fractional Laplacian, it seems difficult to prove regularity for weak solutions of (5.2) belonging to W s, 2 s (R 2 ), so that the assumptions for the symmetry result in Theorem 1.5 can be fulfilled (see [BLS]).However, since the problem is autonomous, it is natural to restrict I α to the radial symmetric setting, by means of Palais' Principle of Symmetric Criticality (see [W]), for which critical points of ).In the sequel we will use some elementary estimates which we collect in the next lemma.
(ii) By Remark 2.3(i) there exists C > 0 such that Hence, by Hardy-Littlewood-Sobolev's inequality and Theorem 2.1, we have (5.5) for some p > 1 and 1 p + 1 p ′ = 1.By Theorem 2.1, we can find p > 1 such that the last factor is bounded independently of u provided u < 3 4 2−s 2 .From (5.5) we deduce that So, let u = ρ > 0 be sufficiently small, and recall the embedding As a consequence of Lemma 5.1, the mountain pass level where )) : γ(0) = 0, γ(1) = e} , turns out to be well defined.Moreover, from the Ekeland Variational Principle, the mountain pass geometry yields the existence of a Cerami sequence at level c α for any fixed α ∈ (0, 1], see e.g.[E].Namely, there exists (u as n → +∞.For the sake of a lighter notation, we will simply use u n := u α n .Remark 5.3.Observe from Lemma 5.2 that there exist two constants a, b > 0 independent of α such that a < c α < b. The next technical Lemma will be crucial in estimating the mountain pass level c α .Let R > 0 and w ∈ C(R 2 ) be such that Lemma 5.4.For all R > 0 and s ∈ (0, 1) Proof.We have w (5.7) Let us now compute the seminorm of w.Since w is radially symmetric, according to the equivalent formulations of the seminorm for radial functions established in [PR, Proposition 4.3], we have and let us next estimate I i , i = 1, . . .4, separately.Recalling that Again using (5.8) we compute the second term I 2 as The third term I 3 can be estimated as Finally, integrating by parts we get

By inspection one has
Eventually we get (5.9) [ w] The desired estimate for the norm of w is obtained combining (5.7) with (5.9).
The estimate of the mountain-pass level obtained in Lemma 5.5 enables us to obtain the following compactness results Lemma 5.6.Assume that (f 1 )- as well as Taking v = u n , we get (5.21) In order to prove the boundedness of (u n ) n , we introduce a suitable test function as follows where τ is the positive constant appearing in (f 3 ).It is easy to check that Remark 2.3(ii) and f (t) = 0 if and only if t ≤ 0. Furthermore, (5.22) by (f 4 ), where ξ n (x, y) ∈ R. Thus v n is well defined in W s, 2 s (R 2 ).Taking v = v n in (5.20) and recalling that f (t) = 0 and F (t) = 0 for any t ≤ 0, we infer Recalling (5.19), this yields (5.23) Similarly to (5.22), we also obtain Combining this and (5.23) we get As a consequence, we have by (5.5), where we note that the constant on the right is independent of n and α.Finally, from (5.19) and (5.21) we immediately obtain (5.18).
Remark 5.7.Thanks to the uniform boundedness of Cerami sequences of Lemma 5.6, from now on we can always suppose that Cerami sequences at level c α are nonnegative.Indeed, ) and thus, following the same argument used in Remark 3.1, one has as n → +∞ and therefore, setting u + n := max{u n , 0}, (u + n ) n is a Cerami sequence of I α at level c α , which we will denote simply by (u n ) n .
On the other hand, invoking the estimate (5.25) with q = 2 s and W = W (x, y) = − 1 > 0, we obtain dx dy .

Requiring
(5.32) Finally, from (5.31), (5.33) and (5.17), we obtain We can now estimate the norm of v n .From (5.29) we get having used Remark 2.3(ii) and the estimate of c α in Lemma 5.5.In order to prove the claim (5.27), we need to check that the constant on the right-hand side is strictly less than 1.This is equivalent to the following (5.34) Defining now assumption (f 3 ) guarantees that both conditions (5.32) and (5.34) are satisfied.
Next we aim at estimating the L 1 norm of a suitable power of (F (u n )) n by using (5.27).By (f 4 ), for any ε > 0 there exists t ε > 0 such that Hence, by (f 3 ) one has (5.36) which implies that for all x ∈ R 2 (5.37) where ε := ε s .Hence, by (f 1 )-(f 2 ) we have that for any given ε > 0, there exists C ε such that (5.38) In view of (5.36) and Remark 2.3(ii), v n ≥ τ s t ε if u n ≥ t ε , and then it follows from (5.38) that (5.39) Since v n 2 s ≤ γ s,τ + σ < 1 for n large enough and σ > 0 small enough by (5.27), the following holds for ε > 0 small enough and κ ∈ 1, 1 γs,τ .As a consequence, from (5.39) we obtain for some C independent of n and α.Similarly, one can also prove that for some C independent of n and α.Now we are in the position to prove the existence of a nontrivial critical point for the approximating functional I α , for α sufficiently small.
So far, we have proved that there exists ᾱ ∈ (0, 1) such that, for all α ∈ (0, ᾱ), the approximating functional I α has a positive critical point in W s, 2 s r (R 2 ).The last step consists in showing that actually the sequence (u α ) α converges, up to a subsequence, to a nontrivial critical point of the original functional I. Thanks to the a-priori bound (5.17), which is independent of α, we have that (u α ) α is uniformly bounded in W s, 2 s (R 2 ) for α ∈ (0, ᾱ) with (5.48) Hence, up to a subsequence, there exists u 0 ∈ W s, 2 s r (R 2 ) such that (5.49) Let us first prove some regularity results for u α .The strategy is similar to the one employed in Section 3 and thus here we just highlight the differences.
Lemma 5.12.Let α ∈ (0, ᾱ) and u α ∈ W s, 2 s (R 2 ) be a weak solution of (5.2), then there exists C > 0 independent of α such that u α ∞ ≤ C. Furthermore, u α ∈ C ν loc (R 2 ) for some ν ∈ (0, 1) and there exists R > 0 such that for |x| ≥ R, for some C > 0 uniformly for α ∈ (0, 4(ω−1) 3ω )., where p is sufficiently large.Noting that all the constants are independent of α, we can follow the argument of Lemma 3.2 and obtain u α ∞ ≤ C, with C independent of α.Finally, the decay estimate and Hölder's regularity can be obtained following step by step the proofs of Lemmas 3.4 and 3.6, paying only attention to the fact that G α substitutes the logarithmic Riesz kernel.
We are finally in a position to prove Theorem 1.6, namely the existence of a weak solution of (Ch s ).Next we prove that the sequence of solutions (u α ) α for the approximated problems (5.2) has a nontrivial accumulation point u 0 ∈ W s, 2 s (R 2 ) satisfying I ′ (u 0 ) = 0.
Proof of Theorem 1.6.Let us divide the proof into two steps.
Step 1.We show that u 0 ∈ W Step 2. To conclude the proof, we need to show that u 0 = 0 and that u α → u 0 in W s, 2 s (R 2 ).Assume on the contrary that u α ⇀ 0 in W s, 2 s (R 2 ), and so u α → 0 in L t (R 2 ) for t ∈ ( 2 s , +∞).Similarly to (5.47), we obtain R 2 f (u α )u α dx = o α (1).Hence, by Lemma 5.1 and Lemma 5.11, we have = o α (1) , which yields a contradiction.Note that in the last inequality we have used Remark 2.3(i) with (5.48) to estimate the first term, and (f 1 ) together with the monotonicity of F and the decay given by Lemma 5.12, to estimate the second term.Finally, similarly to (5.58) and (5.61), by Lemma 5.12 and the Lebesgue dominated convergence theorem, we have from which we conclude that u α → u 0 in W s, 2 s (R 2 ) as α → 0 + .

2 ( 2 s 2 T
11).By the computations of the norm of w carried out in Lemma 5.4, one has w < 1 (s)