On a logarithmic Hartree equation

: We study the existence of radially symmetric solutions for a nonlinear planar Schrödinger-Poisson system in presence of a superlinear reaction term which doesn’t satisfy the Ambrosetti-Rabinowitz condition. The system is re-written as a nonlinear Hartree equation with a logarithmic convolution term, and the existence of a positive and a negative solution is established via critical point theory.


Introduction
In recent past years many papers have been devoted to nding solutions of Schrödinger-Poisson systems of the form and often the main objects were standing wave solutions, i.e. solutions of the form ψ(x, t) = e −iωt u(x), ω ∈ R, so that (SP) reads −∆u + b(x)u + γwu =f (x, t), in R N × R, where b(x) = E(x) + ω andf (x, t) = e iωt f (x, t).
Due to the numerous several Physical applications, the most studied case is N = (or N ≥ ). On the other hand, the 1-dimensional case was considered in [1] when f = , and the existence of a unique ground state was established by decreasing symmetric rearrangements tools. The 2-dimensional case when f = was rst approached in [2], Section 6, only from a numerical point of view, while the rst rigorous existence result was given in [3] by using a shooting method for ODEs.
Moving down to lower dimensions, in particular to N = , introduces several complications, the rst important one is that in this case the Coulomb potential is not positive.
In [4] the authors studied the eigenvalue problem for the Schrödinger operator in a bounded domain of R , with electromagnetic eld E-H that is not assigned; in this case the unknowns are the wave function ψ(x, t) and the gauge potentials A(x, t), ϕ(x, t) related to E-H. In particular, they considered the problem in which A and ϕ do not depend on the time and ψ(x, t) = e −iωt u(x), with ω ∈ R and u real function. With these consideration they assume A = , and thus the system reduces to −∆u − ϕu = ωu, in Ω ⊂ R , Under this hypotheses, they proved the existence of a sequence of solution in a bounded domain of R . Later on, in [5] the authors considered the problem −∆u − ϕu − ωu = |u| p− u, in R , with p ∈ R + and they proved the existence of radially symmetric solutions in R , for p ∈ [ , ), while in [6] they showed the nonexistence of solutions for p ∈ ( , ] or p ∈ [ , ∞). After that, system (1.2) has been object of an intensive study, where generalizations of several type where considered; we refer to [7][8][9][10][11][12][13][14] for other references and improvements on this subject. All these works have been done in the whole of R , while the two dimensional case has remained for a long time a quite open eld of study. Indeed, a theoretical approach in dimension 2 is harder than in higher dimensions due to the lack of positivity of the Coulomb interaction term: precisely, the Coulomb potential is neither bounded from above nor from below.
However, in 2008 Stubbe [15] bypassed this problem giving a suitable variational framework for the problem and proving the existence of ground states, which is a positive spherically symmetric strictly decreasing function, by solving an appropriate minimization problem for the energy functional associated to the system (see also [16]). In some recent works, a local nonlinear terms of the form b|u| p− u, p > has been added; this kind of nonlinearity are frequently used in Schrödinger equations to model the interaction among particles, like recalled above (see [4]). Thus in [17], they studied a Schrödinger-Poisson system of the type −∆u + a(x)u − π ln |x| * |u| u = b|u| p− u in R , with b ≥ , p ≥ and a ∈ L ∞ (R ) and they proved that if p ≥ then the problem has a sequence of solution pairs ±un such that I(un) → ∞ as n → ∞.
In this work we are concerned with the integro-di erential equation where a > and f is a superlinear function. We refer to (P) as the logarithmic Choquard equation. Note that, if compared with (SP), we have chosen γ = − ; since γ represents the charge of the particle that we are studying, it means that we are considering electrons. In order to generalize and include the previous cases, on the reaction term f : R × R → R we assume that it is a Carathéodory function having superlinear growth and not verifying the Ambrosetti-Rabinowitz condition, from now on (AR).
Our main result has the following avour (see Theorem 3.1 for the precise statement): This work is organized as follows. In Section 2 we recall some useful de nitions and results that we shall use, we set up an appropriate variational framework and de ne the energy functional associated to the problem. Moreover, we give an extended results of the estimates given in the Strauss theorem.
In Section 3 we prove the well-posedness and the regularity of our functional and we give a Lemma that plays a fundamental role in the proof of the Cerami condition. Finally, we give the proof of the main existence theorem.

Background and Variational Framework
We provide a suitable variational framework for studying (P): indeed, the associated functional is not well de ned on the natural Sobolev space H (R ), and so we need some adjustments taken from [15], see also [17].
We rst recall an important result for L p -spaces.
Theorem 2.1 (Hardy-Littlewood-Sobolev's inequality, [18]). Let p, q > and < λ < N with p Let f ∈ L p (R N ) and g ∈ L q (R N ). Then there exists a sharp constant C (N, λ, p), independent of f and g, such that The sharp constant satis es In this case there is equality in (2.1) if and only if g ≡ cf with c constant and for some A ∈ R, ≠ γ ∈ R and x ∈ R N . Here |S N− | denotes the area of the unit sphere in R N .
First of all, we endow H (R ) with the scalar product (recall that a > is a constant) and we introduce the space with the norm de ned by Then, we de ne the symmetric bilinear forms since for all r > we have The de nitions above are restricted to measurable functions u, v : R → R such that the corresponding double integral is well de ned in the Lebesgue sense. Finally, for any measurable functions u : We note that, since we have by the Schwarz inequality for all u ∈ L (R ) and from (2.5) we have for all u ∈ L (R ). The energy functional I : X → R associated to (P) is De nition 1. We say that u ∈ X is a weak solution of (P) if thus if u is a critical point for I. Of course, these consideration are only formal, since, without any assumption on f , we cannot di erentiate I. In Section 3 we will give some su cient conditions for I to be of class C in X, while for the moment we continue with formal computations.
We have the following results, the second statement being new, as far as we know, and extending Strauss' Radial Lemma [19] to the space X and its N− dimensional version. Indeed, though later on we shall use only the compact embedding in dimension N = presented below, we can prove an asymptotic result which is valid in any space dimension. Since we believe that this property is of independent interest, we present our result in the general case. For this, let us introduce the sets Proposition 2.1. The following properties hold true: .
Proof. The compact embedding for N = is an application of the Riesz criterion (see [20,Theorem XIII.66]). Indeed, if S is a bounded subset of X, then S is bounded in L q (R ) for any q ∈ [ , ∞), as well. Moreover, for any R > and any u ∈ S we have for some C > . Finally, working as in [21,Theorem 9.16] we conclude.
As for the estimate in dimension N, let u ∈ Xr C ∞ c (R N ) and r > . We have Integrating from r to ∞ we obtain Hence The conclusion follows by density.
We nally introduce the "positive" and "negative" part of the reaction term, namely which are Carathéodory functions if f is, and F±(x, t) = t f±(x, s)ds; moreover, we set

The Existence Theorem
We assume the following hypotheses on the reaction term f : x ∈ R and for all ≤ s ≤ t or t ≤ s ≤ ; , where ζ ∈ L + (R ) andq > . The very last condition is generally a consequence of the usual Ambrosetti-Rabinowitz condition, which here should be assumed a priori, see [23].
We start proving Proof. We do the proof for I, the ones for I± being completely analogous. From hypothesis H(i) we have for some constant C > , so the associated functional is well-de ned. Now we observe that the Gâteaux derivative of B(u , u ) is for all φ ∈ X, so the functional I is the sum of C terms and we have the desired regularity follows.
Our purpose is to prove that both I+ and I− satisfy the assumptions of the mountain pass theorem. While the geometric structure is somehow standard and is obtained exploiting H(i) and H(v), the compactness condition is the delicate part: the lack of the Ambrosetti-Rabinowitz condition makes the bound on Cerami sequences more complicated, and, indeed, by using H(iv) we obtain only a bound in H (R ). Thus we move to radial functions and use Strauss' Lemma to exploit the compact embedding in L q (R ): thanks to the principle of symmetric criticality, a critical point for the functional constrained on the subset of radial functions is a free critical point, see [24]. This permits to recover the desired bound of Cerami sequences in X and nally prove that the Cerami condition holds. Hence, from now on, we consider I : Xr → R, where and we look for critical point for I| Xr . For the sake of simplicity we will continue to denote by I the functional I| Xr . Now we are ready to prove that the (C) d -condition holds. In order to do that, we rst give the following de nition De nition 2. We say that a sequence (un)n ⊂ Xr is a (C) d -sequence if I(un) → d and I (un) X * r + un Xr → as n → ∞.
We say that I satis es the (C) d -condition if any (C) d -sequence admits a convergent subsequence.
We prove that, under suitable hypotheses, a (C) d -sequence in Xr is bounded in H r (R ).

Lemma 3.1. Suppose hypotheses H(i), H(iii) and H(iv) hold and let (un)n ⊂ Xr be a (C) d -sequence for I+ (I− respectively). Then (un)n is bounded in H r (R ).
Proof. We do the proof for the I+, for I− being analogous. Let (un)n ⊂ Xr be a (C) d -sequence. In particular, and using (3.6) we obtain We assume by contradiction that (un)n is unbounded in H r (R ), then by passing to a subsequence, if necessary, we assume that u + n H r (R ) → ∞ as n → ∞.
We set vn = u + n u + n H r (R ) , n ≥ , so we may assume that, by Strauss' Theorem, To reach our goal we show that both v ≠ and v = lead to a contradiction. We start with the case v ≠ . We de ne the set Z(v) = x ∈ R : v(x) = ; then meas R \ Z(v) > and u + n (x) → ∞ as n → ∞ for a.e. x ∈ R \ Z(v). By H(iii) we have and by Fatou's Lemma we obtain ∈ ( , ) for all n ≥ n . Then, by (3.12), for all n ≥ n .

Now we observe that vn L (R )
→ as n → ∞ by Strauss' Theorem, and by (3.13) we have Being λ > arbitrary, we nally nd I(tn u + n ) → ∞ as n → ∞. (3.14) Since ≤ tn u + n ≤ u + n for all n ≥ , from H(iv) we have Thus, (3.14) and (3.16) imply that tn ∈ ( , ) for all n ≥ n ≥ . Hence, by (3.12) we obtain that = tn d dt for all n ≥ n . Replacing (3.18) in (3.15), we obtain for all n ≥ n . Again by H(iv) Using (3.17) the previous inequality reads as Thus, replacing (3.19) in (3.20), since B (tn u + n ) , (tn u + n ) ≤ B (u + n ) , (u + n ) being tn < , we have This last formula, together with (3.14), tells us that Comparing (3.8) and (3.21) we reach a contradiction. So (u + n )n is bounded in H r (R ).
We use this result to nally prove the Cerami compactness condition.
Since un → in L s (R ) for s ∈ ( , ∞), and p > , we have and then u ± n H r (R ) → , and also Hence, but I+(un) → d > , so we reach a contradiction. Thus (3.22) holds. This means that vanishing (see [27]) cannot occur. Moreover, since we use radial functions, dichotomy cannot take place, either. Hence, we can conclude that u ≠ . By [ Hence, from (3.24) un → u in Xr and so the (C) d -condition hold.
Now we are ready to produce two nontrivial solutions of (P) using the Mountain Pass Theorem.

Theorem 3.1. Under hypotheses H(i) -H(v), problem (P) has two nontrivial constant sign solutions.
Proof. We do the proof for the functional I+; for I− it is analogous. First, I+( ) = . By Proposition 3.1 we have the regularity of I+ and by Proposition 3.2 the (C) d -condition is veri ed. Now, takeũ as in H(v), t > and, following [10], we set u t (x) = t ũ(tx). Then and by H(v), lim t→+∞ I+(u t ) = −∞.
In order to complete the proof it only remains to show that I+(u) ≥ α ≥ with u = r, for some r > . By H(i) we have |f (x, t)| ≤ c(x) + d|t| q− and then |F(x, t)| ≤ c(x)|t| + d q |t| q a.e. x ∈ R and for all t ∈ R. (3.25) Hypothesis H(ii) says that for all ε > there exists δ = δ(ε) > such that for |t| < δ we have |f (x, t)| |t| ≤ ε a.e. x ∈ R , thus |F(x, t)| ≤ ε t a.e. x ∈ R and |t| ≤ δ. (3.26) On the other hand, when |t| ≥ δ Combining the inequality above with (3.26) we get that for a.e. x ∈ R and for all t ∈ R Hence, that is R F(x, u)dx ≤ εC t H r (R ) + C δ C t q H r (R ) .
We use this estimates on functional I+: Since u + H r (R ) ≤ u H r (R ) we get u H r (R ) .
Choosing ε ∈ , C and u H r (R ) = r, we have for some C > . We take r such that C − C + B δ C r q− > and so thus we have the Mountain Pass geometry, and we can apply [25,Theorem 5.40].