EQUATIONS DRIVEN BY MIXED LOCAL-NONLOCAL OPERATORS

. In this paper we prove existence of solutions to Schr¨odinger-Maxwell type systems involving mixed local-nonlocal operators. Two diﬀerent models are considered: classical Schr¨odinger-Maxwell equations and Schr¨odinger-Maxwell equations with a coercive potential


Introduction and main results
Since it was introduced in the 1950s, the nonlinear Schrödinger equation could be considered a versatile tool in many concrete applications.Indeed, its first appearances were strictly related to the study of superconductivity [15] and superfluidity [16].It was only later, during the 1960s, with the increasing of its physical importance, that many studies have been spreading in the physical mathematical community, however always connected with practical applications as the diffusion of optical beams [7] in a nonlinear medium.
Nowadays, many of the above quoted applications have become fundamental in many areas of Physics.One of the most interesting problems arising from this kind of studies regards the interaction between the nonlinear Schrödinger equations and the electromagnetic field governed by the Maxwell equations.Some of the models emerging from the system defined by the above type-equations have recently found applications in the electronic characterization of nanodevices [20,21].
From a more theoretical perspective, the basic interpretation of the Schrödinger-Maxwell type systems is provided by the interaction between the electromagnetic field (Maxwell equations) and the particle field (Schrödinger equation).In this work, we shall adopt such a point of view following the rigorous mathematical formalization firstly proposed by Benci and Fortunato [4].
For the sake of completeness (with the aim of providing a clearer explanation), we briefly retrace in the following lines the mathematical argument that has led to the already classic formulation of the Schrödinger-Maxwell equations commonly studied in literature 1 .A similar argument can be found also in D'Avenia [11].
The starting point is the well-known nonlinear Schrödinger equation: (1.1) with p > 2 and where ψ(x, t) : R 3 × R → C is the field describing a non-relativistic charged particle moving in the three dimensional space, denotes the reduced Planck constant, and m is the mass of the particle.
It is possible to interpret equation (1.1) as the Euler-Lagrange equation with respect to the action where and, for any z j = x j + iy j ∈ C, x j , y j ∈ R, with j = 1, 2, and z ∈ C We now introduce the electromagnetic field 2 (E, H), which can be described in terms of gauge potentials by using the first two Maxwell equations, namely Let us suppose that the electromagnetic field is not assigned 3 , so that the interaction between the fields ψ and (E, H) can be expressed by the rule of minimal coupling, i.e., by substituting the ordinary derivatives with the so-called Weyl covariant derivatives [14]: Under these hypotheses, let us consider the following action: in which the two terms L int and L emf represent, respectively, the Lagrangian density of ψ interacting with (E, H), and the Lagrangian density of the electromagnetic field, namely If we now take solutions of the form ψ(x, t) = u(x, t)e iS(x,t)/ , we immediately get The Euler-Lagrange equations with respect the functional S(u, S, φ, A) are Thus, equations (1.3), (1.4), and (1.5) become Note that one can interpret ρ as the charge density and j as the current density.In this view, equation (1.6) is actually the continuity equation, and equations (1.7) and (1.8) are two of the Maxwell equations (precisely Gauss and Ampère equations).
Finally, since we are interested in the study of standing waves in the electrostatic case, we take u = u(x), S = ωt, φ = φ(x), A = 0, with ω > 0.
This choice implies that equations (1.3) and (1.5) are identically satisfied, and the couple of equations (1.2) and (1.4) becomes the so-called Schrödinger-Maxwell equations: The mathematical formulation of the problem, exposed in the previous lines has a dual purpose.On the one hand, we want to highlight the physical value of such a model justifying the algebraic steps providing the physical interpretation.On the other hand, there is no doubt that these kinds of problems, arising from Physics, constitute a stimulating challenge for their intrinsic mathematical properties.
Indeed, many different questions can be addressed starting from the above equations, whether with a clear physical meaning or purely mathematical.This scenario is pretty common, especially in the interdisciplinary domains as the Mathematical Physics.For all these reasons, in this manuscript we try to combine two different approaches: one closely related to a physical model and one purely mathematical.This dual perspective could offer a useful framework for those tackling this type of problems from both a physical and a mathematical point of view.
More specifically, our plan is to replace the classical Laplace operator ∆ with mixed local-nonlocal operators, involving the fractional Laplacian.Without going into technical details, we recall that the introduction of the fractional Schrödinger equation due to Laskin [18] (who was inspired by the Feynman path integral approach to quantum mechanics) has paved the way to a prolific field of studies based on various approaches.
Among other, we want to mention two works regarding the search of ground state solutions: the first one by Secchi [24], who adopted a variational method based on minimization on the Nehari manifold, and the second one written by Ambrosio [2], who considered weaker assumptions (than the Ambrosetti-Rabinowitz condition) for cases where the potential is 1-periodic or is bounded.At the same time, our choice is also strongly motivated by an increasing interest in the study of fractional calculus per se, as well explained by the authors in [5] (see also [17,19] for further references).
Getting to the heart of the work, we shall deal with generalized Schrödinger-Maxwell (SM) type systems of the form (1.9) where > 0 is the reduced Planck constant, m > 0 is the mass of the particle, ω > 0, q ∈ {±1}, and p ∈ (2, 2 * ).Here 2 * denotes the classical Sobolev critical The operator L α is a mixed local-nonlocal one of the following form where α ∈ R, ∆ denotes the classical Laplacian, and (−∆) s , s ∈ (0, 1), denotes the fractional Laplacian, which we shall introduce in the sequel.
To simplify the exposition, we choose q = 1, that is we consider the (SM) type system (1.11) Indeed, if (u, Φ) is a solution of (1.11), then (u, −Φ) is also a solution of (1.9) with q = −1.
Before stating our main results, it seems appropriate to lay out a short survey of the existing literature on the topics and some of the motivations that lead us to focus on the generalization introduced in the system (1.11).
In the recent paper [5], the authors have generalized the Klein-Gordon-Maxwell type systems (KGM) to the setting of mixed local-nonlocal operators, where the nonlocal one is allowed to be nonpositive definite according to a real parameter.They provided a range of parameter values to ensure the existence of solitary waves in terms of Mountain Pass critical points for the associated energy functionals.Following the existing literature they considered two different classes of potentials: constant potentials and continuous, bounded from below, and coercive potentials.
This paper keeps the same spirit replacing the Klein-Gordon equation with the Schrödinger equation.Our aim is to continue on the research line opened by the already mentioned paper of Benci and Fortunato [4], which has been studied heavily alongside KGM.Indeed, at the beginning of the 2000s Coclite and Georgiev [8], following the original work of Benci and Fortunato, proved the existence of a sequence of radial solitary waves for these equations with a fixed L 2 norm and analyzed the asymptotic behavior and the smoothness of such solutions.
However, the above papers treated the linear case, while the majority of the results focused on the nonlinear case, for which important achievements have been obtained on existence, nonexistence, multiplicity, and stability.Specifically, in 2002, D'Avenia [11] proved the existence of non-radially symmetric wave solutions of nonlinear Schrödinger equation coupled with Maxwell equations.Two years later, D'Aprile and Mugnai approached to the SM systems considering the theory of Mountain Pass critical points for the associated energy functional, in order to show the existence of radially symmetric solitary waves [9] and, moreover, they obtain some non-existence results based on a suitable Pokhozhaev's identity [10].Other impressive existence and nonexistence results were provided by Ruiz [23] in 2006, in which he closed the gap of the previous works linked to the range of the parameter p.Since 2008, many authors faced the problems of SM type-systems with different kind of potentials.Standing out among the others the works of Azzollini and Pomponio [3], Chen and Tang [6], and Zhao and Zhao [26].
The purpose of the present manuscript is precisely to generalize this kind of results by considering the mixed local-nonlocal operator L α , introduced in (1.10).In a similar context, other recent papers studied the existence of solutions for analogous generalized system of equations.In particular, the case of nonlinear fractional Schrödinger-Maxwell systems has been addressed by Zhang, do Ó, and Squassina [25] through a perturbation approach in the subcritical and critical case.
We can now proceed with the statements of the main results of this paper.I. Existence results for the Schrödinger-Maxwell equations.We introduce the function α 0 : (0, 1) × (0, ∞) → (0, ∞), which is defined by α 0 (s, t) := s −s (1 − s) s−1 t 1−s for s ∈ (0, 1) and t ∈ (0, ∞), and we denote the Hilbert space (1.12) • If p ∈ (4, 6), then problem (1.11) admits infinitely many radially symmetric solutions II. Existence results for the Schrödinger-Maxwell equations involving a coercive potential.In the last part of the paper, motivated by the recent literature, we study the following variant of the Schrödinger-Maxwell system, involving coercive potentials: (1.14) Here which is trivially satisfied when lim Here the space of solutions for u is Clearly W trivially reduces to H 1 (R 3 ) in the main case.
The second main result of the paper is the following one.

Theorem 1.2. Assume the validity of conditions
The paper is organized as follows.In Section 2, we outline our main assumptions, notations, and the preliminary aided further to both cases (I) and (II).In Sections 3 and 4, we shall study the cases (I) and (II), respectively, providing the proofs of Theorem 1.1 and Theorem 1.2.

Assumptions, notations, and preliminary results
2.1.Functional setting.We recall that the Sobolev space H 1 (R 3 ) is defined by and it is a Hilbert space endowed with the norm ).We denote by F the Fourier transform, defined for all functions ϕ ∈ S(R 3 ) (the Schwartz space of rapidly decreasing smooth functions) by and then extended by density to the space of tempered distributions.By Plancharel theorem, Given any s ∈ (0, 1), the fractional Sobolev space H s (R 3 ) is equivalently defined by Section 3], and it is a Hilbert space when endowed with the norm ) by Plancharel Theorem, since for all u ∈ H 1 (R 3 ) we have Let (−∆) s u denotes the fractional Laplacian of u, which is defined via Fourier transform for functions ϕ ∈ S(R 3 ) by By Plancherel Theorem, we have In particular, for all u ∈ H 1 (R 3 ) and ε > 0, we have Therefore, the fractional Laplacian can be interpreted as an operator Remark 2.1.We recall that the fractional Sobolev space H s (R 3 ) can also be defined via the Gagliardo seminorm [ • ] s,2 as Indeed, we have where the constant C(s) is given by , see e.g.[12, Proposition 3.4 and Proposition 3.6].In particular, the fractional Laplacian can be defined for ϕ ∈ S(R 3 ) as where P.V. denotes the Cauchy principal value, that is and the constant C(s) is the one defined by (2.3).
For the reader's convenience, we recall the definition of the mixed local-nonlocal operator L α , α ∈ R, given in (1.10), that is Here ∆u denotes the classical Laplace operator, while (−∆) s u is the fractional Laplacian.We can then interpret L α as an operator to which we can naturally associate a bilinear form as follows.

2.2.
Preliminaries for the SM equations.Regarding problem (1.11), the space of solutions for u is H 1 (R 3 ).We recall that the embedding H 1 (R 3 ) ֒→ L p (R 3 ) is continuous for all p ∈ [2, 6], being 6 = 2 * the critical Sobolev exponent in dimension n = 3.In particular, for any p ∈ [2,6], there exists a constant C p > 0 such that (2.4) Instead, the space of solutions for Φ is the Hilbert space D 1,2 (R 3 ) introduced in (1.12), and since in the whole space R 3 the Poincaré inequality does not hold, we get In any case, D 1,2 (R 3 ) is continuously embedded into L 6 (R 3 ), i.e., there exists a constant C D > 0 such that We can now introduce the definition of weak solutions of (1.11).
To show that Definition 2.3 makes sense we state and prove the following result.
Lemma 2.4.The system is coherent, whether Proof.Let us show that all the terms in (2.6) and (2.7) are well defined for u, v ∈ H 1 (R 3 ) and Φ, φ ∈ D 1,2 (R 3 ).As observed before, the bilinear form B α is well defined and continuous on H 1 (R 3 ) × H 1 (R 3 ).Moreover, by Hölder inequality, we have < ∞, and ).On the other hand It is easy to see that a regular solution of (1.11) is actually a weak solution, according to Definition 2.3.As usual, a weak solution of (1.11) can be found by studying the critical points of the functional F : We remark that the functional and, for all u, v ∈ H 1 (R 3 ) and Φ, φ ∈ D 1,2 (R 3 ), we have and Unfortunately, even though it seems to be natural to work with the functional F , we are unable to endow the Hilbert space H 1 (R 3 ) × D 1,2 (R 3 ) with a norm suitable to apply the theory of critical points to F .Therefore, we look for another variational characterization of problem (1.11).
Lemma 2.5.For every u ∈ H 1 (R 3 ) there exists a unique Φ(u) ∈ D 1,2 (R 3 ), which is a solution of (2.7).Moreover, we have Proof.The proof of the existence and uniqueness of Φ(u) and of (i) and (ii) can be found in [9, Proposition 3.1].It remains to prove (iii).We recall that a function u : R where g(x) := Ox, with O orthogonal matrix.
We then fix u ∈ H 1 (R 3 ) radially symmetric and g ∈ O(3).By the chain rule, the change of variables formula, and by (2.7), we have ) and u is radially symmetric.
Therefore, by the uniqueness of solutions of (2.7), we get that that is, Φ(u) is radially symmetric.
Remark 2.6.The unique solution Φ(u) of (2.7) can be also directly computed by convolution with the Newton potential and it has the form As a consequence of Lemma 2.5, we can deduce some useful estimates for the solutions of (2.7).Fix u ∈ H 1 (R 3 ) and let Φ u := Φ(u) ∈ D 1,2 (R 3 ) be the unique solution of (2.7).Then, In particular, as a consequence of (2.5), (2.8) and Hölder inequality, we get , which gives (2.9) .
This allows us to introduce the following functional, as done in [9].
) be the unique solution of (2.7).We define the functional J : By the identity (2.8), we have Moreover, by standard arguments, the map u Hence, the functional J is Fréchet differentiable on H 1 (R 3 ) and ) is a weak solution of problem (1.11) if and only if Φ = Φ u and u is a critical points of J. Hence, in order to find a solution of problem (1.11) it is enough to find a critical point of J on H 1 (R 3 ).This is done be applying an equivariant version of the Mountain Pass Theorem (see Theorem 2.9 below), in the case 4 < p < 6, and the original Mountain Pass Theorem (see Theorem 2.10 below), in the case p = 4.In what follows, X denotes an infinite dimensional Banach space and f : X → R.
The following notion of compactness will be required in both cases.
Definition 2.8.Let f ∈ C 1 (X).We say that f satisfies the Palais-Smale condition, (P S) condition in short, if any sequence (u n ) n ⊂ X such that Theorem 2.9 ([22, Theorem 9.12]).Let f ∈ C 1 (X) be an even functional and such that f (0) = 0. Assume that X is decomposable as direct sum of two closed subspaces X = X 1 ⊕ X 2 , with dim X 1 < ∞.Suppose that: (i) there exist δ, ̺ > 0 such that where for any u ∈ Y with u X ≥ R; (iii) f satisfies the (P S) condition.Then f has an unbounded sequence of positive critical values.
(iii) f satisfies the (P S) condition.Then f has a positive critical value.
We notice that, also using Lemma 2.5-(ii), the functional J defined in (2.11) is even, J ∈ C 1 (H 1 (R 3 )) and J(0) = 0.In the next section we prove that a suitable restriction of J satisfies the assumptions of Theorem 2.9, when p ∈ (4, 6), and of Theorem 2.10, when p = 4.
In order to prove the the geometric condition (ii) of Theorem 2.10 for p = 4, it is convenient to introduce the following compact notation.Definition 2.11.Let λ > 1 and β, γ ∈ R be fixed.For every u ∈ L 2 (R 3 ) we define The identity (2.13) can be proved for all ϕ ∈ S(R 3 ), by using (2.1) and the change of variable formula.Then, it can be extended to all u ∈ L 2 (R 3 ) by density.

2.3.
Preliminaries for the SM equations with coercive potentials.As done in [5] for the Klein-Gordon-Maxwell equations with potential V , the space of solutions for u is defined as endowed with the norm while the space of solutions for Φ is the Hilbert space D 1,2 (R 3 ) introduced in (1.12).We recall the following result for W , whose proof follows from [5, Lemma 2.3] and [5,Lemma 4.1].Lemma 2.13.Assume (V 1 )-(V 3 ).Then W is a Hilbert space with respect to • W and the space C ∞ c (R 3 ) ⊂ W is dense in W .Moreover, the embedding W ֒→ L p (R 3 ) is continuous for all p ∈ [2,6] and compact for all p ∈ [2,6).
Similarly to Definition 2.3, we also introduce the notion of weak solutions of problem (1.14).
By the same arguments already used in Lemma 2.4, Definition 2.14 makes sense, and every regular solution of (1.14) is actually a weak solution, according to Definition 2.14.As done before for the SM equations, we look for weak solutions of (1.14) as critical points of the functional The functional F is Fréchet differentiable on W × D 1,2 (R 3 ) and for all u, v ∈ W and Φ, φ ∈ D 1,2 (R 3 ) we have We are again unable to endow the Hilbert space W ×D 1,2 (R 3 ) with a norm suitable to apply the theory of critical points to F. We then fix u ∈ W ⊂ H 1 (R 3 ) and consider the unique solution Φ u ∈ D 1,2 (R 3 ) of (2.7), given by Lemma 2.5.Then, ), and for φ = Φ u we get (2.8).Therefore, we can introduce the following functional.Definition 2.15.Fix any function u ∈ W , let Φ u ∈ D 1,2 (R 3 ) be the unique solution of (2.7).We define the functional J : W → R by (2.18) By the identity (2.8), we have and, by standard arguments, the map Hence, the functional J is Fréchet differentiable on W and Therefore, a pair (u, Φ) ∈ W × D 1,2 (R 3 ) is a weak solution of problem (1.14) if and only if Φ = Φ u and u is a critical points of J .Thus, to find a solution of problem (1.14), we search critical points of J on W , and this is done be applying Theorem 2.9, in the case 4 < p < 6, and Theorem 2.10, in the case p = 4. Indeed, by Lemma 2.5-(ii), the functional J : W → R defined in (2.18) is even, and it satisfies J ∈ C 1 (W ) and J (0) = 0.

The SM equations
To find critical points of the functional J defined in (2.11), we shall restrict it to the subspace of radial functions for any x ∈ R 3 }.This (standard) procedure is allowed by the following result.
The arguments of the proof of [5, Lemma 3.1] apply as well for the functional J defined in (2.11).

Lemma 3.2. Assume the validity of the assumptions of Theorem 1.1. Then
• when p ∈ (4, 6), the functional J satisfies (i) and (ii) of Theorem 2.9 in X = H 1 r (R 3 ), with X 1 = {0} and X 2 = X; • when p = 4, the functional J satisfies (i) and (ii) of Theorem 2.10 in X = H 1 r (R 3 ).Proof.We divide the proof in two steps.
Step 1.The first geometric condition in Theorems 2.9 and 2.10.
We claim that for every p ∈ [4, 6) there exist δ, ̺ > 0 such that inf J(S ̺ ) ≥ δ, where , for any u ∈ H 1 r (R 3 ) and ε > 0, we have where α − := max{−α, 0} denotes the negative part of α.Let us consider the following system The first inequality of (3.1) holds whenever while the second inequality of (3.1) leads us to Therefore, the system (3.1) is satisfied when either α − = 0 or α − > 0 and Since α satisfies (1.13), which implies Hence we get Therefore, by also using Lemma 2.5-(i) and (2.4), for any u ∈ S ̺ we deduce where the last inequality holds for Thus condition (i) is satisfied for both Theorems 2.9 and 2.10.
For example, if we assume the validity of the following system We then look for couples (β, γ) satisfying (3.4).
From the third inequality, we must take γ > 0. The second and the fifth inequalities imply which is satisfied for γ = 2β.This choice satisfies also the first and the forth inequalities, being s ∈ (0, 1).Notice that, by Lemma 2.12, we have where c 1 > 0 and c 2 > 0 are the two constants defined in (3.2).
Hence, by (3.5) and (3.6), we get ).Therefore, there exists a subsequence, not relabeled, and u ∈ H 1 r (R 3 ) such that (u n ) n converges to u weakly in H 1 r (R 3 ), strongly in L p (R 3 ) for any p ∈ (2, 6), and a.e. in R 3 .To conclude, we show that the convergence in H 1 r (R 3 ) turns out to be strong.
Since, by Lemma 2.13, the embedding W ֒→ L 2 (R 3 ) is continuous, dense, and compact, we can apply the spectral decomposition result given in [5,Proposition A.4].
Hence, there exists an increasing sequence (λ k ) k of eigenvalues of L α,V satisfying Moreover, for all k ∈ N, the eigenvalue λ k has finite multiplicity and there exists a sequence of eigenvectors (e k ) k ⊂ W , corresponding to (λ k ) k , which is an orthonormal basis of L 2 (R 3 ).We define the spaces and ue j = 0 for all j = 1, . . ., k − 1 for all k ≥ 2.
Then W is decomposable as the direct sum of these two closed subspace, that is as Let k 0 ∈ N be such that In view of (4.1) and (4.2), there exists a constant c 0 = c 0 (s, α, V 0 ) > 0 satisfying Indeed, for all u ∈ P k 0 , by (4.1) and (4.2) we have Assume the validity of the assumptions of Theorem 1.2.Then • when p ∈ (4, 6), the functional J satisfies (i) and (ii) of Theorem 2.9 in X = W , with X 1 = H k 0 and X 2 = P k 0 , where λ k 0 is given by (4.3); • when p = 4, the functional J satisfies (i) and (ii) of Theorem 2.10 in X = W .
Proof.We divide the proof in two steps.
Step 1.The first geometric condition in Theorems 2.9 and 2.10.We consider two cases.
Case 1.Let p ∈ (4, 6), and let us prove the validity of (i) in Theorem 2.9.By (2.18) and (4.4), for all u ∈ X 2 = P k 0 we have where C p > 0 is the constant satisfying (4.5) u p ≤ C p u W for all u ∈ W .
Indeed, by (2.2), for any u ∈ W and ε > 0 we have Let us consider the following system (4.6) As in the proof of Lemma 3.2, we derive that the system (4.6) is satisfied when either α − = 0 or α − > 0 and Since α satisfies (1.15), which implies Hence we get (4.7) Therefore, by using also Lemma 2.5-(i) and (4.5), for any u ∈ S ̺ , we have where the last inequality is given by eventually setting .
Therefore, condition (i) is satisfied also for Theorem 2.10.
For example, if we assume that (3.4) holds, then we derive that J (u λ,β,γ ) → −∞ as λ → ∞.The proof can then be completed exactly as the proof of Lemma 3.2.Lemma 4.2.Under the assumptions of Theorem 1.2, the functional J satisfies (iii) in Theorems 2.9 and 2.10.
Proof.Let us fix a (P S) sequence (u n ) n ⊂ W .Then, for all p ∈ [4, 6) we have We distinguish between two possible cases.
Case 1 Let p ∈ (4, 6).Then, by (4.1), where Assume, by contradiction, that u n W → ∞ and define w n := un un W for all n ∈ N. Since w n W = 1 for all n ∈ N, by Lemma 2.13 there exist a subsequence, not relabeled, and a function w ∈ W such that, as n → ∞, (w n ) n converges to w weakly in W and strongly in L p (R 3 ) for all p ∈ [2, 6).In particular, since as n → ∞, we have On the other hand, by (2.10) and (2.18), for all n ∈ N we have Case 2. Let p = 4.Then, V 0 > 0 and α satisfies (1.15).Therefore, by (4.7), we have Since (J (u n )) n is bounded in R and (J ′ (u n )) n is bounded in W ′ , being (u n ) n a (P S) sequence, there exist two positive constants K 1 , K 2 such that Hence, by (4.7) and (4.8), we get for any n ∈ N, which implies that (u n ) n is bounded in W .Therefore, there exist a subsequence, not relabeled, and u ∈ W such that (u n ) n converges to u weakly in W , strongly in L p (R 3 ) for any p ∈ [2, 6), and a.e. in R 3 .
To conclude we show that the convergence in W turns out to be strong.By (2.9), the sequence (Φ un ) n ⊂ D 1,2 (R 3 ) is bounded in D 1,2 (R 3 ).Moreover, by (4.1), we have (4.9) Since J ′ (u n ) → 0 in W ′ , u n ⇀ u in W , and u n → u in L 2 (R 3 ) as n → ∞, it follows that the first three terms in the right hand side of (4.9) converge to 0 as n → ∞.Moreover, as n → ∞ Hence the thesis follows.
We can now conclude this section by collecting all the results given in the previous lines in the proof of Theorem 1.2.
Proof of Theorem 1.2.The proof follows by Lemmas 4.1-4.2 and by Theorems 2.9 and 2.10.

Declarations
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Lemma 3 . 1 .
Under the assumptions of Theorem 1.1