Schrodinger-Maxwell systems on compact Riemannian manifolds

In this paper we are focusing to the following Schr\"odinger-Maxwell system $(\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e})$: \[ \begin{cases} -\Delta_{g}u+\beta(x)u+eu\phi=\Psi(\lambda,x)f(u)&\mathrm{in}\ M -\Delta_{g}\phi+\phi=qu^{2}&\mathrm{\mathrm{in}\ M} \end{cases} \] where $(M,g)$ is a 3-dimensional compact Riemannian manifold without boundary, $e,q>0$ are positive numbers, $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, $\beta\in C^{\infty}(M)$ and $\Psi\in C^{\infty}(\mathbb{R}_{+}\times M)$ are positive functions. By various variational approaches, existence of multiple solutions of the problem $(\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e})$ is established.


Introduction and statement of the main results
We are concerned with the nonlinear Schrödinger-Maxwell system where (M, g) is a 3-dimensional compact Riemannian manifold without boundary, e, q > 0 are positive numbers, f : R → R is a continuous function, β ∈ C ∞ (M) and Ψ ∈ C ∞ (R + ×M) are positive functions. From physical point of view, the Schrödinger-Maxwell systems − 2 2m ∆u + ωu + euφ = f (x, u) in R 3 , −∆φ = 4πeu 2 in R 3 , (1.1) describe the statical behavior of a charged non-relativistic quantum mechanical particle interacting with the electromagnetic field. More precisely, the unknown terms u : R 3 → R and φ : R 3 → R are the fields associated to the particle and the electric potential, respectively, the nonlinear term f models the interaction between the particles and the coupled term φu concerns the interaction with the electric field. Note that the quantities m, e, ω and are the mass, charge, phase, and Planck's constant. In fact, system (1.1) comes from the evolutionary nonlinear Schrödinger equation by using a Lyapunov-Schmidt reduction.
The Schrödinger-Maxwell system (or its variants) has been the object of various investigations in the last two decades, the existence/non-existence of positive solutions, sign-changing solutions, ground states, radial and non-radial solutions and semi-classical states has been investigated by several authors. Without sake of completeness, we recall in the sequel some important contributions to the study of system (1.1). Benci and Fortunato [BF02] considered the case of f (x, s) = |s| p−2 s with p ∈ (4, 6) by proving the existence of infinitely many radial solutions for (1.1); their main step relies on the reduction of system (1.1) to the investigation of critical points of a "one-variable" energy functional associated with (1.1).
Based on the idea of Benci and Fortunato, under various growth assumptions on f further existence/multiplicity results can be found in Ambrosetti and Ruiz [AR08], Azzolini [Azz10], in [AdP10] Azzollini, d'Avenia and Pomponio were concerned with the existence of a positive radial solution to system (1.1) under the effect of a general nonlinear term, in [d'A02] the existence of a non radially symmetric solution was established when p ∈ (4, 6), by means of a Pohozaev-type identity, d'Aprile and Mugnai [DM04a,DM04b] proved the nonexistence of non-trivial solutions to system (1.1) whenever f ≡ 0 or f (x, s) = |s| p−2 s and p ∈ (0, 2] ∪ [6, ∞), the same authors proved the existence of a non-trivial radial solution to (1.1), for p ∈ [4.6). Other existence and multiplicity result can be found in the works of Cerami and Vaira [CV10], Kristály and Repovs [KR12], Ruiz [Rui06], Sun, Chen and Nieto [SCN12], Wang and Zhou [WZ07], and references therein.
In the last five years Schrödinger-Maxwell systems has been studied on n−dimensional compact or non-compact Riemannian manifolds (2 ≤ n ≤ 5) by Druet and Hebey [DH10], Farkas and Kristály [FK16], Hebey and Wei [HW13], Ghimenti and Micheletti [GM14b,GM14a] and Thizy [Thi16a,Thi16b]. More precisely, in the aforementioned papers various forms of the system has been considered, where (M, g) is a Riemannian manifold. The aim of this paper is threefold. First we consider the system (SM e Ψ(λ,·) ) with Ψ(λ, x) = λα(x), where α is a suitable function and we assume that f is a sublinear nonlinearity (see the assumptions (f 1 ) − (f 3 ) below). In this case we prove that if the parameter λ is small enough the system (SM e λ ) has only the trivial solution, while if λ is large enough then the system (SM e Ψ(λ,·) ) has at least two solutions, see Theorem 1.1. It is natural to ask what happens between this two threshold values. In this gap interval we have no information on the number of solutions (SM e Ψ(λ,·) ); in the case when q → 0 these two threshold values may be arbitrary close to each other. Similar bifurcation type result for a perturbed sublinear elliptic problem was obtained by Kristály, see [Kri12].
Second, we consider the system (SM λ Ψ(λ,·) ) with Ψ(λ, x) = λα(x)+µ 0 β(x), where α and β are suitable functions. In order to prove a new kind of multiplicity for the system (SM λ Ψ(λ,·) ) (i.e., e = λ), we show that certain properties of the nonlinearity, concerning the set of all global minima's, can be reflected to the energy functional associated to the problem, see Theorem 1.2.
Third, as a counterpart of Theorem 1.1 we will consider the system (SM e Ψ(λ,·) ) with Ψ(λ, x) = λ, and f here satisfies the so called Ambrosetti-Rabinowitz condition. This type of result is motivated by the result of Anello [Ane07b] and Ricceri [Ric00], where the authors studied the classical Ambrosetti -Rabinowitz problem, without the assumption lim t→0 f (t) t = 0, i.e., the authors proved that if the nonlinearity f satisfies the so called (AR) condition and a subcritical growth condition, then if λ is small enough the problem has at least two weak solutions in H 1 0 (Ω). In the sequel we present precisely our results. As we mentioned before, we first consider a continuous function f : [0, ∞) → R which verifies the following assumptions: 4F (s) 2s 2 + eqs 4 are well-defined and positive. Now, we are in the position to state the first result of the paper. In order to do this, first we recall the definition of the weak solutions of the problem Our first result reads as follows: Theorem 1.1. Let (M, g) be a 3−dimensional compact Riemannian manifold without boundary, and let β ≡ 1. Assume that Ψ(λ, x) = λα(x) and α ∈ C ∞ (M) is a positive function.
If the continuous function f : ) has only the trivial solution; ) has at least two distinct non-zero, nonnegative weak solutions in H 1 g (M) × H 1 g (M). Remark 1.1.
(b) (f 1 ) and (f 2 ) mean that f is superlinear at the origin and sublinear at infinity, respectively. Typical functions which fulfill hypotheses By a three critical points result of Ricceri [Ric11], one can prove that the number of solutions of the problem (SM e Ψ(λ,·) ) for λ >λ is stable under small nonlinear perturbations g : R → R of subcritical type, i.e., g(s) = o(|s| 2 * −1 ) as |s| → ∞, 2 * = 2N N −2 , N > 2. In order to obtain new kind of multiplicity result for the system (SM λ Ψ(λ,·) ) (with the choice e = λ), instead of the assumption (f 1 ) we require the following one: (f 4 ) There exists µ 0 > 0 such that the set of all global minima of the function has at least m ≥ 2 connected components. In this case we can state the following result.
) has at least m + 1 weak solutions, m of which satisfy the inequality Remark 1.2. Taking into account the result of Cordaro [Cor07] and Anello [Ane07a] one can prove the following: Consider the following system: where α ∈ L ∞ (M) with essinf α > 0, f : R → R is a continuous function and g : M × R → R, besides being a Carathéodory function, is such that, for some p > 3(= dim M), sup |s|≤t g(·, s) ∈ L p (M) and g(·, t) ∈ L ∞ (M) for all t ∈ R. If the set has m ≥ 2 bounded connected components, then the system has at least m + m 2 weak solutions. For the proof, one can use a truncation argument combining with the abstract critical point theory result of Anello [Ane04, Theorem 2.1]. Note that, the similar truncation method which was presented in [Cor07] fails, due to the extra term M φ u u 2 . To overcome this difficulty, one can use the same method as in [FK16, Proposition 3.1 (i)&(ii)] (see also [KM10]).
Note also that, similar multiplicity results was obtained by Kristály and Rǎdulescu in [KR09], for Emden-Fowler type equations.
Our abstract tool for proving the Theorem 1.2 is the following abstract theorem that we recall here (see [Ric04]): Theorem A. Let H be a separable and reflexive real Banach space, and let N , G : H → R be two sequentially weakly lower semi-continuous and continuously Gateaux differentiable functionals., with N coercive. Assume that the functional N + λG satisfies the Palais-Smale condition for every λ > 0 small enough and that the set of all global minima of N has at least m connected components in the weak topology, with m ≥ 2. Then, for every η > inf H N , there exists λ > 0 such that for every λ ∈ (0, λ) the functional N + λG has at least m + 1 critical points, m of which are in N −1 ((−∞, η)).
As a counterpart of the Theorem 1.1 we consider the case when the continuous function f : [0, +∞) → R satisfies the following assumptions: where p ∈ (2, 6); (f 2 ) there exists η > 4 and τ 0 > 0 such that Riemannian manifold without boundary, and let β ≡ 1. Assume that Ψ(λ, x) = λ. Let f : R → R be a continuous function, which satisfies hypotheses (f 1 ), (f 2 ). Then there exists λ 0 such that for every 0 < λ < λ 0 the problem (SM e λ ) has at least two weak solutions. Our abstract tool for proving the previous theorem is the following abstract theorem that we recall here (see [Ric00]): Theorem B. Let E be a reflexive real Banach space, and let Φ, Ψ : E → R be two continuously Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous and coercive. Further, assume that Ψ is sequentially weakly continuous. In addition, assume that, for each µ > 0, the functional J µ := µΦ − Ψ satisfies the classical compactness Palais-Smale condition. Then for each ρ > inf E Φ and each the following alternative holds: either the functional J µ has a strict global minimum which lies in Φ −1 ((−∞, ρ)), or J µ has at least two critical points one of which lies in Φ −1 ((−∞, ρ)).

Proof of the main results
Let β ∈ C ∞ (M) be a positive function. For every u ∈ C ∞ (M) let us denote by The Sobolev space H 1 β is defined as the completion of C ∞ (M) with respect to the norm · β . Clearly, H 1 β is a Hilbert space. Note that, since β is positive, the norm · β is equivalent to the standard norm, i.e., we have that 6); the Sobolev embedding constant will be denoted by κ p .
We define the energy functional J λ : It is easy to see that the functional J λ is well-defined and of class C 1 on H 1 g (M) × H 1 g (M).
By using standard variational arguments, one has that the pair (u, φ) ∈ H 1 g (M) × H 1 g (M) is a critical point of J λ if and only if u is a critical point of E λ and φ = φ u , see for instance [FK16]. Moreover, we have that 2.1. Schrödinger-Maxwell systems involving sublinear nonlinearity. In this section we set Ψ(x, λ) = λα(x) + µ 0 β(x). Recall that In order to apply variational methods, we prove some elementary properties of the functional E λ : Lemma 2.1. The energy functional E λ is coercive, for every λ ≥ 0.
. Therefore, up to a subsequence, then {u j } j converges weakly in H 1 g (M) and strongly in L p (M), p ∈ (2, 2 * ), to an element u ∈ H 1 g (M). First we claim that, for all u, v ∈ H 1 g (M) we have that This inequality is equivalent with the following one: On the other hand using the Cauchy-Schwarz inequality, we have, that Taking into account the following algebraic inequality (xy) 1/2 (x+y) ≤ (x 2 +y 2 ), (∀)x, y ≥ 0, we have that which proves the claim. Now, using inequality (2.4) one has , the first two terms at the right hand side tend to 0. Let p ∈ (2, 2 * ).
By the assumptions on f , for every ε > 0 there exists a constant C ε > 0 such that for every s ∈ R. The latter relation, Hölder inequality and the fact that as j → ∞. Therefore, u j − u 2 H 1 g (M ) → 0 as j → ∞, which proves our claim. Before we prove Theorem 1.1 we prove the following lemma: Note that s 0 > 0. Fix t 0 ∈ (0, s 0 ), in particular 4F (t 0 ) < θ(2t 3 0 + eqt 4 0 ). On the other hand, from the definition of c f , one has f (t) ≤ θ(s + eqs 3 ). Therefore .
, then the last inequality gives u = 0. By the Maxwell's equation we also have that φ = 0, which concludes the proof of (a). (b) By using assumptions (f 1 ) and (f 2 ), one has can guarantee the existence of a suitable truncation function u T ∈ H 1 g (M) \ {0} such that F (u T ) > 0. Therefore, we may define The above limits imply that 0 < λ 0 < ∞. Since H 1 g (M) contains the positive constant functions on M, we have For every λ > λ 0 , the functional E λ is bounded from below, coercive and satisfies the Palais-Smale condition (see Lemma 2.1, Lemma 2.2). If we fix λ > λ 0 one can choose a function w ∈ H 1 g (M) such that F (w) > 0 and In particular, The latter inequality proves that the global minimum u 1 λ ∈ H 1 g (M) of E λ on H 1 g (M) has negative energy level.
It is also clear that the function q → max s>0 4F (s) 2s 2 + eqs 4 is non-increasing. Let a > 1 be a real number. Now, consider the following function where g : [1, +∞) → R is a continuous function with the following properties (g 1 ) g(1) = −1; (g 2 ) the function s → g(s) s is non-decreasing on [1, +∞); In this case the Thus, a simple calculation shows that We also claim that Indeed, It is clear that, it is enough to show that the maximum of the function 2F (s) s 2 is achieved on the interval s ≥ a, i.e., sg(s) − 2G(s) > −1, s > 1. a 2 (a 2 −2 ln a−1) . Proof of Theorem 1.2. We follow the idea presented in [KR09]. First we claim that the set of all global minima's of the functional N : has at least m connected components in the weak topology on H 1 g (M). Indeed, for every u ∈ H 1 β (M) one has Moreover, if we consider u =t for a.e. x ∈ M, wheret ∈ R is the minimum point of the function t → Φ µ 0 (t), then we have equality in the previous estimate. Thus, On the other hand if u ∈ H 1 g (M) is not a constant function, then |∇ g u| 2 > 0 on a positive measure set in M, i.e., Consequently, there is a one-to-one correspondence between the sets and Let ξ be the function that associates to every t ∈ R the equivalence class of those functions which are a.e. equal to t on the whole M. Then ξ : Min(N ) → Min (Φ µ 0 ) is actually a homeomorphism, where Min(N ) is considered with the relativization of the weak topology on H 1 g (M). On account of (f 4 ), the set Min (Φ µ 0 ) has at least m ≥ 2 connected components. Therefore, the same is true for the set Min(N ), which proves the claim. Now we are in the position to apply Theorem A with H = H 1 g (M), N and Now we prove that the functional G is sequentially weakly lower semicontinuous. To see this, it is enough to prove that the map To prove this, let us fix u, v ∈ H 1 β (M) and t, s ≥ 0 such that t + s = 1. Then we have that Then using a comparison principle it follows that Then multiplying the equations −∆ g φ u + φ u = qu 2 by φ v and −∆ g φ v + φ v = qv 2 by φ u , after integration, we obtain that that m elements among the solutions belong to the set N −1 ν 0 ((−∞, τ )), which proves that m solutions satisfy the inequality 1 2 M |∇ g u| 2 + β(x)u 2 dv g − µ 0 M β(x)F (u)dv g < τ.
Using a Hölder inequality Lemma 2.4. Every (PS) sequence for the functional E λ is bounded in H 1 g (M). Proof. We consider a Palais-Smale sequence (u j ) j ⊂ H 1 g (M) for E λ , i.e., {E λ (u j )} is bounded and (E λ ) ′ (u j ) H 1 g (M ) * → 0 as j → ∞. We claim that (u j ) j is bounded in H 1 g (M). We argue by contradiction, so suppose the contrary. Passing to a subsequence if necessary, we may assume that u j H 1 g (M ) → ∞, as j → ∞.