Potential well and multiplicity of solutions for nonlinear Dirac equations

In this paper we consider the semi-classical solutions of a massive Dirac equations in presence of a critical growth nonlinearity \begin{document}$ -i\hbar \sum\limits_{k = 1}^{3}\alpha_k\partial_k w+a\beta w+V(x)w = f(|w|)w. $\end{document} Under a local condition imposed on the potential \begin{document}$ V $\end{document} , we relate the number of solutions with the topology of the set where the potential attains its minimum. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

1. Introduction and main result. In these last years, attentions have been drawn to the study of standing wave solutions for the nonlinear Dirac equation of the form − i ∂ t ϕ = ic α · ∇ϕ − mc 2 βϕ − V (x)ϕ + g(x, ϕ), (t, x) ∈ R × R 3 (1.1) where ϕ(t, x) ∈ C 4 is a spinor function, is a small positive constant which corresponds to the Plank's constant, m, c > 0 are constants representing the mass of a electron and the speed of light, α = (α 1 , α 2 , α 3 ) and α · ∇ = 3 k=1 α k ∂ k with α 1 , α 2 , α 3 and β being the 4 × 4 complex Pauli matrices: and Moreover, in Eq. (1.1), V is a potential function and g is the nonlinearity modeling some self-interaction in Quantum electrodynamics. In particle physics, (1.1) models 588 YU CHEN, YANHENG DING AND TIAN XU many physical problems in the self-interacting scalar theories, where the nonlinear function g can be both a polynomial and non-polynomial. Various nonlinearities are considered as possible basis models for unified field theories, we just refer to [30,31,34] for more physical background. A solution of the form ϕ(t, x) = exp(−iωt/ )w(x) is called a standing wave. Assume that g(x, exp(−iθ)ξ) = exp(−iθ)g(x, ξ) for θ ∈ R and ξ ∈ C 4 , a change of notation (in particular ε instead of ) leads to an equation of the form − iεα · ∇w + aβw + V (x)w = g(x, w) w ∈ H 1 (R 3 , C 4 ). (1.2) This type of particle-like solution does not change its shape as it evolves in time, and thus has a soliton-like behavior.
It should be pointed out here that, in quantum mechanics, the existence and multiplicity of solutions to a dynamical equation in terms of an asymptotic representation as the Plank's constant tends to zero is of particular importance. To some extent, this corresponds to a deformation of quantum mechanics and quantum field theory to classical mechanics and classical field theory. Such deformation is parameterized by the Planck's constant and, in this deformation, solutions to dynamical equations are usually referred as semiclassical states. In the case of non-relativistic quantum field theories, standing wave solutions for the nonlinear Schrödinger equa- have been in the focus of nonlinear analysis since decades. Particularly, semiclassical states that concentrate near a critical point of the potential V have been widely investigated ever since the influential paper [33] by Floer and Weinstein who treated the cubic nonlinearity |ψ| 2 ψ in one-dimension. An incomprehensive list of references are [2,3,6,10,11,19,20,21,22,36,37,38], in which the authors used Lyapunov-Schmidt type methods, penalization and variational techniques to establish the concentration phenomenon of the semiclassical states for the Schrödinger equations. Much less is known for the nonlinear Dirac equation (1.1) which arises in relativistic field theories. So far only a few results are available for the concentration phenomenon of semiclassical states around a minima x 0 of V ; see [26,27]. Related results, i.e., concentration of semiclassical states under the influence of nonlinear potentials, can be found for similar equations in [24,25,26]. Lyapunov-Schmidt type methods do not seem to be applicable to (1.2) because even for the homogeneous nonlinearity g(x, w) = |w| p−2 w nothing is known about uniqueness or nondegeneracy of the least energy solution of which appears as limit equation for (1.2). As for variational methods, a major difference between nonlinear Schrödinger and Dirac equations is that the Dirac operator is strongly indefinite in the sense that both the negative and positive parts of the spectrum are unbounded and consist of essential spectrum. It follows that the quadratic part of the energy functional associated to (1.2) has no longer a positive sign, moreover, the Morse index and co-index at any critical point of the energy functional are infinite. It is not clear whether one can develop a penalization technique to find semiclassical states. And moreover, beyond the existence and concentration results in [26,27], it is interesting to ask whether one can obtain a multiplicity of semiclassical solutions to Eq. (1.2). Very recently, in [40] Wang and Zhang obtained an interesting result in this direction. By using the symmetric structure of Eq. (1.2), they constructed an infinite sequence of bound state solutions for small values of ε, particularly, these solutions are of higher topological type. In this paper, letting M be a set of local minima of the potential V , we are interested in the following aspects which have not been dealt with before and is new in the case of Dirac equations: (1) to show the multiplicity of semiclassical solutions concentrating around M is influenced by the topology of the level sets of the potential V in a bounded domain; (2) to apply variational methods, concentration-compactness and rescaling techniques to deal with nonlinearities more general than |w| p−2 w, in particular, g(x, w) grows critically as |w| → ∞.
We mention here that starting from the paper of Bahri and Coron [5], many papers are devoted to study the effect of the domain topology on the existence and multiplicity of solutions for semilinear elliptic problems. We refer to [7,8,9,12,16,17] for related studies for Dirichlet and Neumann boundary value problems. We also refer to [4,13,14,15] for the study of semiclassical states of Schrödinger equations. Now, in order to state our results precisely, let us consider the following equation Throughout the paper, we assume that the potential V satisfies And we denote M := {x ∈ Λ : V (x) = ω}. For the nonlinear function f , we make the following assumptions: . Condition (f 1 ) implies that s → f (s)s is superlinear and strictly increasing, an important role in our approach. If κ > 0 in (f 2 ), then F (s) ∼ κs 3 as s → ∞ is of critical growth. This terminology is befitting because the form domain of the quadratic form associated to the Dirac operator is H 1 2 (R 3 , C 4 ). This space embeds into the corresponding L q -spaces for 2 ≤ q ≤ 3. And if κ = 0 then the problem is subcritical. (f 3 ) is a technical assumption, and (f 4 ) is the Ambrosetti-Rabinowitz condition.
Letting cat X (A) denote the Lusternick-Schnirelmann category of A in X for any topological pair (X, A), our main result can be stated as follows Eq. (1.4) has at least cat M δ (M ) solutions w k ε , k = 1, . . . , cat M δ (M ), for sufficiently small ε > 0. These solution have the following properties: (2) The rescaled function v k ε (x) = w k ε (εx + x k ε ), converges in H 1 as ε → 0 to a least energy solution v : The proof will be done by variational techniques. Since we have no information on the potential V at infinity, we employ the truncation trick explored in [20]. It consists in making a suitable modification on the nonlinearity f , solving a modified problem and then check that, for ε small enough, the solutions of the modified problem are indeed solutions of the original one. We emphasize here that, in the usual concept, the truncation tricks are well adapted for the study of the subcritical variational problems, see for instance [10,11,19,21,22] for the studies of Schrödinger equations. However, due the strongly indefinite character of the Dirac operator, we note that it is not easy to obtain compactness in view of the critical growth of the nonlinearity even for the modified problem. To overcome this, we will need a delicate analysis for the limit problem (1.3) on the ground state energy level and use a version of the concentration-compactness principle originated from Lions [35] to control the factor κ > 0 in the critical growth. As a matter of fact, the truncation trick we adapt here is essentially depending on the factor κ as we will see in the Remark 4.3 in Section 4.
To obtain multiple solutions of the modified problem, the main ingredient is to make precisely comparisons between the category of some sublevel sets of the modified functional and the category of the set M . This kind of argument for the Schrödinger equations has been appeared in [8,13,14,15], where subcritical problems were considered.
The remainder part of the paper is organized as follows. In Sect. 2 we first present the variational settings of the problem, both in the original and in the extended variables, and we truncate the original problem. For the sake of completeness, we collect some useful results which are needed in our proof. In Sect. 3, we investigate the associated autonomous problem. This study allow us to show the role which the critical factor κ plays in the ground state energy level. And the Palais-Smale condition, which does not hold in general case since we allow critical growth, will then be studied in Sect. 4. Next, in Sect. 5, we provide the main components of our proof. The first point is we introduce the min-max scheme that can be applied to the truncated problem. And as the second point, we construct two maps in terms of the truncated problem such that their composition is homotopically equivalent to the embedding j : M → M δ . Finally, the main results are proved in Sect. 6..

Notations, known facts and main ingredients
We shall in the sequel focus on this equivalent problem.
In what follows, by | · | q we denote the usual L q -norm, and (·, ·) 2 the usual L 2 -inner product. Let L = −iα · ∇ + aβ denote the self-adjoint operator on where σ(·) and σ c (·) denote the spectrum and the continuous spectrum. Thus the space L 2 possesses the orthogonal decomposition: and the induced norm u = u, u 1/2 , where |L | and |L | 1/2 denote respectively the absolute value of L and the square root of |L |. Since σ(L ) = R \ (−a, a), one has Note that this norm is equivalent to the usual H 1/2 -norm, hence E embeds continuously into L q for all q ∈ [2, 3] and compactly into L q loc for all q ∈ [1, 3). It is clear that E possesses the following decomposition orthogonal with respect to both (·, ·) 2 and ·, · inner products. And remarkably, this decomposition of E induces also a natural decomposition of L q for every q ∈ (1, +∞): Proposition 2.1 (see [27]). Let E + ⊕ E − be the decomposition of E according to the positive and negative part of σ(L ). Then, set Remark 2.2. It is of great importance for the projections from H 1/2 := E = E + ⊕ E − onto E + (or E − ) to be continuous in the L q 's and not only in H 1/2 . This is not the case for every direct sum in H 1/2 . In fact, the proof of Proposition 2.1 implies on the splitting of L q 's that: For every q ∈ (1, ∞), L q can be split into topologically direct sum of two (infinite dimensional) subspaces which, accordingly, are the positive and negative projected spaces of the Dirac operator L .
In what follows, we define the energy functional Standard arguments show that, under our assumptions, Φ ε ∈ C 2 (E, R) and critical point of Φ ε is a (weak) solution to (2.1).
To establish the multiplicity of solutions, we will adapt for our case an argument explored by the penalization method introduced by Del Pino and Felmer [20]. To this end, we need to fix some notations.
We first let δ 0 ∈ 0, a−|V |∞ and the corresponding energy functional One should keep in mind here that Λ has to be rescaled when we consider the modified rescaled equation (2.1). It is well-known that such truncation trick will be helpful in both bringing compactness to the problem and locating the maximum points of the solutions, see [20,21] and [27] for subcritical problems. Since we address here the critical growth, we remark that, in order to recover the compactness, δ 0 should be chosen even smaller and this will be seen in the proof of Proposition 4.1 where a implicit upper bound is established accordingly to the critical fact κ: if κ = 0, then δ 0 ≤ a−|V |∞ 2 is enough; if κ > 0, then δ 0 needs to be properly smaller.
It is elementary to check that (f 1 ) and (f 3 ) implies that g is a Carathéodory function and it satisfies Here we used the notation G(x, s) = 1 2 g(x, s)s 2 − G(x, s). In what follows, we shall collect some properties of Φ ε when the assumptions on V and f hold. First, similar as that in [27], we give the following geometric behaviors of Φ ε . Lemma 2.3. For c ≥ 0, any (P.S.) c -sequence for Φ ε is bounded independent of ε > 0.

Then we have
and And by (f 1 ) and (f 2 ) we have f (s)s 3 2 ≤ Cf (s)s 2 for some C > 0, and hence it follows from (f 4 ) and (2.7)-(2.9) that which implies the boundedness. Moreover, we can see from the above inequalities that u n → 0 in E if and only of c = 0.
In order to describe further the critical values, let us recall some known facts on a Lyapunov-Schmidt type reduction for Φ ε . Such reduction technique depends on the convexity of the nonlinearities, specifically, it requires that the second order derivative of Φ ε is negative definite on E − . And by the anti-coercion and concavity properties of Φ ε | E − , we can define ε : E + → E − to be the bounded reduction map correspondingly such that, for any u ∈ E + , . And denote I ε (u) = Φ ε u + ε (u) , we shall call ( ε , I ε ) : E + × E + → E − × R the reduction couple associated to Φ ε on E + (for details we refer to [1,27]). Then, it is all clear that I ε ∈ C 2 (E + , R) and critical points of I ε and Φ ε are in one-to-one correspondence via the injective map u → u + ε (u) from E + to E. Now, on E + , let us consider the functional I ε .
Moreover, we have Lemma 2.6. For all ε > 0, let Then N ε is a C 1 manifold, and there exist θ, µ > 0 both independent of ε such that for any u ∈ N ε u ≥ θ and I ε (u) ≥ µ; Moreover, critical points of I ε constrained on N ε are free critical points of I ε in E + .
Remark 2.7. In general, letting u ∈ E + \ {0}, we find there exists at most one nontrivial critical point t ε = t ε (u) > 0 which realizes the maximum of the function t → I ε (tu). It can be also seen, that N ε can be rewritten as It is worth pointing out that the set N ε is slightly different from the usual concept of the Nehari manifold associated to the reduced functional I ε . In fact, N ε is no longer expected to be homeomorphic to the sphere S + := {u ∈ E + : u = 1} due to the truncated nonlinear part is not superlinear at infinity for certain directions. The details of the above lemmas can be found in relevant material from [27], and we omit it. A general discussion of the properties of N ε in an abstract setting can be found in [28,Section 4] Lastly, let us remind the definition of the Ljusternik-Schnirelman category and a classical result of the related critical point theory.
Definition 2.8. Let X be a topological space and let Y = ∅ be a closed subset of X. The category of Y in X, cat X (Y ), is the smallest integer n such that where for each k = 1, . . . , n, A k is a closed set contractible in X. If such a integer does not exist, then cat X (Y ) = +∞. And set cat X (∅) = 0.
In this context, the category of X in itself, cat X (X), is simply denoted by cat(X). Theorem 2.9 (see [32]). Let W be a complete C 1 manifold and let Φ ∈ C 1 (W, R) be bounded from below on W and satisfying the Palais-Smale compactness condition. Denoted by, for c ∈ R, Φ c = u ∈ W : Φ(u) ≤ c Then Φ has at least cat(Φ c ) distinct critical points in Φ c .

3.
Variational framework for superlinear problems. In this section we establish some preliminary results which are needed for the proof of our main theorems. Given ω ∈ (−a, a) and κ ≥ 0, we consider the equation and the associated energy functional And denoted by ( , I) the reduction couple for Φ and set N = {u ∈ E + \ {0} : I (u)[u] = 0}. Then we have N is a smooth manifold of codimension 1 in E + , and N is diffeomorphic to S + by a C 1 diffeomorphism. Particularly, the function t → I(tu) attains its unique critical point t = t(u) > 0 for each u ∈ E + \ {0}, and t : S + → R is a C 1 function. If denoted by it can be also seen that γ(ω, κ) = inf N I > 0.
Proposition 3.1. Set ω * = min{ω, 0}, then γ(ω, κ) is attained provided that where S denotes the best Sobolev constant for the embedding Before proving this proposition, we begin with some preliminary materials. Let us first consider the following functional , and the minimax scheme We remark that S|u| 2 6 ≤ |∇u| 2 2 , and if denote by F : L 2 → L 2 the Fourier transform, there holds Then, by virtue of Calderón-Lions interpolation theorem, we have It follows that, in the Fourier domain, we have u 2 = R 3 (a 2 + |ξ| 2 ) 1 2 |F u(ξ)| 2 dξ. And hence, we have for any u ∈ E + \ {0} Taking into account that inf |ξ|>0 (a 2 + |ξ| 2 ) with ω * = min{ω, 0}. Next, let us consider the equation and the corresponding functional
Proof of Proposition 3.1. We only give the proof when κ > 0 since it is much easier for the case κ = 0. Let {u n } ⊂ N be a minimizing sequence for I. It is not difficult to check that {w n = u n + (u n )} is bounded in E. Then by Lion's result (see [35]) it follows that {w n } is either vanishing or non-vanishing.
If {w n } is non-vanishing then we are done, so let us assume contrarily that {w n } is vanishing. Then |w n | s → 0 for all s ∈ (2, 3). And thus we have where we used the notationŵ n :=t n u n +ˆ (t n u n ) witht n =t(u n ) be such that t n u n ∈N .
By the above observation, and Φ(w n ) = I(u n ) = γ(ω, κ)+o n (1), we easily deduce from Lemma 3.2 and (3.3) that which contradicts to (3.2). Therefore we have {w n } is non-vanishing, and this ends the proof.
Let us denote by γ ω the corresponding ground state critical level for T ω , that is, Then we have that γ ω > 0 is achieved provided the factor κ is small. Indeed, by (f 3 ), we have T ω (u) ≤ Φ(u) for all u ∈ E. Moreover, thanks to the linking structure (see for example [39]), we have Clearly, by using the fact γ(ω, κ) decreases dependently with respect to κ, we can infer that γ(ω, κ) ≤ γ(ω, 0) and the condition (3.2) is valid when κ is not large, say Therefore, by using the invariance by translation of the problem and the concentration-compactness argument, we see that the conclusion follows. Moreover, it is evident to check that is a compact set in E (similar results can be found in [26]). (2) The upper bound for κ in (3.8) is explicitly defined. We may apply the argument in [27, Section 3] to deduce that the map (−a, a) → R + , ω → γ(ω, 0) is increasing. And as a consequence, the upper bound for κ increases as ω approaches 0 from the right side. For negative ω's, the picture becomes unclear. Our argument do allow critical growth f (s)s ∼ κs 2 at infinity but the factor κ cannot go too large. (3) As before, we can introduce the reduction couple (J ω , J ω ) for T ω as and set M ω = {u ∈ E + \ {0} : J ω (u)[u] = 0}. Then we have 4. The Palais-Smale condition. Due to the non-compactness of the Sobolev embedding H , it is not difficult to see that Φ ε does not satisfy the Palais-Smale condition on E = H 1 2 (R 3 , C 4 ). However, it will satisfy such compactness condition for certain energy levels. In this section, for notation convenience, let us denote Λ ε = {x ∈ R 3 : εx ∈ Λ}, g ε (x, s) = g(εx, s) and G ε (x, s) = G(εx, s). Inspired by the priori bound for the factor κ in (3.8), our compactness result can be stated as follows. Proof. Let {w n } ⊂ E be a (P.S.) c -sequence for Φ ε , where c ≤ c 0 , i.e., , Φ ε (w n ) → 0 as n → ∞.
By obtaining this, we can get where we have used the fact Φ ε (w) = 0. The above estimation shows Φ ε (z n ) → 0 as n → ∞ as was claimed.
To prove the second statement, it suffices to show that Indeed, by using the covariation of I ε under translation, Since ξ ∈ M ⊂ intΛ, χ Λε (εx + ξ) → 1 a.e. in R 3 as ε → 0. Thus , by the fact that V (εx + ξ) → ω as ε → 0 uniformly on bounded set of R 3 and the boundedness of {t ξ,ε }, we have where in the last inequality we have used the fact γ ω = J ω (w + ) = max t>0 J ω (tw + ).
This completes the proof of (5.1). Therefore from now on, according to Remark 4.3, we can precisely fix δ 0 and set κ in a range as so that I ε satisfies the Palais-Smale condition at the level c ε as ε goes to zero.

5.2.
The function ζ ε . Now we introduce the function ζ ε : E + \ {0} → R 3 to construct the homotopy, which is used to relate multiplicity of solutions to the topology of M . We define where η ε (x) = η(εx) is the cut-off function: In what follows, we will divide our proof into four steps: Step 1. The sequence {u n } is non-vanishing.
Set v n (x) = u n (x + x n ), then v n satisfies L v n +V ε (x)v n = g(ε n (x + x n ), |v n |)v n (5.5) whereV ε (x) = V (ε n (x + x n )). Moreover, we have v n v in E and v n → v in L q loc for q ∈ [1, 3) for some v = 0. Now assuming without loss of generality that V (ε n x n ) → V ∞ , and using ψ ∈ C ∞ c (R 3 , C 4 ) as a test function in (5.5), one gets And hence v ∈ E is a non-trivial solution to However, using the test function v + − v − in (5.6), we have Therefore, we have v = 0, which is a contradiction.
Then ε n x n → x 0 ∈ M as n → ∞.
In order to see this, first of all, we choose x n ∈ Λ εn , i.e., ε n x n ∈ Λ. And suppose that, up to a subsequence, ε n x n → x 0 ∈Λ as n → ∞. Then, as argued in Step 2, it is possible to show that the sequence v n (·) := u n (· + x n ) weakly converges to some v in E which satisfies 7) with g ∞ in the form of g ∞ (x, s) = χ ∞ · f (s) + (1 − χ ∞ ) ·f (s). Here χ ∞ is either a characteristic function of a half-space in R 3 provided lim sup n→∞ dist(x n , ∂Λ εn ) < +∞ or χ ∞ ≡ 1 (this is due to the fact χ ∞ is the pointwise limit of the function χ Λ (ε n (·+ x n )) as n → ∞).
Denote S ∞ to be the associate energy functional to (5.