EXISTENCE OF AT LEAST FOUR SOLUTIONS FOR SCHRODINGER EQUATIONS WITH MAGNETIC POTENTIAL INVOLVING AND SIGN-CHANGING WEIGHT FUNCTION

. We consider the elliptic problem − ∆ A u + u = a λ ( x ) | u | q − 2 u + b µ ( x ) | u | p − 2 u, for x ∈ R N , 1 < q < 2 < p < 2 ∗ = 2 N/ ( N − 2), a λ ( x ) is a sign-changing weight function, b µ ( x ) satisﬁes some additional conditions, u ∈ H 1 A ( R N ) and A : R N → R N is a magnetic potential. Exploring the Bahri-Li argument and some preliminary results we will discuss the existence of a four nontrivial solutions to the problem in question.


Introduction
In this work we are interested in studying the existence of a fourth solution for the concave-convex elliptic problem −∆ A u + u = a λ (x)|u| q−2 u + b µ (x)|u| p−2 u in R N , u ∈ H 1 A (R N ), (1.1) where N ≥ 3, −∆ A = (−i∇ + A) 2 , 1 < q < 2 < p < 2 * = 2N N −2 , a λ (x) is a family of functions that can change signs, b µ (x) is continuous and satisfies some additional conditions, u : R N → C with u ∈ H 1 A (R N ) (such space will be defined later), λ > 0 and µ > 0 are real parameters, and A : R N → R N is a magnetic potential in L 2 loc (R N , R N ). For the relevance of this equation to the magnetic Laplacian in Physics, the reader is referred to Alves and Figueiredo [1] and Arioli and Szulkin [3].
In [12] the authors showed the existence of three solutions for this problem and proved their regularity. In this paper we show the existence of a fourth solution.
There are many works on problems similar problem to (1.1), but with A = 0. Ambrosetti, Brezis, and Cerami [2] considered the problem −∆u + u = λu q−1 + u p−1 in Ω, u > 0 in Ω, where Ω is a bounded regular domain of R N (N ≥ 3), with smooth boundary and 1 < q < 2 < p ≤ 2 * . Combining the method of sub and super-solutions with the variational method, the authors proved the existence of λ 0 > 0 such that there are two solutions when λ ∈ (0, λ 0 ), one solution when λ = λ 0 , and no solution when λ > λ 0 .
Wu [26] studied the concave-convex problem −∆u + u = λf (x)u q−1 + u p−1 in Ω, u > 0 in Ω, with f ∈ C(Ω) a sign changing function and 1 < q < 2 < p < 2 * . It was proved that the problem has at least two positive solutions for λ small enough. For p, q as above, many studies have been devoted to the existence and multiplicity of solutions to concave-convex elliptic problems in bounded domains; see for example Brown [8], Brown and Wu [6], Brown and Zhang [7], Hsu [18], Hsu and Lin [17], and their references.
For an unbounded domains, we can cite Chen [10], Huang, Wu and Wu [19], who studied a similar problems in R N . Wu [25] studied the problem with 1 < q < 2 < p < 2 * , g µ ≥ 0, and f λ being able to change sign. Wu [25] showed the existence of at least four solutions to the problem when λ and µ small enough. This result was extend in [12], investigating if it would be possible to obtain similar consequences when we use the magnetic Laplacian in place of the usual Laplacian. In this work we will show the existence of a fourth solution for this problem.
The first results for non-linear Schrödinger equations with A = 0 can be attributed to Esteban and Lions [14]. They obtained the existence of stationary solutions for the equation with V = 1 and p ∈ (2, ∞) using a minimization method with constant magnetic field. This was done for the general case.
Chabrowski and Szulkin [9] worked with this operator in the critical case, that is p = 2 * , and with the electric potential V being able to change signs. Already Cingolani, Jeanjean and Secchi [11] considered the existence of multi-peak solutions in the subcritical case.
Alves and Figueiredo [1] considered the problem with µ > 0 and 2 ≤ q < 2 * . They related number of solutions with the topology of Ω. The authors in [12] studied non-zero A case with a weight function that changes signs in the concave-convex case, just like the problem stated in this work. They proved the existence of three solutions for the problem. Now we would like to show the existence of a fourth solution. There the authors used category theory, the Nehari manifold, and the fibering map.
In what follows, we will present a set of preliminary results. Observe that is the functional associated with problem (1.1) and is of class C 1 in H 1 A (R N ) as can shown in [22]. Also, the critical points of J λ,µ (u) are weak solutions of problem (1.1). We will use the following hypotheses: We assume a(x) ∈ L q (R N ), q = p p−q , and a ± = ± max{±a(x), 0} = 0. Let and assume that (A1) a(x) ∈ L q (R N ), q = p p−q , and there existsĉ > 0 and r a− > 0, such that The above hypotheses were used in [12]. We define In [12], under assumptions (A1)-(A3), it was proved that (1.1) has at least one solution, provided that holds for each λ > 0 and µ > 0. Then, assuming that the potential is asymptotic to a constant at infinity, they prove the existence of at least two solutions u + and In the previous result, the existence is valid for all λ and µ satisfying inequality (1.4). So, if we additionally set values of λ and µ conveniently small we obtain the multiplicity result, that is, there exist at least three solutions. Actually they showed the existence of λ 0 > 0 and µ 0 > 0 with such that for all λ ∈ (0, λ 0 ) and µ ∈ (0, µ 0 ), problem (1.1) has at least three solutions.
In this work, we observe that for the problem in question, the numbers λ 0 and µ 0 as previously mentioned are independent of the value of a − . However, considering some additional hypotheses and taking values of a − q sufficiently small we obtain another solution. Before stating this result we present the following hypotheses: Theorem 1.1. Suppose that the potential A(x) converges to some constant d ∈ N N as |x| → ∞. Assuming (A1)-(A5), there are positive valuesλ 0 ≤ λ 0 ,μ 0 ≤ µ 0 , and ν 0 such that for λ ∈ (0,λ 0 ), µ ∈ (0,μ 0 ), and a − q < ν 0 , problem (1.1) has at least four solutions.
For the first three solutions of this problem, the Nehari method was used together with category theory. We will use variational methods to prove the above theorem. We will work under a few more assumptions to estimate different energy levels and will use the Bahri-Li min-max argument to show that for very small values of a − q , the problem has at least four distinct solutions.

Initial considerations
According to Tang [23], we denote by H A (R N ) the Hilbert space obtained by the closing of ). The norm induced by this product is Esteban and Lions [14, Section II] proved that that for all u ∈ H 1 A (R N ) the diamagnetic inequality holds, i.e.
we have that |u| belongs to the usual Sobolev space H 1 0 (R N ). 2.1. Preliminary results. To obtain the existence, we introduced the Nehari manifold The Nehari manifold is linked to the functions F u : t → J λ,µ (tu), (t > 0), called fibering maps. Note that the fabering map it was defined and depends on u, λ, and µ; so that proper notation should be F u,λ,µ , but to simplify the notation, we write F u . 3) The following remark relates the Nehari manifold and the Fibering map.
Remark 2.1. Let F u be the application defined above and u ∈ H 1 A (R N ). Then: From the previous remark we conclude that the elements in M λ,µ , correspond to the critical points of the Fibering map. Thus, as F u (t) ∈ C 2 (R + , R), we can divide the Nehari manifold into three parts Then M 0 λ,µ = ∅. As shown in [12], under certain conditions on λ and µ, we have a minimizer in M + λ,µ and another in M − λ,µ . The minimum levels of energy will be denoted respectively by To establish the existence of the first two solutions and compare with the energy level of the fourth solution, we will need the following result that was shown in [12].
Our next lemma shows that these points are well defined, and i prove can be found in [17, Lemma 2.1].
Lemma 2.4. The functional J λ,µ is coercive and bounded from below in M λ,µ .
For the next results we need some estimates on m ± λ,µ . To do this, from (2.4) we have Therefore, By has been seen, we will show the following results on the values of m ± λ,µ .
The proof of the above lemma is similar to one in [25, Theorem 3.1]; we omit it. By Lemma 2.5, we can conclude that for every u ∈ H 1

Existence of m ∞
In this section we define the energy level of the limit problem and make some estimates for energy levels of the solutions in the Nehari manifold. Then, we will have tools to show that the fourth solution has a different level than other solutions. For this, consider the semilinear elliptic problem (3.1) We define J ∞ (u) = 1 2 u 2 A − 1 p u p p , as the functional associated with problem (3.1). Then J ∞ is a C 2 functional in H 1 A (R N ). The Nehari manifold associated with (3.1) is . In this problem we can observe that if u ∈ N ∞ , then u 2 A = u p p . Now consider the minimization problem In [12] it was shown that there existsū ∈ H 1 . From these considerations we will show the following result that gives us a description of a sequence (PS) of J λ,µ .
there is a subsequence {u n } and u 0 ∈ H 1 A (R N ), with a non zero u 0 , such that u n = u 0 + o n (1) strong in H 1 A (R N ) and J λ,µ (u 0 ) = β. Moreover, u 0 is a solution of (1.1).
Proof. From (A1)-(A3), we obtain by a standard argument that {u n } is a bounded sequence in H 1 A (R N ). Then there is a subsequence {u n } and Denoting by B(0, 1) the ball centered on the origin of radius 1, we have in B(0, 1) the strong convergence By the Dominated Convergence Theorem we obtain Then, by Hölder and the integrability of a λ it follows that As > 0 it is arbitrary, we have On the other hand, by (A2) and (A3) and the Brezis-Lieb lemma (see [24]), we can conclude that µ b 2 (x)|v n | p = o n (1), (1 − b 1 (x))|v n | p = o n (1) and b µ (x)(|u n | p − |v n | p − |u 0 | p ) = o n (1), which together with the above inequality gives us In a similar way we obtain that J ∞ (v n )v n = J λ,µ (u n )u n − J λ,µ (u 0 )u 0 + o n (1). By hypothesis J λ,µ (u n ) → 0 strong in H 1 A (R N ) −1 and u n u 0 weak in H 1 A (R N ) as n → ∞ and so we have J λ,µ (u 0 ) = 0. Now, define δ = lim sup n→∞ sup y∈R N B(y,1) |v n | p . So we have two cases: (i) δ > 0, and (ii) δ = 0. Suppose that (i) happens. Then there will be a sequence {y n } ⊂ R N such that B(yn,1) |v n | p ≥ δ 2 and for all n ∈ N. Defineṽ n (x) = v n (x + y n ). We have that {ṽ n } is bounded andṽ n v weak and almost everywhere. Making a change of variables we obtain Then giving us v = 0. But, v n 0 weakly; then We see that Likewise, For each n ∈ N, we can get t n such that t n v n ∈ M ∞ . So we build a sequence {t n } ⊂ R N with t n → t 0 as n → ∞, such that t n v n ∈ M ∞ , that is, such that J ∞ (t n v n )t n v n = 0. We see also that and With this, we have 2c . With that and by (3.6) we obtain that (1−t p−2 n ) → 0, giving us that t n → 1. Now, see that v n 0 weak in H 1 A (R N ) as n → ∞. With this and by the fact t n → 1, we can conclude that Note that by hypotheses J λ,µ (u n ) = β + o n (1) with β < m ∞ + m + λ,µ . From there we obtain We have already seen that J λ,µ (u n ) converges strongly to zero, therefore we obtain J λ,µ (u 0 ) = 0. Thus u 0 ∈ M λ,µ . Still, by Lemma 2.2, M 0 λ,µ = ∅ and by Lemma 2.5, we conclude that m + > 0 and m − < 0. Then which contradicts what we have concluded in (3.7). We have proved that (ii) occurs. In this case, {v n } such that |v n | p → 0 if n → ∞.
As we already have A − v n p p and |v n | p → 0, we conclude that v n 2 → 0 giving us u n → u 0 strong in H 1 A (R N ). See also that u 0 = 0. In fact, note that if u 0 = 0 soṽ n = v n = u n and B(0,1) |u n | p ≥ δ 4 , which we have already seen to be an absurd.
To address the existence of a second solution to (1.1), certain considerations need to be made. Note that equation is such that a λ (x) → 0 and b µ (x) → 1 as |x| → ∞. Adding the hypothesis of A → d with d constant as |x| → ∞, problem (3.8) converges to the problem Furthermore, the equations (3.9) and (3.10) share the same least energy. Specifically, we have where J ∞ and I ∞ represent the corresponding functionals associated with the aforementioned problems. According to Berestick, Lions [5] or Kwong [21], equation (3.10) has a unique solution z 0 symmetric, positive, and radial. By [15,Theorem 2], for all > 0, exists A , B 0 and C positive such that According to Kurata [20,Lemma 4], defining w 0 = z 0 e −idx , we have that w 0 is the unique, symmetrical, positive and radial solution of (3.9). So we will have J ∞ (w 0 ) = m ∞ . We see also that z 0 = |w 0 |, which together with (3.11) gives us the inequalities Next, To prove the existence of a second solution, we make some estimates on the minimum energy levels in the Nehari Manifold. Not to overload the notation, we write u + := u + λ,µ . Considering J(u + ) = m + , m − = inf u∈M − λ,µ J λ,µ (u), and m ∞ = inf u∈M∞ J ∞ (u) = J ∞ (w 0 ), we will make the following estimate for such energy levels.
The proof of the above proposition is similar to that of [12, Proposition 6.1]; we omit it.

Third solution
To obtain the third solution of problem (1.1), we need some additional results. For λ = 0 and µ = 0 we define the sets Lemma 4.1. With the above notation we have Proof. Let w k be as defined above. Because λ = 0, we have a(x) = λa As w 0 is a solution of problem (3.10) and remembering that the functional associated with (3.10) is Being w 0 solution of problem (3.10) follows that w k (x) = w 0 (x + ke). With this and I (w 0 )w 0 = 0, we have I (w k )w k = 0. So that It is known that w n is bounded in L r and w n → 0 a.e., by Theorem [16,Theorem 13.44] that w n 0 weakly in L r . By the condition (A1), a − ∈ (L r ) = L r we obtain In addition, by (A2) and (A3) we have as |w k | → ∞. By (4.1), (4.3), and (4.4) we have that t − (w k ) → 1 as k → ∞. Likewise lim Thus We also have u ∈ M a0,b0 , by Lemma 2.3(i), J a0,b0 (u) = sup t≥0 J a0,b0 (tu), and furthermore, there is a single t ∞ > 0 such that t ∞ u ∈ M ∞ . So By (4.5) and (4.6), we have inf To obtain the fourth solution of the problem, we need a lemma that establishes suitable values of λ and µ.

Fourth solution
In this section we will work to estimate of the energy levels of the functional associated with the main problem, to prove the existence of a solution whose energy level satisfies the conditions of Proposition 3.1(ii); that is, to find a distinct solution from the three solutions in the previous sections. For α > 0, we define : J 0,αb0 (u), u = 0}. We now define the following subset of the unitary ball : u ≥ 0 and u A = 1}. Let us recall that for every u ∈ H 1 A (R N ) \ {0} there exists a unique t − (u) > 0 and t 0 (u) > 0 such that t − (u) ∈ M − a λ ,bµ and t 0 (u) ∈ M 0,b0 . To apply the minimax argument of Bahri-Li we present the following result. .
Proof of Theorem 1.1. With the result of Theorem 5.3 we can complete the proof of Theorem 1.1. For λ ∈ (0,λ 0 ), µ ∈ (0,μ 0 ) and a − q * < ν 0 , also using the results presented in the introduction about the existence of the first three solutions and Theorem 5.3, we obtain that the equation (1.1) admits at least four solutions.