Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 47, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: https://doi.org/10.58997/ejde.2023.47 EXISTENCE OF AT LEAST FOUR SOLUTIONS FOR SCHRODINGER EQUATIONS WITH MAGNETIC POTENTIAL INVOLVING AND SIGN-CHANGING WEIGHT FUNCTION FRANCISCO ODAIR DE PAIVA, SANDRA MACHADO DE SOUZA LIMA, OLÍMPIO HIROSHI MIYAGAKI In memory of Prof. Alan C. Lazer with admiration Abstract. We consider the elliptic problem −∆Au + u = aλ(x)|u|q−2u + bµ(x)|u|p−2u, for x ∈ RN , 1 < q < 2 < p < 2∗ = 2N/(N − 2), aλ(x) is a sign-changing weight function, bµ(x) satisfies some additional conditions, u ∈ H1 A(RN ) and A : RN → RN 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. 1. Introduction In this work we are interested in studying the existence of a fourth solution for the concave-convex elliptic problem −∆Au+ u = aλ(x)|u|q−2u+ bµ(x)|u|p−2u in RN , u ∈ H1 A(RN ), (1.1) where N ≥ 3, −∆A = (−i∇+A)2, 1 < q < 2 < p < 2∗ = 2N N−2 , aλ(x) is a family of functions that can change signs, bµ(x) is continuous and satisfies some additional conditions, u : RN → C with u ∈ H1 A(RN ) (such space will be defined later), λ > 0 and µ > 0 are real parameters, and A : RN → RN is a magnetic potential in L2 loc(RN ,RN ). 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 = λuq−1 + up−1 in Ω, 2020 Mathematics Subject Classification. 35J20, 35Q60, 35Q55, 35B38. Key words and phrases. Magnetic potential; Nehari method; sign-changing function; variational method. ©2023. This work is licensed under a CC BY 4.0 license. Submitted January 13, 2022. Published July 11, 2023. 1 2 F. O. DE PAIVA, S. M. DE SOUZA LIMA, O. H. MIYAGAKI EJDE-2023/47 u > 0 in Ω, u = 0 on ∂Ω, where Ω is a bounded regular domain of RN (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 λ0 > 0 such that there are two solutions when λ ∈ (0, λ0), one solution when λ = λ0, and no solution when λ > λ0. Wu [26] studied the concave-convex problem −∆u+ u = λf(x)uq−1 + up−1 in Ω, u > 0 in Ω, u = 0 on ∂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 RN . Wu [25] studied the problem −∆u+ u = fλ(x)uq−1 + gµu p−1 in RN , u ≥ 0 in RN , u ∈ H1(RN ), with 1 < q < 2 < p < 2∗, gµ ≥ 0, and fλ 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 6= 0 can be attrib- uted to Esteban and Lions [14]. They obtained the existence of stationary solutions for the equation −∆Au+ V u = |u|p−2u, u 6= 0, u ∈ L2(RN ), 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 −∆Au = µ|u|q−2u+ |u|2 ∗−2u, u 6= 0,Ω ⊂ RN , 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 EJDE-2023/47 SCHRODINGER EQUATION WITH MAGNETIC POTENTIAL 3 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 Jλ,µ(u) = 1 2 ∫ RN (|∇Au|2 +|u|2) dx− 1 q ∫ RN aλ(x)|u|q dx− 1 p ∫ RN bµ(x)|u|p dx, (1.2) is the functional associated with problem (1.1) and is of class C1 in H1 A(RN ) as can shown in [22]. Also, the critical points of Jλ,µ(u) are weak solutions of problem (1.1). We will use the following hypotheses: We assume a(x) ∈ Lq′(RN ), q′ = p p−q , and a± = ±max{±a(x), 0} 6= 0. Let aλ(x) = λa+(x) + a−(x). and assume that (A1) a(x) ∈ Lq′(RN ), q′ = p p−q , and there exists ĉ > 0 and ra− > 0, such that a−(x) > −ĉ exp(−ra− |x|) for all x ∈ RN . We assume that bµ(x) = b1(x) + µb2(x), where (A2) b1(x) > 0 in continuous in RN , with b1(x)→ 1 as |x| → ∞ and there exists rb1 > 0, such that 1 ≥ b1(x) ≥ 1− c0 exp(−rb1 |x|) for some c0 < 1 and all x ∈ RN . (A3) b2(x) > 0 is continuous in RN , b2(x) → 0 as |x| → ∞ and exists rb2 > 0, with rb2 < min{ra− , rb1 , q} such that b2(x) ≥ d0 exp(−rb2 |x|) for some d0 < 1 and all x ∈ RN . The above hypotheses were used in [12]. We define Υ0 = (2− q)2−q ( p− 2 ‖a+‖q′ )p−2( Sp p− q )p−q , where Sp = inf u∈H1 A(RN\{0}) ( ∫ RN |∇Au| 2 + u2dx )1/2( ∫ RN |u|pdx )2/p > 0. (1.3) In [12], under assumptions (A1)–(A3), it was proved that (1.1) has at least one solution, provided that λp−2(1 + µ‖b2‖∞)2−q < (q 2 )p−2 Υ0 (1.4) 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 u− with Jλ,µ(u+) < 0 < Jλ,µ(u−). 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 λ0 > 0 and µ0 > 0 with λp−2 0 (1 + µ0‖b2‖∞)2−q < (q 2 )p−2 Υ0, such that for all λ ∈ (0, λ0) and µ ∈ (0, µ0), problem (1.1) has at least three solutions. 4 F. O. DE PAIVA, S. M. DE SOUZA LIMA, O. H. MIYAGAKI EJDE-2023/47 In this work, we observe that for the problem in question, the numbers λ0 and µ0 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: (A4) b1(x) < 1 in RN in a positive measure set; (A5) rb1 > 2. Theorem 1.1. Suppose that the potential A(x) converges to some constant d ∈ NN as |x| → ∞. Assuming (A1)–(A5), there are positive values λ̃0 ≤ λ0, µ̃0 ≤ µ0, and ν0 such that for λ ∈ (0, λ̃0), µ ∈ (0, µ̃0), and ‖a−‖q′ < ν0, 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. 2. Initial considerations According to Tang [23], we denote by HA(RN ) the Hilbert space obtained by the closing of C∞0 (RN ,C) with the inner product 〈u, v〉A = Re ∫ R (∇Au∇Av + uvdx), where ∇Au := (D1u,D2u, . . . ,DNu) and Dj := −i∂j−Aj(x), with j = 1, 2, . . . , N , and with A(x) = (A1(x), . . . , AN (x)). The norm induced by this product is ‖u‖2A := ∫ R (|∇Au|2 + u2dx). Esteban and Lions [14, Section II] proved that that for all u ∈ H1 A(RN ) the dia- magnetic inequality holds, i.e. |∇|u|(x)| = ∣∣Re ( ∇u u |u| )∣∣ = ∣∣Re ( (∇u− iAu) u |u| )∣∣ ≤ |∇Au(x)|. So, if u ∈ H1 A(RN ) we have that |u| belongs to the usual Sobolev space H1 0 (RN ). 2.1. Preliminary results. To obtain the existence, we introduced the Nehari manifold Mλ,µ = {u ∈ H1 A(RN ) \ {0} : 〈J ′λ,µ(u), u〉 = 0}, where 〈·, ·〉 denotes the usual duality between H1 A(RN ) and its dual H1 A(RN )∗. The Nehari manifold is linked to the functions Fu : t→ Jλ,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 Fu,λ,µ, but to simplify the notation, we write Fu. If u ∈ H1 A(RN ), we have Fu(t) = t2 2 ‖u‖2A − tq q ∫ RN aλ(x)|u|q dx− tp p ∫ RN bµ(x)|u|p dx, (2.1) F ′u(t) = t‖u‖2A − tq−1 ∫ RN aλ(x)|u|q dx− tp−1 ∫ RN bµ(x)|u|p dx, (2.2) F ′′u (t) = ‖u‖2A − (q − 1)tq−2 ∫ RN aλ(x)|u|q dx− (p− 1)tp−2 ∫ RN bµ(x)|u|p dx. (2.3) EJDE-2023/47 SCHRODINGER EQUATION WITH MAGNETIC POTENTIAL 5 The following remark relates the Nehari manifold and the Fibering map. Remark 2.1. Let Fu be the application defined above and u ∈ H1 A(RN ). Then: (i) u ∈Mλ,µ if, and only if, F ′u(1) = 0; (ii) more generally tu ∈Mλ,µ, and only if, F ′u(t) = 0. From the previous remark we conclude that the elements in Mλ,µ, correspond to the critical points of the Fibering map. Thus, as Fu(t) ∈ C2(R+,R), we can divide the Nehari manifold into three parts M+ λ,µ = {u ∈Mλ,µ : F ′′λ,µ(1) > 0}, M−λ,µ = {u ∈Mλ,µ : F ′′λ,µ(1) < 0}, M0 λ,µ = {u ∈Mλ,µ : F ′′λ,µ(1) = 0}. The lemma below shows that under some conditions, M0 λ,µ is empty, as shown in [6, Lemma 2.2]. Lemma 2.2. Let µ ≥ 0 and λ > 0 such that λp−2(1 + µ‖b2‖∞)2−q < Υ0. (2.4) Then M0 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 m+ λ,µ = inf u∈M+ λ,µ Jλ,µ(u), m−λ,µ = inf u∈M−λ,µ Jλ,µ(u). 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]. Lemma 2.3. For each u ∈ H1 A(RN ) \ {0} and µ > 0 we have: (i) If ∫ RN aλ(x)|u|q dx ≤ 0, there is a single t−(u) > tmax(u) such that t−(u)u ∈ M−λ,µ. Also, Fu(t) is increasing in (0, t−(u)), decreasing in (t−(u),+∞) and Fu(t)→ −∞ as t→ +∞. (ii) If ∫ RN aλ(x)|u|q dx > 0 and λ is such that λp−2(1 + µ‖b2‖∞)2−q < Υ0, so there is 0 < t+(u) < tmax(u) < t−(u) such that t±(u)u ∈M±λ,µ. Also, Fu(t) is decreasing in (0, t+(u)), increasing in (t+(u), t−(u)) and decreasing in (t−(u),+∞). Furthermore, Fu(t)→ −∞ as t→ +∞. 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 Jλ,µ is coercive and bounded from below in Mλ,µ. For the next results we need some estimates on m±λ,µ. To do this, from (2.4) we have ‖u‖2A < p− q p− 2 ∫ RN aλ(x)|u|q dx ≤ Υ 1/(p−2) 0 p− q p− 2 S−q/2p ‖a+‖Lq′‖u‖ q A. 6 F. O. DE PAIVA, S. M. DE SOUZA LIMA, O. H. MIYAGAKI EJDE-2023/47 Therefore, ‖u‖A ≤ ( Υ 1/(p−2) 0 p− q p− 2 S−q/2p ‖a+‖Lq′ )1/(2−q) ‖u‖qA, (2.5) for all u ∈ M+ λ,µ. Also, if λ = 0, then (2.4) is satisfied, so that by Lemma 2.3(i), M+ λ,µ = ∅, and we have Mλ,µ = M−λ,µ for all µ ≥ 0. By has been seen, we will show the following results on the values of m±λ,µ. Lemma 2.5. (i) If λp−2(1 + µ‖b2‖∞)2−q < ( q2 )p−2Υ0, then m − λ,µ > 0; (ii) For λ > 0 and µ ≥ 0 with λp−2(1 + µ‖b2‖∞)2−q < Υ0, then m + λ,µ < 0. In particular, if λp−2(1 + µ‖b2‖∞)2−q < ( q2 )p−2Υ0, then m+ λ,µ = inf Mλ,µ Jλ,µ(u). 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 ∈ H1 A(RN ) \ {0} Jλ,µ(t−(u)u) = max t≤0 Jλ,µ(tu), (2.6) whenever λp−2(1 + µ‖b2‖∞)2−q < ( q 2 )p−2 Υ0, with λ ≥ 0 and µ > 0. 3. 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 −∆Au+ u = |u|p−2u in RN , u ∈ H1 A(RN ). (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 C2 functional in H1 A(RN ). The Nehari manifold associated with (3.1) is M∞ = {u ∈ H1 A(RN ) \ {0} : J ′∞(u)u = 0}. In this problem we can observe that if u ∈ N∞, then ‖u‖2A = ‖u‖pp. Now consider the minimization problem m∞ = inf M∞ J∞(u). (3.2) In [12] it was shown that there exists ū ∈ H1 A(RN ) such that m∞ = infN∞ J∞(u) = J∞(ū). From these considerations we will show the following result that gives us a description of a sequence (PS) of Jλ,µ. Lemma 3.1. Let {un} ⊂M−λ,µ be a (PS)β sequence in H1 A(RN ) of Jλ,µ, this is, a sequence satisfying Jλ,µ(un) = β + on(1) and J ′λ,µ(un) = on(1) in H−1 A as n→∞, where m+ λ,µ +m∞ < β < m−λ,µ +m∞, then there is a subsequence {un} and u0 ∈ H1 A(RN ), with a non zero u0, such that un = u0 + on(1) strong in H1 A(RN ) and Jλ,µ(u0) = β. Moreover, u0 is a solution of (1.1). EJDE-2023/47 SCHRODINGER EQUATION WITH MAGNETIC POTENTIAL 7 Proof. From (A1)–(A3), we obtain by a standard argument that {un} is a bounded sequence in H1 A(RN ). Then there is a subsequence {un} and u0 ∈ H1 A(RN ) such that un ⇀ u0 weakly in H1 A(RN ) as n→∞. Taking vn = un−u0, we have vn ⇀ 0 weak in H1 A(RN ) as n→∞. Denoting by B(0, 1) the ball centered on the origin of radius 1, we have in B(0, 1) the strong convergence∫ B(0,1) |un|q → ∫ B(0,1) |u0|q. By the Dominated Convergence Theorem we obtain∫ B(0,1) aλ‖un|q − |u0|q| → 0, as n→∞. Then, by Hölder and the integrability of aλ it follows that∣∣ ∫ aλ(x)(|un|q − |u0|q) ∣∣ ≤ on(1) + ∫ Bc(0,1) aλ(x)‖un|q − |u0|q| ≤ on(1) + (∫ Bc(0,1) aλ(x)q ∗ )1/q∗ (‖un‖qp + ‖u0‖qp) ≤ on(1) + εC. As ε > 0 it is arbitrary, we have∫ aλ(x)(|un|q − |u0|q) = on(1). On the other hand, by (A2) and (A3) and the Brezis-Lieb lemma (see [24]), we can conclude that µ ∫ b2(x)|vn|p = on(1), ∫ (1− b1(x))|vn|p = on(1) and ∫ bµ(x)(|un|p− |vn|p − |u0|p) = on(1), which together with the above inequality gives us Jλ,µ(un) = J∞(vn) + Jλ,µ(u0) + on(1). In a similar way we obtain that J ′∞(vn)vn = J ′λ,µ(un)un − J ′λ,µ(u0)u0 + on(1). By hypothesis J ′λ,µ(un)→ 0 strong inH1 A(RN )−1 and un ⇀ u0 weak inH1 A(RN ) as n→ ∞ and so we have J ′λ,µ(u0) = 0. Now, define δ = lim supn→∞ supy∈RN ∫ B(y,1) |vn|p. So we have two cases: (i) δ > 0, and (ii) δ = 0. Suppose that (i) happens. Then there will be a sequence {yn} ⊂ RN such that∫ B(yn,1) |vn|p ≥ δ 2 and for all n ∈ N. Define ṽn(x) = vn(x+ yn). We have that {ṽn} is bounded and ṽn ⇀ v weak and almost everywhere. Making a change of variables we obtain ∫ B(0,1) |ṽn|p ≥ δ 4 . Then ∫ B(0,1) |v|p ≥ δ 4 , (3.3) giving us v 6= 0. But, vn ⇀ 0 weakly; then∫ RN |vn|p ≥ ∫ B(yn,1) |vn|p ≥ δ 2 > 0. (3.4) 8 F. O. DE PAIVA, S. M. DE SOUZA LIMA, O. H. MIYAGAKI EJDE-2023/47 We see that J∞(vn) = 1 2 ∫ (|∇Avn|2 + v2 n)dx− 1 p ∫ |vn|pdx. Likewise, Fvn(t) = J∞(tvn) = t2 2 ‖vn‖2A − tp p ‖vn‖p. For each n ∈ N, we can get tn such that tnvn ∈ M∞. So we build a sequence {tn} ⊂ RN with tn → t0 as n → ∞, such that tnvn ∈ M∞, that is, such that J ′∞(tnvn)tnvn = 0. We see also that J ′∞(vn)vn = ‖vn‖2A − ‖vn‖p = on(1) and F ′vn(t) = J ′∞(tvn)vn = t‖vn‖2A − tp−1‖vn‖p = on(1). (3.5) With this, we have (tn − tp−1 n )‖vn‖2A = tn(1− tp−2 n )‖vn‖2A = on(1). (3.6) From (3.3) we know that ‖vn‖2A 6→ 0 (that is, vn does not converge to zero). Also note that t2−pn = ∫ |vn|p ‖vn‖2A ≥ δ 2c . With that and by (3.6) we obtain that (1−tp−2 n )→ 0, giving us that tn → 1. Now, see that vn ⇀ 0 weak in H1 A(RN ) as n → ∞. With this and by the fact tn → 1, we can conclude that Jλ,µ(un) = J∞(tnvn) + Jλ,µ(u0) + on(1) ≥ m∞ + Jλ,µ(u0). Note that by hypotheses Jλ,µ(un) = β + on(1) with β < m∞ + m+ λ,µ. From there we obtain β + on(1) = Jλ,µ(un) = J∞(tnvn) + Jλ,µ(u0) + on(1) ≥ m∞ + Jλ,µ(u0), giving us m∞ + Jλ,µ(u0) ≤ β + on(1) < m∞ +m+ λ,µ + on(1); therefore Jλ,µ(u0) < m+ λ,µ + on(1). (3.7) We have already seen that J ′λ,µ(un) converges strongly to zero, therefore we obtain J ′λ,µ(u0) = 0. Thus u0 ∈Mλ,µ. Still, by Lemma 2.2, M0 λ,µ = ∅ and by Lemma 2.5, we conclude that m+ > 0 and m− < 0. Then Jλ,µ(u0) ≥ inf Mλ,µ Jλ,µ(u) = inf M+ λ,µ Jλ,µ(u) = m+, which contradicts what we have concluded in (3.7). We have proved that (ii) occurs. In this case, {vn} such that ∫ |vn|p → 0 if n→∞. As we already have J ′∞(vn)vn = on(1) with J ′∞(vn)vn = ‖vn‖2A − ‖vn‖pp and∫ |vn|p → 0, we conclude that ‖vn‖2 → 0 giving us un → u0 strong in H1 A(RN ). See also that u0 6= 0. In fact, note that if u0 = 0 so ṽn = vn = un and ∫ B(0,1) |un|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 −∆Au+ u = aλ(x)|u|q−2u+ bµ(x)|u|p−2u (3.8) EJDE-2023/47 SCHRODINGER EQUATION WITH MAGNETIC POTENTIAL 9 is such that aλ(x)→ 0 and bµ(x)→ 1 as |x| → ∞. Adding the hypothesis of A→ d with d constant as |x| → ∞, problem (3.8) converges to the problem −∆du+ u = |u|p−2u., (3.9) where −∆d = (−i∇+ d)2. Now, by a result of Ding and Liu [13, Lemma 2.5], u is a least energy solutions of Problem (3.9) if and only if v(x) := |u(x)| ∈ H1 it is a least energy solution to the problem −∆v + v = vp−1; v > 0. (3.10) Furthermore, the equations (3.9) and (3.10) share the same least energy. Specifi- cally, we have J∞(u) = I∞(v) = m∞, where J∞ and I∞ represent the corresponding functionals associated with the afore- mentioned problems. According to Berestick, Lions [5] or Kwong [21], equation (3.10) has a unique solution z0 symmetric, positive, and radial. By [15, Theorem 2], for all ε > 0, exists Aε, B0 and Cε positive such that Aε exp(−(1 + ε)|x|) ≤ z0(x) ≤ B0 exp(−|x|), (3.11) |∇z0(x)| ≤ Cε exp(−(1− ε)|x|). (3.12) According to Kurata [20, Lemma 4], defining w0 = z0e −idx, we have that w0 is the unique, symmetrical, positive and radial solution of (3.9). So we will have J∞(w0) = m∞. We see also that z0 = |w0|, which together with (3.11) gives us the inequalities Aε exp(−(1 + ε)|x|) ≤ |w0(x)| ≤ B0 exp(−|x|), (3.13) |∇w0(x)| ≤ Cε exp(−(1− ε)|x|). (3.14) 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− = infu∈M−λ,µ Jλ,µ(u), and m∞ = infu∈M∞ J∞(u) = J∞(w0), we will make the following estimate for such energy levels. Proposition 3.2. For all λ > 0 and µ > 0 satisfying λp−2(1 + µ‖b2‖∞)2−q < Υ0, we have m− < m+ +m∞. The proof of the above proposition is similar to that of [12, Proposition 6.1]; we omit it. 4. 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 M−a0,b0 = {u ∈ H1 A(RN ) \ {0} : 〈J ′a0,b0(u), u〉 = 0} where Ja0,b0 = 1 2 ∫ RN (|∇Au|2 + |u|2) dx− 1 q ∫ a0(x)|u|qdx− 1 p ∫ b0(x)|u|pdx = 1 2 ∫ RN (|∇Au|2 + |u|2) dx− 1 q ∫ a−(x)|u|qdx− 1 p ∫ b1(x)|u|pdx. 10 F. O. DE PAIVA, S. M. DE SOUZA LIMA, O. H. MIYAGAKI EJDE-2023/47 Lemma 4.1. With the above notation we have inf u∈M−a0,b0 Ja0,b0(u) = inf u∈M∞ J∞(u) = m∞. Proof. Let wk be as defined above. Because λ = 0, we have a(x) = λa+(x) + a−(x) = a−(x) < 0 from where ∫ RN a−|t −(wk)wk|qdx ≤ 0, hence by Lemma 2.3(i) there is only one t−(wk) > ( 2−q p−q ) 1 p−2 such that t−(wk)wk ∈ M−a0,b0 for all k > 0; that is, J ′a0,b0(t−(wk)wk) = 0, giving us ‖t−(wk)wk‖2A = ∫ RN a−|t−(wk)wk|qdx+ ∫ RN b−|t−(wk)wk|pdx. (4.1) As w0 is a solution of problem (3.10) and remembering that the functional associ- ated with (3.10) is I(u) = 1 2‖u‖ 2 A − 1 p‖u‖ p p, and I ′(u) = ‖u‖2A − ‖u‖pp we have I ′(w0)w0 = ‖w0‖2A − ‖w0‖pp = 0. Therefore, m∞ = I(w0) = 1 2 ‖w0‖2A − 1 p ‖w0‖pp = 1 2 ‖w0‖2A − 1 p ‖w0‖2A = p− 2 2p ‖w0‖.A Being w0 solution of problem (3.10) follows that wk(x) = w0(x + ke). With this and I ′(w0)w0 = 0, we have I ′(wk)wk = 0. So that ‖wk‖2A = ∫ RN |wk|qdx = 2p p− 2 m∞ for all k ≥ 0. (4.2) It is known that wn is bounded in Lr ′ and wn → 0 a.e., by Theorem [16, Theorem 13.44] that wn ⇀ 0 weakly in Lr ′ . By the condition (A1), a− ∈ (Lr ′ )′ = Lr we obtain ∫ RN a−|wk|qdx→ 0 as k →∞. (4.3) In addition, by (A2) and (A3) we have∫ RN (1− b1)|wk|qdx = ∫ B(0,R) (1− b1)|wk|qdx+ ∫ Bc(0,R) (1− b1)|wk|qdx→ 0, (4.4) as |wk| → ∞. By (4.1), (4.3), and (4.4) we have that t−(wk) → 1 as k → ∞. Likewise lim k→∞ Ja0,b0(t−(wk)wk) = lim k→∞ J∞(t−(wk)wk) = m∞. Thus m∞ = inf u∈M∞ J∞(u) = lim k→∞ J∞(t−(wk)wk) ≥ inf u∈M−a0,b0 Ja0,b0(u). (4.5) We also have u ∈ Ma0,b0 , by Lemma 2.3(i), Ja0,b0(u) = supt≥0 Ja0,b0(tu), and furthermore, there is a single t∞ > 0 such that t∞u ∈M∞. So Ja0,b0(t∞u) = 1 2 ‖t∞u‖2A − (t∞)q q ∫ RN a−|u|qdx− (t∞)p p ∫ RN b1|u|pdx ≥ 1 2 ‖t∞u‖2A − (t∞)p p ∫ RN |u|pdx = J∞(t∞u) ≥ m∞; EJDE-2023/47 SCHRODINGER EQUATION WITH MAGNETIC POTENTIAL 11 therefore inf u∈Ma0,b0 Ja0,b0(t∞u) ≥ m∞. (4.6) By (4.5) and (4.6), we have inf u∈Ma0,b0 Ja0,b0(u) = inf u∈M∞ J∞(u) = m∞. � To obtain the fourth solution of the problem, we need a lemma that establishes suitable values of λ and µ. Lemma 4.2. Exist λ0 > 0 and µ0 > 0 with λp−2 0 (1 + µ0‖b1‖∞)2−q < (q 2 )p−2 Υ0, such that for all λ ∈ (0, λ0) and all µ ∈ (0, µ0), we have∫ RN x |x| (|∇u|2 + u2)dx 6= 0 for all u ∈M−aλ,bµ with Jλ,µ(u) < m+ aλ,bµ +m∞. The above lemma can be obtained arguing as in [12, Lemma 7.6]; we omit its proof. 5. 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 J0,αb0(u) = 1 2 ∫ RN |∇Au|2 + u2dx− 1 p ∫ RN αb0|u|pdx, M0,αb0 = {u ∈ H1 A(RN ) \ {0} : 〈J ′0,αb0(u), u〉 = 0}. We now define the following subset of the unitary ball B = {u ∈ H1 A(RN ) \ {0} : u ≥ 0 and ‖u‖A = 1}. Let us recall that for every u ∈ H1 A(RN ) \ {0} there exists a unique t−(u) > 0 and t0(u) > 0 such that t−(u) ∈ M−aλ,bµ and t0(u) ∈ M0,b0 . To apply the minimax argument of Bahri-Li we present the following result. Lemma 5.1. For each u ∈ B we will have (i) There is a unique tα0 = tα0 (u) > 0 such that tα0u ∈M0,αb0 and sup t≥0 J0,αb0(tu) = J0,αb0(tα0u) = p− 2 2p (∫ RN αb0|u|pdx )−2/(p−2) . (ii) For ρ ∈ (0, 1), Jaλ,bµ(t−(u)u) ≥ (1− ρ) p p−2 (1 + µ‖b2/b1‖∞) 2 p−2 J0,b0(t0(u)u)− 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗) 2 2−q and Jaλ,bµ(t−(u)u) ≤ (1 + ρ) p p−2 2 J0,b0(t0(u)u) + 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗ + ‖a−‖q∗) 2 2−q . 12 F. O. DE PAIVA, S. M. DE SOUZA LIMA, O. H. MIYAGAKI EJDE-2023/47 Proof. (i) For each u ∈ B, we consider Ku(t) = J0,αb0(tu) = 1 2 t2 − 1 2 tp ∫ RN αb0|u|pdx, so Ku(t)→ −∞ as t→∞, and K ′u(t) = t− tp−1 ∫ RN αb0|u|pdx. Thus, K ′u(tα0 ) = 0, and tα0u ∈M0,αb0 as tα0 = tα0 (u) = (∫ RN αb0|u|pdx ) 1 2−p > 0. Moreover, K ′′u(t) = 1− (p− 1)tp−2 ∫ RN αb0|u| pdx. So, for tα0 (u) we have K ′′u(tα0 ) = 2− p < 0, that is, tα0 is a maximum point of Ku. Then, there exists a unique tα0 = tα0 (u) > 0 such that tα0u ∈M0,αb0 and also by definition Ku(t) = J(tu) we obtain sup t≥0 J0,αb0(tu) = J0,αb0(tα0u) = p− 2 2p (∫ RN αb0|u|pdx ) −2 2−p . (ii) We consider α = (1+µ‖b2/b1‖∞)/(1−ρ). Then, for each u ∈ B and ρ ∈ (0, 1), we have∫ RN aλ|tα0u|qdx ≤ λS−q/2p ‖a+‖q∗‖tα0u‖ q A ≤ 2− q 2 ( (ρSp) −q 2 λ‖a+‖q∗ ) 2 2−q + q 2 ( (ρ) q 2 ‖tα0u‖A )2/q = 2− q 2 (ρSp) q q−2 (λ‖a+‖q∗) 2 2−q + qρ 2 ‖tα0u‖2A. (5.1) Then, from part (i) and by (5.1), we have sup t≥0 Jaλ,bµ(tu) ≥ Jaλ,bµ(tα0u) ≥ 1− ρ 2 ‖tα0u‖2A − 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗) 2 2−q − (1 + µ‖b2/b1‖∞) p ∫ RN b0|tα0u|pdx = (1− ρ)J0,αb0(tα0u)− 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗) 2 2−q = (p− 2)(1− ρ) p p−2 2p((1 + µ‖b2/b1‖∞) ∫ RN b0|u|pdx) 2 p−2 − 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗) 2 2−q = (1− ρ) p p−2 (1 + µ‖b2/b1‖∞) 2 p−2 J0,αb0(t0(u)u)− 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗) 2 2−q . By Lemma 2.3 and by Theorem 2.5, we have sup t≥0 Jaλ,bµ(tu) = Jaλ,bµ(t−(u)u). EJDE-2023/47 SCHRODINGER EQUATION WITH MAGNETIC POTENTIAL 13 Thus, Jaλ,bµ(t−(u)u) ≥ (1− ρ) p p−2 (1 + µ‖b2/b1‖∞) 2 p−2 J0,αb0(t0(u)u)−2− q 2q (ρSp) q q−2 (λ‖a+‖q∗) 2 2−q . Furthermore, by Hölder, Sobolev, and Young’s inequalities,∣∣ ∫ RN aλ|tu|qdx ∣∣ ≤ ∫ RN aλ|tu|qdx ≤ (λ‖a+‖q∗ + ‖a−‖q∗)S−q/2p ‖tu‖qA ≤ 2− q 2 (ρSp) q q−2 (λ‖a+‖q∗ + ‖a−‖q∗) 2 2−q + qρ 2 ‖tu‖2A. Also Jaλ,bµ(tu) ≤ (1 + ρ) 2 t2 + 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗ + ‖a−‖q∗) 2 2−q − 1 p ∫ RN b0|tu|pdx ≤ (1 + ρ) p p−2 2 J0,b0(t0(u)u) + 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗ + ‖a−‖q∗) 2 2−q . Then Jaλ,bµ(t−(u)u) ≤ (1 + ρ) p p−2 2 J0,b0(t0(u)u) + 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗ + ‖a−‖q∗) 2 2−q . As we wanted to prove. � Note that as m−aλ,bµ > 0 for all λ ∈ (0, λ0) and µ ∈ (0, µ0), we can define Iaλ,bµ(u) = sup t≥0 Jaλ,bµ(tu) = Jaλ,bµ(t−(u)u) > 0, where t−(u)u ∈ M−aλ,bµ . We can see that if λ, µ and ‖a−‖q∗ are sufficiently small, we can use the minimax Bahri-Li’s argument [4] for our functional Jaλ,bµ . Let Γaλ,bµ = {γ ∈ C(BN (0, k),B) : γ|∂BN (0,k) = wk/‖wk‖A} be for values of l large enough. We define naλ,bµ = inf γ∈Γaλ,bµ sup x∈RN Iaλ,bµ(γ(x)), n0,b0 = inf γ∈Γ0,b0 sup x∈RN I0,b0(γ(x)) By Lemma 5.1(ii), for 0 < ρ < 1, we have naλ,bµ ≥ (1− ρ) p p−2 (1 + µ‖b2/b1‖∞) 2 p−2 n0,b0 − 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗) 2 2−q , (5.2) naλ,bµ ≤ (1 + ρ) p p−2n0,b0 + 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗ + ‖a−‖q∗) 2 2−q . (5.3) We will use the following estimates of the energy levels. Lemma 5.2. m∞ < n0,b0 < 2m∞. Proof. From the results by Bahri and Li [4] we have that (1.1), with aλ = 0 and bλ = b0, admits at least one solution u0 with J0,b0(u0) = n0,b0 < 2m∞. In addition, by (A4), problem (1.1), with aλ = 0 and bλ = b0, does not have a minimum energy solution; this implies the lower estimates. � 14 F. O. DE PAIVA, S. M. DE SOUZA LIMA, O. H. MIYAGAKI EJDE-2023/47 Theorem 5.3. Let λ0 and µ0 be as in Lemma 4.2. Then there will be positive values λ̃0 ≤ λ0, µ̃0 ≤ µ0 and ν̃0 ≤ ν0 such that for λ ∈ (0, λ̃0), µ ∈ (0, µ̃0) and ‖a−‖q∗ < ν0, we have m+ aλ,bµ +m∞ < naλ,bµ < m−aλ,bµ +m∞. In addition, (1.1) admits a solution vaλ,bµ with Jaλ,bµ(vaλ,bµ) = naλ,bµ . Proof. By Lemma 5.1(ii), for 0 < ρ < 1 we have m−aλ,bµ ≥ (1− ρ) p p−2 (1 + µ‖b2/b1‖∞) 2 p−2 m∞ − 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗) 2 2−q , m−aλ,bµ ≤ (1 + ρ) p p−2m∞ + 2− q 2q (ρSp) q q−2 (λ‖a+‖q∗ + ‖a−‖q∗) 2 2−q . For each ε > 0 there are positive values λ̃1 ≤ λ0, µ̃1 ≤ µ0 and ν1 such that λ ∈ (0, λ̃1), µ ∈ (0, µ̃1), and ‖a−‖q∗ < ν1, we have m∞ − ε < naλ,bµ < m∞ + ε. Then 2m∞ − ε < naλ,bµ +m∞ < 2m∞ + ε. Using 5.2 and 5.3, for all δ > 0 there will be positive values λ̃2 ≤ λ0, µ̃2 ≤ µ0, and ν2, such that for λ ∈ (0, λ̃2), µ ∈ (0, µ̃2) and ‖a−‖q∗ < ν2, we have n0,b0 − δ < naλ,bµ < n0,b0 + δ. Fixing small values of 0 < ε < (2m∞ − n0,b0)/2, and being m∞ < n0,b0 < 2m∞, and choosing δ > 0 so that for λ < λ̃0 = min{λ̃1, λ̃2}, µ < µ̃0 = min{µ̃1, µ̃2} and ‖a−‖q∗ < ν0 = min{ν1, ν2}, we will have m+ aλ,bµ +m∞ < m∞ < naλ,bµ < 2m∞ − ε < m−aλ,bµ +m∞. Thus, by Proposition 3.1(ii), we obtain that problem (1.1) has a solution vaλ,bµ with Jaλ,bµ(vaλ,bµ) = naλ,bµ . � Proof of Theorem 1.1. With the result of Theorem 5.3 we can complete the proof of Theorem 1.1. For λ ∈ (0, λ̃0), µ ∈ (0, µ̃0) and ‖a−‖q∗ < ν0, 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. � Acknowledgments. F. O. Paiva received research grants from FAPESP 17/ 16108- 6. S. M. S. Lima was supported by CAPES/Brazil. This article was completed while she was visiting the Departament of Mathematics of UFJF, whose hospi- tality she gratefully acknowledges. O. H. 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MIYAGAKI EJDE-2023/47 Francisco Odair de Paiva Departamento de Matemática, Universidade Federal de São Carlos (UFSCar), São Car- los - SP, CEP 13565-905, Brazil Email address: odair@dm.ufscar.br Sandra Machado de Souza Lima Departamento de Ciências Exatas, Biológicas e da Terra, INFES-UFF, Santo Antônio de Pádua - RJ, Brazil Email address: sandra.msouzalima@gmail.com Oĺımpio Hiroshi Miyagaki Departamento de Matemática, Universidade Federal de São Carlos (UFSCar), São Car- los - SP, CEP 13565-905, Brazil Email address: olimpio@ufscar.br 1. Introduction 2. Initial considerations 2.1. Preliminary results 3. Existence of m 4. Third solution 5. Fourth solution Acknowledgments References