Small Perturbations for Nonlinear Schrödinger Equations with Magnetic Potential

We are concerned with the qualitative analysis of solutions for three classes of nonlinear problems driven by the magnetic Laplace operator. We are mainly interested in the perturbation effects created by two reaction terms with different structure. Two equations are studied on bounded domains (under Dirichlet boundary condition) while the third problem is on the entire Euclidean space. Our main results establish that if a certain perturbation is sufficiently small (in a prescribed sense) then the problems have at least two distinct solutions in a related magnetic Sobolev space. The proofs combine variational, topological and analytic methods.


Introduction
In quantum mechanics, the Hamiltonian for a nonrelativistic charged particle in an electromagnetic field is defined by (−i∇ − A) 2 + b, where b : D ⊂ R N → R is the electric (or scalar) potential, A : D ⊂ R N → R N and the open set D ⊂ R N is the region that constrains the particles. The vector (magnetic) potential A = (A 1 , A 2 , . . . , A N ) is a source for the magnetic field B = curl A, where curl A is the N × N skew-symmetric matrix with entries B jk = ∂ j A k − ∂ k A j . The magnetic potential A can also be expressed in the language of differential forms, namely if A where λ is a positive parameter, N ≥ 3 and 1 < q < 2. Finally, we study a perturbation problem in R N , but from another point of view. More exactly, we will analyze the effect of a small perturbation g in the semilinear elliptic problem where N ≥ 2, 2 < q < 2 * , and A : R N → R N is the magnetic potential. We assume that A and b, K ∈ L ∞ (R N , R) satisfy the following conditions: (A) For all y ∈ Z N there exists ϕ y ∈ H 1 loc R N , R such that A(x + y) − A(x) = ∇ϕ y (x); (B) ess lim |x|→+∞ b(x) = b ∞ ∈ R + and there exists a constant b 1 > 0 such that b 1 ≤ b(x) ≤ b ∞ a.e. in R N ; (K) ess lim |x|→+∞ K(x) = K ∞ ∈ R + and K(x) ≥ K ∞ a.e. in R N ; We recall that the Schrödinger operator with magnetic potential is defined by The problems we study in this paper are related with the existence of solitary waves, namely solutions of the form φ(z, t) := e −i E t u(z), with E ∈ R, for the nonlinear Schrödinger equation where t > 0, N ≥ 2, is the Planck constant, and A is a magnetic potential associated with a given magnetic field B, U (z) is a real electric potential and the nonlinear term l is a superlinear function. A direct calculation states that φ is a solitary wave of problem (4) if and only if u is a solution of the problem Vol.88 (2020) Small Perturbations for Nonlinear Schrödinger Equations 481 Small Perturbations for Nonlinear Schrödinger Equations 3 where b(z) = U (z) − E. It is significant to study the existence and the shape of such solutions in the semiclassical limit, that is, as → 0. The significance of this research relies on this fact that the transition from Quantum Mechanics to Classical Mechanics can be formally performed by sending the Planck constant to zero. In the last decades, many authors have conducted extensive research on perturbations of problem (5) and of boundary value problems without magnetic potential (namely A ≡ 0 and = 1). We first give the elementary example −∆u = |u| q−2 u in Ω, where Ω is a smooth bounded domain in R N (N ≥ 2) and 2 < q < 2 * . A classical result, based on a Z 2 symmetric version of the Mountain Pass Theorem see Ambrosetti and Rabinowitz [3] , shows that problem (6) admits infinitely many solutions in H 1 0 (Ω, R). A natural question is to see what happens if the above problem is affected by a certain perturbation. Consider the problem −∆u = |u| q−2 u + η(x) in Ω, u = 0 on ∂Ω.
To the best of our knowledge, the first result in the presence of a magnetic potential A ≡ 0 seems to be obtained by Esteban and Lions [12]. They have made use of the concentration-compactness principle and minimization arguments to obtain solution for > 0 fixed and dimensions N = 2 or N = 3. We also would like to cite the papers [1,11,4] for other results related to the problems (1) and (3) in the presence of magnetic field when the nonlinearity has a subcritical growth. We also refer to the papers [2,4,13,14] for the critical case. Now, we assume that A, A ∈ L s loc (R N , R N ) for some s ∈ [1, ∞) and curl A = B = curl A (in the sense of distributions). Then A−A = ∇ϕ for some ϕ ∈ W 1,s loc (R N ), see [17,Lemma 1.1] . We can easily deduce that if u = e iϕ , then is given in Section 2. The above properties are called the gauge invariance and the transformation T : u → u the change of gauge. The condition (A) is inspired by hypothesis A2 introduced by Arioli and Szulkin [4]. By hypothesis (A), we can define a different "transformation" T (see Subsection 5.1) satisfying the above gauge invariance.
Motivated by the works of Rȃdulescu-Smets [22] and Cîrstea-Rȃdulescu [10], we are concerned with the qualitative analysis of solutions for three classes of nonlinear problems driven by the magnetic Laplace operator. Due to the existence of the magnetic potential A, problems (1), (2) and (3) cannot be converted into pure real-valued problems, thus we are supposed to directly treat these problems with 482 Y. Zhang, X. Tang and V.D. Rȃdulescu Vol.88 (2020) 4 Y. Zhang, X. Tang and V.D. Rȃdulescu complex values. But in this way, we will encounter some new challenges when dealing with our problems. In particular, we refer to the combined effects of the lack of compactness of problems (2) and (3) and the presence of a magnetic potential. Since we deal with different problems where the functions are complex-valued, it is necessary to make a careful analysis of technical estimates used in this paper. By establishing new threshold estimates we overcome the lack of compactness in problem (2). Furthermore, we overcome the lack of compactness of problems (3) by employing a variant of the Mountain Pass theorem without the Palais-Smale condition (see Brezis and Nirenberg [9, Theorem 2.2]), combined with a generalization of the Brezis-Lieb lemma [8, Theorem 1]. We also refer to Pucci and Rȃdulescu [20] for a survey concerning the Mountain Pass theorem.
If we perturb problems (1), (2) and (3) such that the perturbations do not exceed some levels, then we will show in this paper that problems (1), (3) have at least two solutions while problem (2) has at least one nontrivial solution. More precisely, if λ and g b,−1 (see Section 2) are sufficiently small, then, problem (2) has a mountain pass solution, while problems (1), (3) have local minimums near the origin, whereas, the second solution is obtained as a mountain pass. Since problem (3) is investigated in R N , we need to take advantage of the hypotheses (B), (K) and (M ) in order to conclude the existence of the mountain pass solution for problem (3), while the existence of a simple solution (the local minimum) will follow without these stronger assumptions for problem (3).
We refer to Laptev et al. [15,16] for recent advances in the study of magnetic differential operators and to Papageorgiou, Rȃdulescu and Repovš [19] for some of the abstract methods used in this paper.

Abstract framework and the main results
In this section, we outline the variational framework and give the statement of the main results. We refer to Esteban-Lions in [12] and Lieb-Loss in [18] for more details on the complex-valued function Sobolev spaces with magnetic potential.
Let U be an open set in R N . For u : U → C, we define Let Ω be a domain with smooth boundary in R N , we shall denote by H A (Ω) the Hilbert space obtained by the closure of C ∞ 0 (Ω, C) under the scalar product where Re(ω) denotes the real part of ω ∈ C, ω is its complex conjugated. Moreover, we shall denote by u A the norm induced by this scalar product, that is, For u : R N → C, let us define and, for N ≥ 3, The spaces H A R N and D 1,2 A R N are Hilbert spaces endowed with scalar product, respectively, Let u H A (R N ) and u D 1,2 A (R N ) denote the norms induced by the scalar products, namely By [12, Section 2] and [18,Theorem 7.22], [12], the function space D 1,2 A R N has been defined as the closure of C ∞ 0 R N , C with respect to the norm u D 1,2 A (R N ) . Finally, the spaces H A R N and Let H b (R N ) and H b∞ R N be the Sobolev spaces defined as the completion of C ∞ 0 R N , C with respect to the norms respectively. The spaces H b (R N ) and H b∞ R N are Hilbert spaces endowed with scalar products respectively.

484
Y. Zhang, X. Tang and V.D. Rȃdulescu Vol.88 (2020) 6 Y. Zhang, X. Tang and V.D. Rȃdulescu We shall denote by · −1 and · b,−1 the norms of H −1 A (Ω) and Throughout this work we suppose that Definition 2.1. We say that a function u ∈ H A (Ω) is a weak solution of (1) if Definition 2.2. We say that a function u ∈ H A (Ω) is a weak solution of (2) if Re Our main results are the following. (1) has at least two solutions. Theorem 2.5. There exists λ * > 0 such that problem (2) has at least one nontrivial solution for all λ ∈ (0, λ * ).   respectively. The following well-known diamagnetic inequality is proved by Esteban-Lions in [12] and Lieb-Loss in [18].
. Thus, the weak solutions of problems (1) and (2) are precisely the critical points of J and J , respectively.
The following result shows that H b R N is continuously embedded in L q R N , C . Applying this fact and (K) we deduce that the functional Ψ is well- Proof. By Lemma 2.7 and the hypotheses of b, it is obvious that Proposition 2.8 holds true. This proof is now complete.
In this paper we denote by " " the weak convergence and by "→" the strong convergence in an arbitrary Banach space X. Remark 2.9. Let u n be a sequence that converges weakly to some For all u ∈ H b R N , we define the functionals Υ : H b R N → R and Υ ∞ :

we can easily see that the norms of the spaces
respectively. A direct calculation shows that Ψ, Υ, Υ ∞ ∈ C 1 H b R N , R and their derivatives are given by Brezis and Lieb established in [8, Theorem 1] a subtle refinement of Fatou's lemma. Our following result is a weighted variant of the Brezis-Lieb lemma. This proof uses some ideas found in Cîrstea and Rȃdulescu [10, Lemma 2]. We give the details of the proof for the convenience of the reader.

Lemma 2.10. Let u n be a sequence which is weakly convergent to
Proof. By Proposition 2.8 and the boundedness of u n in H b R N we can see that u n is a bounded sequence in L q R N , C . For given ε > 0 we take R ε > 0 such that We first observe that where 0 ≤ θ(x) ≤ 1. On the other hand, by relation (10) and Hölder's inequality we can deduce that for some constants c, c > 0 independent of n and ε. Next, using relation (9), It follows from relations (10)-(13) that lim sup Thanks to ε > 0 is arbitrary we conclude that which completes our proof. Proof. By a simple computation we obtain that Let ε be a positive number. The hypothesis (K) yields that there is R ε > 0 such that for some constant M > 0 independent of n and ε. It follows that To prove (14) and (15) we need only to show that For this purpose, notice that for every R > 0 we can obtain that Vol.88 (2020) Small Perturbations for Nonlinear Schrödinger Equations 489 Small Perturbations for Nonlinear Schrödinger Equations 11 By (B) we get that, for every ε > 0, there exists R ε > 0 such that But, it follows from Remark 2.9 that H b R N is continuously embedded in L 2 R N , C . Furthermore, using (9) we see that v n → 0 in L 2 loc R N , C . Thus, by relations (17) and (18) we deduce that there is a positive number L which is independent of n and ε such that Due to ε > 0 is a arbitrary, it follows that (16) holds true. This proof is now complete.

Proof of Theorem 2.4
We first prove that the weak limit (if this exists) of any (P S) c sequence of the energy functional J is a solution of problem (1).
Proof. Consider an arbitrary function ξ ∈ C ∞ 0 (Ω, C) and set Θ = supp ξ. Clearly, the fact that J (u n ) → 0 in H −1 A (Ω) implies J (u n ), ξ → 0 as n → ∞, that is, It follows that since u n u 0 in H A (Ω). The boundedness of u n in H A (Ω) and the continuous embedding H A (Ω) → L q (Ω, C) imply that {|u n | q−2 u n } is bounded sequence in L q/(q−1) (Ω, C). Combining this with the convergence (up to a subsequence) we can get (see [7]) that |u 0 | q−2 u 0 is the weak limit of the sequence |u n | q−2 u n in L q/(q−1) (Ω, C). Hence, By relations (19), (20) and (21) we conclude that By density, this equality holds for any ξ ∈ H A (Ω) which implies that J (u 0 ) = 0. This proof is now complete.
Lemma 3.2. For any 0 < ε < 1 there exist R = R(ε) > 0 and C = C(ε) > 0 such that for all f = 0 and 0 < λ Proof. Fix 0 < ε < 1. Then for any u ∈ H A (Ω), using the Sobolev and Young's inequalities we obtain where C 0 > 0 is a positive Sobolev constant given by the continuous embedding For instance, we can take Since f = 0 and λ > 0, we have c 0 < J(0) = 0. The set B R becomes a complete metric space with respect to the distance Additionally, J is weakly lower semi-continuous and bounded from below on B R . Thus, similarly with the proof of Corollary I.5.3 of Struwe [24] (see also [25, Corollary 2.5]), we can deduce that there exists a minimizing sequence {u n } of J with u n A < R such that But the fact that u n A < R, for fixed R, implies that u n converges weakly (up to a subsequence) in H A (Ω). Combining the compact embedding H A (Ω) → L q (Ω, C), relation (22) and Lemma 3.1 we obtain that for some u 0 ∈ H A (Ω) J (u 0 ) = 0. Combining (22)-(23) and the weakly lower semi-continuity of J, we can deduce that Due to u 0 ∈ B R , we see that J(u 0 ) = c 0 . This proof is now complete.
3.1. The second solution of Theorem 2.4 Now we show that the functional J satisfies the mountain pass geometry (see [25]).
Lemma 3.3. For any fixed 0 < ε 1 < 1 and f = 0, the functional J satisfies the following properties: Proof. (i) Using the conclusion and proof of Lemma 3.2, we can easily get this result.
(ii) For each 0 = u ∈ H A (Ω) and t > 0, one has Since q > 2, this relation shows that for all λ ∈ (0, C 1 f −1 ) we can find t λ > 0 such that J(t λ u) < 0. So, we get the conclusion.
This proof is now complete.

So, we have
Note that 2 < q < 2 * and λ ∈ (0, C 1 f −1 ), for n big enough we can find that there exists a constant C > 0 such that It follows immediately that that u n is bounded in H A (Ω). Thus, up to a subsequence, we may assume that there exists u ∈ H A (Ω) such that  Using this fact and Hölder's inequality, it is easy to see that This proof is now complete.
On the other hand, it is easy to see that J (

Proof of Theorem 2.4 completed
Consider R 1 > 0, C 1 = C 1 (R 1 ) > 0 and δ R 1 > 0 given by Lemma 3.3. Thus, in light of its proof, we are able to obtain that for all f = 0 and 0 < λ < C 1 f −1 the conclusion of Lemma 3.2 also holds true. Hence, we get the existence of a solution u 0 ∈ H A (Ω) of problem (1) such that J(u 0 ) = c 0 < 0.
Clearly, u 0 = u . This proof of Theorem 2.4 is now complete.

Proof of Theorem 2.5
The energy functional associated to problem (2) is J : Let S denote the best constant of the continuous embedding of H A (Ω) into L 2 * (Ω, C), that is, It follows that First of all, note that 1 < q < 2. Fix ε > 0 small enough. Combining the Hölder's and Young's inequalities we obtain for all u ∈ H A (Ω) where Combining relations (24) and (25) we have Since ε > 0 can be chosen sufficiently small, we deduce that u n ⊂ H A (Ω) is bounded. Taking n → ∞, it follows that for all v ∈ H A (Ω) Re Since J (u n ) −1 = o(1) as n → ∞, these relations show that Therefore Choose ε = 2 (2−q)N +2q . Then relation (26) yields the following lower estimate where A(q, N ) is a positive constant. Fix λ > 0 such that We already know that Next, the Brezis-Lieb theorem implies and Since {u n } is a (P S) c sequence, relations (29), (30) and (31) imply that Next, using the fact that J (u n ) −1 = o(1) as n → ∞ in conjunction with J (u 0 ), u 0 = 0 we obtain Combining relations (32) and (33) we deduce that there exists ≥ 0 such that We claim that if c < c * (where c * is defined in (28)) then = 0, which implies that the sequence u n is convergent (up to a subsequence) to u 0 in H A (Ω).
Fix c < c * and assume that > 0. Then relation (33) implies that which contradicts the choice of c. This proves our claim and the proof is now complete.
By relation (25) we have for all u ∈ H A (Ω) Taking ε ∈ (0, 1/2), we can choose two positive numbers λ * and r so that both relation (28) and hold. This property establishes the existence of a "mountain" near the origin and for small perturbations (that is, small positive values of the parameter). Next, we prove the existence of a "valley". For this purpose, let φ 1 be a positive element in H A (Ω). It follows that for all t > 0 This relation shows that for all λ ∈ (0, λ * ) we can find t λ > 0 such that J (t λ φ 1 ) < 0, hence c λ := inf{J (u); u A ≤ r} < 0 < inf{J (u); u A = r}.
Therefore, similarly with the proof of Corollary I.5.3 of Struwe [24] (see also [25,Corollary 2.5]), we can deduce that there exists a minimizing sequence {u n } of J with u n A < r such that J (u n ) → c λ and J (u n ) −1 → 0.
Combining Lemma 4.2 with the fact that c λ < 0, we deduce that this minimizing sequence is relatively compact in H A (Ω). It follows that its limit is a solution of problem (2). This solution is nontrivial, since c λ < 0. The proof of Theorem 2.5 is now complete.

Proof of Theorem 2.6
We start by proving that the weak limit (if this exists) of any (P S) c sequence of Ψ is a solution of problem (3).  Proof. Consider an arbitrary function ξ ∈ C ∞ 0 (R N , C) and set Θ = supp ξ . Obvi- From the boundedness of u n in H b (R N ) and Proposition 2.8 we know that |u n | q−2 u n is bounded sequence in L q/(q−1) R N , C . Combining this with the convergence which is a consequence of (9) |u n | q−2 u n → |u 0 | q−2 u 0 a.e. in R N we conclude (see [7]) that |u 0 | q−2 u 0 is the weak limit of the sequence |u n | q−2 u n in From (35), (36) and (37) we deduce that By density, this equality holds for any ξ ∈ H b R N which means that Ψ (u 0 ) = 0. This concludes our proof.
Proof. Fix 0 < ε < 1. Then for any u ∈ H b R N , by (K) and Young's inequality we have where M q > 0 is a positive constant given by Proposition 2.8. The above estimate states the existence of R = R (ε) > 0, C = C (ε) > 0 and δ = δ(R ) > 0 such that Ψ(u)| ∂B R ≥ δ > 0 for all g ≡ 0 with g b,−1 ≤ C . For example, we can choose Since g ≡ 0, c 0 < J(0) = 0. The set B R becomes a complete metric space with respect to the distance On the other hand, Ψ is weakly lower semi-continuous and bounded from below on B R . So, similarly with the proof of Corollary I.5.3 of Struwe [24] (see also [25,Corollary 2.5]), we can deduce that there exists a minimizing sequence {u n } of Ψ with u n b < R such that But u n b < R , for the fixed R , we shows that u n converges weakly (up to a subsequence) in H b R N . Therefore, (9), (38) and Lemma 5.1 imply that, for We prove that Ψ(u 0 ) = c 0 . By (38) and (39) we have

By (38)-(40) and Fatou's lemma we have
Since u 0 ∈ B R , it follows that Ψ(u 0 ) = c 0 . This proof is now complete. Firstly, we consider the following constrained minimization problem By virtue of (A) we can define a different "translation" T : (x) . Note that in general T y 1 +y 2 = T y 2 T y 1 , hence T is not a group action of Z N . That the operator T is well-defined is a consequence of the following property.
In particular, for each y ∈ Z N the operator T is an isometry.
Proof. Similarly with the proof of Lemma 4.1 of Arioli and Szulkin [4], we can get the conclusion. For the convenience of readers, we give the details. Indeed, from condition (A), it follows that Going if necessary to a subsequence, we may assume the existence of y n ⊂ R N such that So, using the above inequality we can easily obtain Let us define v n := T [yn] u n . Applying Lemma 5.3 we see that Since v n is bounded in H b∞ R N , we may assume, going if necessary to a subsequence Using the Brezis-Lieb lemma as in [8,25] we have where w n := v n − v. Therefore, similarly with the proof of Theorem 1.34 of Willem [25] we have This proof is now complete.
Consider the following Nehari manifold Proof. For all ϕ ∈ H b R N \ 0 , let us define Since 2 < q < 2 * , it is clear that k(s) has a maximum which is a unique critical point for k on s : s ≥ 0 . Moreover, taking any ϕ on the set one immediately finds a unique element such that s ϕ ϕ ∈ S . From Lemma 5.4 we can see that the minimum of problem (41) is achieved by v ∈ H b∞ R N , and then using (B) we can easily deduce that v ∈ H b R N . So, we can conclude that there exists a unique Now Ψ ∞ can be easily computed in terms of m ∞ . Indeed, using (B) we can obtain . This proof is now complete.
Lemma 5.6. Assume u n is (P S) c sequence of Ψ that converges weakly to u 0 in H b R N . Then the following alternative holds: either u n converges strongly in Proof. Since u n is a (P S) c sequence and u n u 0 in H b R N we get Ψ(u n ) = c + o(1) and Ψ (u n ), u n = o(1). We set v n = u n − u 0 . Then v n 0 in H b R N which yields We rewrite the above relations as By (44), (45) and Lemmas 5.1, 2.10, it follows that If v n → 0 in H b R N , then combining this fact that v n 0 in H b R N we may assume that v n b → l . Then (46) and Lemma 2.11 yield where lim n→∞ µ n = 0, In light of (47), it remains to prove that If we show the existence of a sequence s n with s n ≥ 0, s n → 1 and On account of α n → l ≥ l 2 > 0, lim n→∞ µ n = 0 and q > 2, then for n sufficiently large, we define δ n + = 2|µ n |/(qα n − 2α n ) and δ n − = −2|µ n |/(qα n − 2α n ), which verify the following properties: Lemma 5.7. There are R 1 > 0, C = C(R 1 ) > 0 and δ R 1 > 0 such that for all g with g b,−1 < C we have Ψ | ∂B R 1 ≥ δ R 1 and c 1 < c 0 + Ψ ∞ , where c 1 is given by (51) and c 0 = inf B R 1 Ψ(u).

Proof of Theorem 2.6 completed
Consider R 1 > 0, C = C (R 1 ) > 0 and δ R 1 > 0 given by Lemma 5.7, in view of its proof, we get that for all g ≡ 0 with g b,−1 < C the conclusion of Lemma 5.2 also holds. So, we get the existence of a solution u 0 ∈ H b R N of problem (3)  This yields Therefore u n is bounded sequence in H b R N and, passing to subsequence, we may suppose that u n u 1 in H b R N for some u 1 ∈ H b R N . Thus, using Lemma 5.1, u 1 is a weak solution of problem (3). Finally, we prove that J(u 0 ) = J(u 1 ). Indeed, from Lemma 5.6, either u n → u 1 in H b R N which gives Ψ(u 1 ) = lim n→∞ Ψ(u n ) = c 1 > 0 > c 0 = Ψ(u 0 ) and the conclusion follows, or If we assume that Ψ(u 1 ) = Ψ(u 0 ) = c 0 , then c 1 ≥ c 0 + Ψ ∞ which contradicts Lemma 5.7.
This proof of Theorem 2.6 is now complete.