Multiple solutions to a nonlinear Schrödinger equation with Aharonov-Bohm magnetic potential

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1 Multiple solutions to a nonlinear Schrödinger equation with Aharonov-Bohm magnetic potential Mónica Clapp and Andrzej Szulkin Abstract. We consider the magnetic nonlinear Schrödinger equations ( i + sa) 2 u + u = u p 2 u, p (2, 6), ( i + sa) 2 u = u 4 u, in = O R, where O an open subset of R 2 {0}, s R, and A : R 3 R 3 is the Aharonov-Bohm magnetic potential A(x 1, x 2, x 3) := 1 x x2 2 ( x 2, x 1, 0). We prove multiplicity results and describe the symmetry properties of the solutions obtained. Mathematics Subject Classification (2000). Primary 35Q55; Secondary 35J60. Keywords. Magnetic nonlinear Schrödinger equation, Aharonov-Bohm potential, gauge invariance, ground state, symmetry properties. 1. Introduction In quantum mechanics the Hamiltonian for a non-relativistic charged particle in an electromagnetic field is given by ( i + A) 2 + V, where V : R N R is the electric (or scalar) potential and A : R N R N is the magnetic (or vector) potential which is a source for the magnetic field B = curl A. By definition, if A = (A 1,..., A N ) then curl A is the N N skew-symmetric matrix with entries B jk = j A k k A j, or in the language of differential forms, if A is the 1-form A 1 dx A N dx N then B = da. M. Clapp was supported by CONACYT and PAPIIT IN

2 2 Mónica Clapp and Andrzej Szulkin Consider the magnetic nonlinear Schrödinger problem { ( i + sa) 2 u + u = u p 2 u, (,s ) u H 1 (, C), where = O R with O an open subset of R 2 {0}, s R, p (2, 6), and A : R 3 R 3 is the Aharonov-Bohm magnetic potential 1 A(x 1, x 2, x 3 ) := x ( x 2, x 1, 0). x2 2 The space H 1 (, C) is the closure of C c (, C) with respect to the norm ( u sa := ( sa u 2 + u 2)) 1/2, where sa u := u + isau. We note that curl A = 0 in while sa = 2πs if γ is a properly oriented γ simple closed curve enclosing the x 3 -axis. This integral is called the magnetic flux and describes the influence of a magnetic potential on a charged quantummechanical particle moving in a region where the magnetic field is 0 (the so-called Aharonov-Bohm effect, see e.g. [23]). In our setting the particle is confined to a region outside a thin solenoid extending along the x 3 -axis. If O is the annulus O a,b := {(x 1, x 2 ) : a < x x 2 2 < b}, 0 a < b, we write a,b := O a,b R. We shall prove the following results. Theorem 1.1. Let = a,b with 0 a < b. For every n Z there exists a nontrivial solution u n of problem (,s ) with the following properties: (a) u 0 = u 0 and u n > 0 in, (b) u n u m if s + m s + n, (c) u n (g(x 1, x 2 ), x 3 ) = g n u n (x 1, x 2, x 3 ) for every g SO(2), x a,b, (d) u n (x 1, x 2, x 3 ) = u n (x 1, x 2, x 3 ) for every x a,b, (e) u m sa < u n sa if s + m < s + n, (f) lim n u n sa =. Here SO(2) denotes the group of rotations of the plane R 2 C or, equivalently, the group of unit complex numbers acting by multiplication on C. Let G be a closed subgroup of SO(2). A subset O of R 2 is said to be G-invariant if gx O for every g G and x O, and a function u : C is said to be G-invariant if u(g(x 1, x 2 ), x 3 ) = u(x 1, x 2, x 3 ) for every g G. The closed subgroups of SO(2) consist of the cyclic groups G n := {e 2πik/n : k = 0,..., n 1} generated by the rotation by angle 2π/n, n N, and G := SO(2) itself. Note that the solutions given by Theorem 1.1 are symmetry breaking: if n 0, then u n is G n -invariant but it is not G m -invariant for any m > n. Moreover, properties (a) and (c) assert that the map u n u n : a,b S 1 := {z C : z = 1}

3 NLS equation with Aharonov-Bohm magnetic field 3 is well defined and has degree n in the sense that, for an appropriate orientation of a,b, the induced homomorphism of the fundamental groups π 1 ( a,b ) π 1 (S 1 ) is multiplication by n. Theorem 1.2. Let G n = {e 2πik/n : k = 0,..., n 1}, n N. If O is bounded and G n -invariant, then problem (,s ) has at least n nontrivial solutions u 0,..., u n 1 which satisfy u m (g(x 1, x 2 ), x 3 ) = g m u m (x 1, x 2, x 3 ) g G, x, m = 0,..., n 1. The above result applied to the trivial group G 1 = {1} asserts the existence of at least one solution for every domain = O R with O bounded and every s R. We also consider the problem with critical nonlinearity { ( ( i + sa) 2 u = u 4 u,,s) u D 1,2 (, C), where D 1,2 (, C) is the closure of the space C c (, C) with respect to the norm ( ) 1/2 u sa, := sa u 2. We prove the following. Theorem 1.3. If either = 0, or = a,b with 0 < a < b < then for every n Z there exists a nontrivial solution u n of problem (,s ) with the following properties: (a) u 0 = u 0 and u n > 0 in, (b) u n u m if s + m s + n, (c) u n (g(x 1, x 2 ), x 3 ) = g n u n (x 1, x 2, x 3 ) for every g SO(2), x a,b, (d) u n (x 1, x 2, x 3 ) = u n (x 1, x 2, x 3 ) for every x a,b, (e) u m sa, < u n sa, if s + m < s + n. The first existence results for problems of this type with a nonsingular magnetic potential were obtained by Esteban and Lions in [12]. In particular, for A(x 1, x 2, x 3 ) = ( x 2, x 1, 0) they proved the existence of a sequence of solutions to equation ( i + A) 2 u + u = u p 2 u in R 3. Further existence results for nonsingular magnetic potentials may be found for example in [2, 22, 5, 10]. Existence and multiplicity of semiclassical solutions are given e.g. in [16, 9, 6, 3, 7, 8]. Very recently, Abatangelo and Terracini obtained existence results for problems with critical nonlinearity and singular magnetic and electric potentials [1]. Problems (,s ) and (,s ) do not always have a ground state in H1 (, C) and D 1,2 (, C) respectively, see Propositions 2.1, 2.2 and 3.4. However, they may have ground states on appropriate subspaces. In particular, the solution u 0 in Theorems 1.1 and 1.3 will be obtained by constrained minimization in the subspaces of H 1 (, C) and D1,2 (, C) consisting of SO(2)-invariant functions. The other solutions are obtained as a consequence of the gauge invariance of problems (,s )

4 4 Mónica Clapp and Andrzej Szulkin and (,s ). Observe that, unlike the case of nonsingular magnetic potentials in RN whose curl is 0, the Aharonov-Bohm potential A is not the gradient of a function defined on. However, it is the gradient of a function which is defined locally. Namely, if θ(x 1, x 2, x 3 ) denotes the polar angle of x 1 + ix 2 then θ = A. The function e iθ is uniquely defined (while θ is only unique up to an integer multiple of 2π) and a direct computation shows that if v s+n is a solution of (,s+n ) then e inθ v s+n is a solution of (,s ), and similarly for (,s ). We use this fact to construct u n by taking v s+n to be an SO(2)-invariant solution of (,s+n ) and setting u n := e inθ v s+n. A more detailed study of gauge invariance in domains which are not simply connected may be found in [10]. In what follows θ = θ(x 1, x 2, x 3 ) denotes the polar angle of x 1 + ix 2. (1.1) Semiclassical solutions having the kind of symmetry described by (c) in the theorems above have been recently considered in [7]. We will show that the solution u n in Theorems 1.1 and 1.2 is a ground state for problem (,s ) in the subspace of H 1 (, C) consisting of functions which satisfy the symmetry property (c). A similar statement holds for the solution u n in Theorem 1.3, see Propositions 2.6 and 3.8 below. This paper is organized as follows. In Section 2 we discuss the subcritical case and prove Theorems 1.1 and 1.2. In Section 3 we discuss the critical case and prove Theorem The subcritical case 2.1. Preliminaries Let = O R with O an open subset of R 2 {0}, s R, and let A : R 3 R 3 be the Aharonov-Bohm magnetic potential Thoughout this section we fix p (2, 6). For 0 a < b we set 1 A(x 1, x 2, x 3 ) := x ( x 2, x 1, 0). x2 2 O a,b := {(x 1, x 2 ) : a < x x 2 2 < b} and a,b := O a,b R. We start by recalling some useful inequalities. If u H 1 (, C), then the absolute value u of u is in H0 1 (, R) and u (x) sa u(x) for a.e. x. (2.1) This is called the diamagnetic inequality [18]. Together with the Sobolev inequality it yields ( ) 1/q u q := u q C q u sa q [2, 6], (2.2)

5 NLS equation with Aharonov-Bohm magnetic field 5 where C q > 0 is a constant depending on q. Hence, H 1 (, C) is continuously embedded in L q (, C) for all q [2, 6]. Inequalities (2.1) and (2.2) hold for more general vector potentials. For our specific A more is true. It was shown in [17] that the Aharonov-Bohm potential in R 2 satisfies R 2 2 ϕ x x2 2 C R 2 ϕ + i( x 2, x 1 ) x ϕ x2 2 2 ϕ C c (O 0,, C) for some positive constant C. Following [1], we integrate this inequality with respect to x 3 and obtain Hence, R 3 2 ϕ x x2 2 C A ϕ 2 R 3 ϕ Cc ( 0,, C). ( ) ϕ 2 c sa ϕ 2 + saϕ 2 R 3 R 3 R ( 3 ) = c sa ϕ s 2 ϕ R 3 R x x2 2 C s R 3 sa ϕ 2 ϕ C c ( 0,, C). This inequality implies that ϕ 2 := ( ϕ 2 + ϕ 2 ) C s ( sa ϕ 2 + ϕ 2 ) = C s ϕ 2 sa ϕ Cc (, C). (2.3) If 0 / then A L () and both norms are equivalent. But in general they are not. However, this last inequality asserts that H 1 (, C) is continuously embedded in the nonmagnetic Sobolev space H0 1 (, C). As has been observed in [1], these spaces are in general not equal. This can be seen by taking u(x) = χ(x) x 1, where χ Cc (R 3, [0, 1]) is such that χ = 0 for x 1 and χ = 1 for 2 x 3. Then u H 1 (R 3, C). Note that, since a line in R 3 has capacity 0, H0 1 ( 0,, C) = H 1 (R 3, C) (see Theorem in [13] and Theorem in [20]). On the other hand, u H 1 ( 0,, C) unless s = 0. The same conclusion remains valid for the spaces D 1,2 ( 0,, C) and D 1,2 (R 3, C). The nontrivial solutions of (,s ) are in one-to-one correspondence with the critical points of the functional on the manifold I s (u) := u 2 sa Σ,s,p := {u H 1 (, C) : u p = 1}.

6 6 Mónica Clapp and Andrzej Szulkin More precisely, u Σ,s,p is a critical point of I s iff I s (u) 1 p 2 u is a nontrivial solution of (,s ). Set S,s,p := inf I s (u) = u Σ,s,p inf u H 1 (,C) u 0 u 2 sa It follows from inequality (2.2) that S,s,p > 0. Observe that sa u = sa u, where u stands for the complex conjugate of u. Hence I s (u) = I s (u) and, consequently, S,s,p = S, s,p Existence and nonexistence of ground states Next we investigate whether S,s,p is attained. Let ω H 1 (R 3, R) be the positive radially symmetric ground state of problem { v + v = v p 2 v, v H 1 (R 3 (2.4), R). Let S p := ω 2. The following holds. ω 2 p u 2 p Proposition 2.1. If O O a, for some a (0, ) then S,s,p = S p. Proof. Let ψ C (R 2, R) be radially symmetric and such that 0 ψ 1, ψ(x 1, x 2 ) = 0 if x x 2 2 a 2 and ψ(x 1, x 2 ) = 1 if x x 2 2 2a 2. Define ω n (x 1, x 2, x 3 ) := ω(x 1 n, x 2, x 3 ), v n (x 1, x 2, x 3 ) := ψ(x 1, x 2 )ω n (x 1, x 2, x 3 ). Since v n is real-valued, using the fact that A(x 1, x 2, x 3 ) 0 as x x 2 2, we obtain v n 2 sa ( v = n 2 + sav n 2 + v n 2) ( ) = ψ ω n + ω n ψ 2 + ψ 2 saω n 2 + ψ 2 ωn 2 R 3 ( ) = ψ 2 ω n 2 + ωn 2 (ψ ψ) ωn 2 + ψ 2 sa 2 ωn 2 R 3 R 3 R ( 3 ω 2 + ω 2) + o(1), R 3 v n p p R = ω p + (ψ p 1) ω n p = ω p + o(1). 3 R 3 R 3 Therefore, v n 2 sa v n 2 p ω 2 + o(1) ω 2 p + o(1) = S p + o(1)..

7 NLS equation with Aharonov-Bohm magnetic field 7 On the other hand, the diamagnetic inequality (2.1) yields S p v 2 v 2 p We conclude that S,s,p = S p. v 2 sa v 2 p v H 1 (, C) {0}. (2.5) As we have mentioned in the introduction, the function θ given by (1.1) is defined only locally in 0, while if m Z then the function x e imθ(x) is globally defined. A straightforward computation shows that e imθ (s+m)a (ϕ) = sa (e imθ ϕ) ϕ C c (, C), m Z. (2.6) Proposition 2.2. Assume S,s,p = S p. Then S,s,p is attained if and only if = 0, and s Z. Proof. If = 0, then, as we have noticed earlier, H 1 0 (, C) = H 1 (R 3, C). So the unique positive radially symmetric solution ω of (2.4) belongs to H 1 0 (, C). For m Z set u m := e imθ ω where θ(x 1, x 2, x 3 ) is the polar angle of x 1 + ix 2. It follows from (2.6) that u m H 1 ma,0 ( 0,, C) and u m 2 ma u m 2 p = ω 2 ω 2 p = S p = S,m,p. Next assume there exists v Σ,s,p such that I s (v) = S,s,p = S p. Then the diamagnetic inequality S p v 2 v 2 sa = S p becomes an equality. So, up to translation, Sp 1/(p 2) v = ω. In particular, = 0, and v(x) > 0 for all x. The equality v 2 = v 2 sa is equivalent to sa = Im v (2.7) v (cf. the proof of the diamagnetic inequality in [18] and [2], and also the argument of Theorem 1.1 in [2]). Let γ(t) := (cos t, sin t, 0), t [0, 2π], and u(t) := v(γ(t)). Identifying (x 1, x 2, 0) with z := x 1 + ix 2 we may consider u as a curve in C {0}. Since u (t) u(t) = v(γ(t)) v(γ(t)) γ (t), it follows from (2.7) that 2πs = sa = Im γ γ v v 2π = Im 0 u (t) dz dt = Im u(t) γ z = 2πm, where m is the winding number of u with respect to 0. Hence s = m Z. Note that the above result can be rephrased by saying that a necessary and sufficient condition for the existence of a ground state in H 1 (, C) is that = 0, and the magnetic potential can be gauged away.

8 8 Mónica Clapp and Andrzej Szulkin 2.3. Existence of G-invariant ground states If O is invariant under the action of some closed subgroup G of SO(2) and = O R, then G acts on H 1 (, C) by (g, u) u g, where This is an orthogonal action, that is, u g (x 1, x 2, x 3 ) := u(g 1 (x 1, x 2 ), x 3 ). (2.8) u g, v g sa = u, v sa g G, u, v H 1 (, C), where u, v sa := Re sa u sa v + uv is the inner product associated to the norm u sa. Moreover, u g p = u p g G, u L p (, C). Therefore, Σ,s,p is G-invariant, i.e. u g Σ,s,p iff u Σ,s,p, and the functional I sa satisfies I sa (u g ) = I sa (u). Set H 1 (, C) G := {u H 1 (, C) : u g = u g G}, Σ G,s,p := Σ,s,p H 1 (, C) G. By the principle of symmetric criticality [21] (see also Theorem 1.28 in [26]), the critical points of the restriction of I sa to Σ G,s,p correspond to the nontrivial G- invariant solutions to problem (,s ), i.e. to solutions u of (,s ) which satisfy Set u(g(x 1, x 2 ), x 3 ) = u(x 1, x 2, x 3 ) g G, x. (2.9) S G,s,p := inf I s (u) = u Σ G,s,p inf u H 1 (,C)G u 0 u 2 sa Proposition 2.3. I s attains its minimum on Σ G,s,p for every s R if one of the following two conditions holds: (D1) O is bounded and G is an arbitrary closed subgroup of SO(2). (D2) O = O a, := {(x 1, x 2 ) : x x 2 2 > a}, a [0, ), and G = SO(2). Proof. Let (v n ) be a sequence in Σ G,s,p such that lim n I s (v n ) = S,s,p G. The diamagnetic inequality (2.1) implies that ( v n ) is bounded in H0 1 (, R), so if lim sup v n 2 = 0 n x R 3 B 1(x) then by P.L. Lions lemma [19] (see also Lemma 1.21 in [26]) we would have that v n 0 in L p (R 3 ) which is impossible because v n p = 1. Hence there exist δ > 0 and ξ n R 3 such that, after passing to a subsequence, v n 2 δ. B 1(ξ n) u 2 p.

9 NLS equation with Aharonov-Bohm magnetic field 9 Here B r (ξ) := {x R 3 : x ξ < r}. Write ξ n = (ξ n1, ξ n2, ξ n3 ) and define w n (x 1, x 2, x 3 ) := v n (x 1, x 2, x 3 + ξ n3 ). Then w n 2 sa = v n 2 sa, w n p p = v n p p = 1, and w n 2 δ. B 1(ξ n1,ξ n2,0) We claim that the sequence (ξ n1, ξ n2, 0) is bounded. This is trivially true if O is bounded. So assume that G = SO(2) and O = O a,. Let m = m(n) be the largest number of elements g 1,..., g m SO(2) such that B 1 (g j (ξ n1, ξ n2 ), 0) B 1 (g k (ξ n1, ξ n2 ), 0) = if j k. Since w n (g(x 1, x 2 ), x 3 ) = w n (x 1, x 2, x 3 ) for all g SO(2), we have that w n 2 sa > w n 2 m w n 2 mδ. B 1(ξ n1,ξ n2,0) If ξn1 2 + ξn2 2, then m, contradicting the fact that (w n ) is bounded in H 1 (, C). Consequently, if either (D1) or (D2) holds, there exists R > 0 such that w n 2 δ. (2.10) B R (0) Now, after passing to a subsequence, we have that w n w weakly in H 1 (, C) G, w n (x) w(x) a.e. in, w n w strongly in L 2 loc(, C). Inequality (2.10) implies that w 0. Since w n p = 1, using the Brézis-Lieb lemma [4] (see also Lemma 1.32 in [26]) we obtain 1 lim n w n w p p = w p p and hence S,s,p G = lim w n 2 n sa = lim w n w 2 n sa + w 2 sa [ ( ) 2/p ( ) ] 2/p S,s,p G lim w n w p n p + w p p [ (1 = S,s,p G ) w p 2/p ( ) ] p + w p 2/p p S G,s,p ( 1 w p p + w p p) 2/p = S G,s,p. Since w 0, this implies that lim n w n w p p = 0. Hence, w p p = 1 and S G,s,p = S G,s,p w 2 p w 2 sa lim n w n 2 sa = SG,s,p. Consequently, w Σ G,s,p and I s(w) = S G,s,p.

10 10 Mónica Clapp and Andrzej Szulkin 2.4. Proof of Theorems 1.1 and 1.2 Next we prove our main results for the subcritical case. Proof of Theorem 1.2. Let G := {e 2πik/n : k = 0,..., n 1}, n N. Since O is bounded, Proposition 2.3 asserts that S,t,p G is attained for every t R. Let w t Σ G,t,p be such that w t 2 ta = SG,t,p. Then v t := (S,t,p G ) 1 p 2 wt is a solution of (,t ). For m = 0,..., n 1 define u m := e imθ v s+m where θ is given by (1.1). Using equality (2.6) it is easy to see that u m is a nontrivial solution of (,s ). Note that θ(e 2πik/n (x 1, x 2 ), x 3 ) = θ(x 1, x 2, x 3 ) + 2πk/n. Therefore, since v s+m is G-invariant, we have that u m (g(x 1, x 2 ), x 3 ) = g m u m (x 1, x 2, x 3 ) g G, x. (2.11) We claim that these solutions are all different. Indeed, if u k = u m, then equality (2.11) implies that g k m u k (x) = u k (x) for all g G and all x. Hence k = m. In the remaining part of this subsection we make the following Assumption. G = SO(2) and = a,b with 0 a < b. If ϕ C c (, C) satisfies (2.9), then ϕ is constant on every circle {(x 1, x 2, x 3 ) : x x 2 2 = r 2, x 3 = c}. Therefore, ϕ(x) A(x) = 0 x. It follows that ϕ + isaϕ 2 = ϕ 2 + saϕ 2, so u 2 sa = u 2 + sa 2 u 2 u H(, 1 C) G. (2.12) Let H 1 (, R) be the subspace of real-valued functions in H1 (, C) and Σ G,s,p,R := Σ G,s,p H 1 (, R) be the submanifold of all real-valued functions in Σ G,s,p. Lemma 2.4. One has that inf I s (u) = inf I s (u) =: S,s,p. G (2.13) u Σ G,s,p,R u Σ G,s,p Moreover, if u is a minimizer of I s on Σ G,s,p, then u is a minimizer of I s on Σ G,s,p,R. Proof. Since Σ G,s,p,R ΣG,s,p, we have that inf I s (u) inf I s (u). u Σ G,s,p,R u Σ G,s,p

11 NLS equation with Aharonov-Bohm magnetic field 11 On the other hand, applying the diamagnetic inequality (2.1) and (2.12) we obtain ( ) u 2 sa = u 2 + sa u 2 ( ) u 2 + sa u 2 = u 2 sa. Therefore, inf I s (u) inf I s (u). u Σ G,s,p,R u Σ G,s,p Thus, equality (2.13) holds. Moreover, if u Σ G,s,p and I s(u) = S,s,p G, then I s ( u ) = S,s,p G. Therefore, u is a minimizer of I s on Σ G,s,p,R. Lemma 2.5. (a) S,s,p G < SG,t,p if s2 < t 2, (b) The set {S,s,p G : s 0} is unbounded. Proof. (a) By Proposition 2.3, for every s R there exists v s Σ G,s,p I s (v s ) = S,s,p G. Formula (2.12) yields such that v t 2 ta = v t 2 sa + ( t 2 s 2) A 2 v t 2 > v t 2 sa if t 2 > s 2. Hence, S,s,p G < SG,t,p if t2 > s 2. (b) Arguing by contradiction, assume that sup s 0 S,s,p G =: ĉ R. Let v n Σ G,n,p be such that v n 2 na = SG,n,p ĉ. Arguing as in the proof of Proposition 2.3 we conclude that there exist R > 0 and z n R such that, after passing to a subsequence, we have v n 2 δ > 0. It follows that Therefore B R (0,0,z n) B R (0,0,z n) ( ) A 2 v n 2 inf x B R (0,0,z A(x) 2 v n 2 δ n) B R (0,0,z n) R 2. S G,n,p = v n 2 na = v n 2 + n 2 and sup s 0 S,s,p G =, contradicting our assumption. A 2 v n 2 n 2 δ R 2 Proof of Theorem 1.1. Let w s be a positive minimizer of I s on Σ G,s,p,R. Applying the moving plane method [14] (see also Appendix C in [26]), we may assume that w s (x 1, x 2, x 3 ) = w s (x 1, x 2, x 3 ). Set v s := (S,s,p G ) 1 p 2 ws. Note that, v s v t iff s 2 t 2. Otherwise, v = v s = v t would satisfy ( ) ( ) v + sa v = v p 2 v = v + ta v, which implies s 2 = t 2. Define u n := e inθ v s+n, n Z.

12 12 Mónica Clapp and Andrzej Szulkin Then u n = v s+n satisfies (a) and (b). Clearly, (c) and (d) hold. Using equality (2.6) it is easy to see that u n is a solution of problem (,s ) satisfying Lemma 2.5 yields (e) and (f). u n 2 sa = v s+n 2 (s+n)a = SG,s+n,p. (2.14) Equality (2.14) and Lemma 2.5 say that, among the solutions u n of (,s ) given by Theorem 1.1, the minimal value of I s ( un u n ) is S G p,σ,p where σ [ 1 2, 1 2 ] and σ s Z. So ( ) u σ s I s = S,σ,p G S,s,p S p. u σ s p Propositions 2.1 and 2.2 assert that S G,σ,p > S,s,p = S p if either a 0 and b = or a = 0, b = and s / Z. If a = 0 and b = then S,0,p = S p is attained at a radial function. Therefore S G,0,p = S p. Since σ = 0 for s Z it follows that S G,σ,p = S,s,p = S p if a = 0, b = and s Z Minimizing properties of the solutions Next we obtain some additional information on the solution u m. As we have seen it lies in the space X m := {u H 1 (, C) : u(g(x 1, x 2 ), x 3 ) = g m u(x 1, x 2, x 3 ) g G, x }. Moreover, the following holds. Proposition 2.6. The solution u m in Theorems 1.1 and 1.2 satisfies u m 2 sa u m 2 p = inf u X m u 0 Proof. The proof of Proposition 2.3 shows that u 2 sa inf u X m u 2 p u 0 u 2 sa u 2. (2.15) p (2.16) is attained. The only difference is that for the sequence (w n ) constructed there we now have w n (g(x 1, x 2 ), x 3 ) = g m w n (x 1, x 2, x 3 ). Since both sides of this equality have the same absolute value, the argument of this proposition remains valid. Hence (2.16) is attained at some u X m. Let v := e imθ u. Recall from the proof of Theorem 1.1 that u m := e imθ v s+m. Since v is G-invariant, S G,s+m,p v 2 (s+m)a v 2 p = u 2 sa u 2 p u m 2 sa u m 2 p = v s+m 2 (s+m)a v s+m 2 p = S G,s+m,p. Therefore u m satisfies (2.15).

13 3. The critical case NLS equation with Aharonov-Bohm magnetic field Preliminaries Let D 1,2 (, C) be the closure of C c (, C) with respect to the norm ( ) 1/2 u sa, := sa u 2. Inequality (2.3) asserts that ( ) 1/2 ϕ := ϕ 2 C s ϕ sa, ϕ Cc (, C). Hence D 1,2 (, C) is continuously embedded in D1,2 0 (, C). If G is a closed subgroup of SO(2) and O is G-invariant we define D 1,2 (, C)G := {u D 1,2 (, C) : u g = u g G}, with u g as in (2.8). Set I s, (u) := u 2 sa,, Σ,s, := {u D 1,2 (, C) : u 6 = 1}, ΣG,s, := Σ,s, D 1,2 (, C)G, S,s, := Note that 6 = 2N N 2 inf I s, (u) and S,s, G := inf I s, (u). u Σ,s, u Σ G,s, is the critical Sobolev exponent in dimension N = Existence of G-invariant ground states Let M(R 3 ) be the space of finite measures in R 3. A positive measure ν M(R 3 ) is concentrated at a single G-orbit of R 3 if there exists ζ R 3 such that ν(gζ) = ν where Gζ := {(g(ζ 1, ζ 2 ), ζ 3 ) : g G} is the G-orbit of ζ = (ζ 1, ζ 2, ζ 3 ). Note that ν is concentrated at a single G-orbit of R 3 iff ν(x) {0, ν } for every G-invariant open subset X of R 3. Lemma 3.1 (Concentration compactness). Let (u n ) be a sequence in D 1,2 (, C)G such that u n u weakly in D 1,2 (, C), sa (u n u) 2 µ weakly in M(R 3 ), u n u 6 ν weakly in M(R 3 ), u n (x) u(x) a.e. in R 3, and let µ := lim R lim sup sa u n 2, ν := lim n x R R Then, the following hold: (a) S G,s, ν 1/3 µ. (b) S G,s, (ν ) 1/3 µ. (c) lim sup n u n 2 sa, = u 2 sa, + µ + µ. (d) lim sup n u n 6 6 = u ν + ν. lim sup u n 6. n x R

14 14 Mónica Clapp and Andrzej Szulkin (e) If u = 0 and S,s, G ν 1/3 = µ then µ and ν are concentrated at a single finite G-orbit of R 3 or are zero. Proof. The proof of assertions (a)-(d) is completely analogous to the one given in [2] for the nonequivariant case (see also Lemma 1.40 in [26]). Further, one can show that if u = 0 and S,s, G ν 1/3 = µ then µ and ν are concentrated at the G-orbit of some point ζ R 3. One needs only to observe that the test functions h Cc (R 3, R) which occur in [2] and [26] must be G-invariant in our case, cf. [25]. Next we show that Gζ is finite. Let ϕ Cc (R 3, R). Since u n 0 in L 2 loc (R3, C) we have that S,s, ( ) 1/3 ϕu n 6 sa (ϕu n ) 2 = ϕ sa (u n ) + u n ϕ 2 R 3 R 3 R 3 2 ϕ sa (u n ) ϕ 2 u n 2 R 3 supp(ϕ) = 2 ϕ sa (u n ) 2 + o(1). R 3 Passing to the limit as n we obtain 1/3 S,s, ϕ dν) (R 6 2 ϕ 2 dµ ϕ Cc (R 3, R). 3 R 3 By Proposition 4.2 in [25] there exist an at most countable set {x j : j J} of points in R 3 and numbers ν j (0, ) such that ν = j J ν jδ xj. Since ν{(g(x 1, x 2 ), x 3 )} = ν{(x 1, x 2, x 3 )} for every g G, it follows that Gζ = {x j : j J}. Therefore Gζ must be finite. Proposition 3.2. If = 0, and G = SO(2) then S G,s, is attained. Proof. Let (u n ) be a sequence in Σ G,s, such that u n 2 sa, SG,s,. Set Q n (r) := sup u n 6. z R B r(0,0,z) Then Q n (r) 0 as r 0 and Q n (r) 1 as r, for each fixed n N. Hence, there exist r n (0, ) such that Q n (r n ) = 1 2 and z n R such that u n 6 = u n 6 = 1 2. Let w n (x) := r 1/2 n sup z R B rn (0,0,z) B rn (0,0,z n) u n (r n x+(0, 0, z n )). Then w n Σ G,s,, w n 2 sa, = u n 2 sa, and w n 6 = w n 6 = 1 2. (3.1) sup z R B 1(0,0,z) B 1(0)

15 NLS equation with Aharonov-Bohm magnetic field 15 Recall that D 1,2 (, C) D1,2 0 (, C). So passing to a subsequence, we have that w n w weakly in D 1,2 (, C)G, w n (x) w(x) a.e. in R 3, sa (w n w) 2 µ and w n w 6 ν weakly in M(R 3 ). Since lim n w n 2 sa, = S G,s, lim w n 2 6, n using Lemma 3.1 and the definition of S,s, G we obtain w 2 sa, + µ + µ = S,s, G ( ) w 6 1/3 6 + ν + ν ( ) S,s, G w ν 1/3 + ν 1/3 w 2 sa, + µ + µ. Hence ( ) w 6 1/3 6 + ν + ν = w ν 1/3 + ν 1/3. It follows that exactly one of the quantities w 6, ν, ν is 1 and the other two are 0. Equality (3.1) implies that ν 1, hence ν = 0. Assume ν = 1. Then w = 0 and S,s, G ν 1/3 = µ, so ν is concentrated at the G-orbit of a point ζ = (0, 0, ζ 3 ) R 3 (other orbits are infinite). But (3.1) yields 1 2 = w n 6 lim w n 6 = ν = 1. n B 1(0) B 1(0,0,ζ 3) This is a contradiction. Therefore ν = 0, w 6 = 1 and w 2 sa, = SG,s,. If O is bounded, Poincaré s inequality holds for. Together with the diamagnetic inequality this yields ( u 2 + u 2 ) C u 2 sa, u D 1,2 (, C) (3.2) for some constant C > 0. The proof of the following proposition is completely analogous to that of Theorem 1.1 in [11]. We include it here for the reader s convenience. Proposition 3.3. If = a,b with 0 < a < b < and G = SO(2) then S G,s, is attained. Proof. Let (u n ) be a sequence in Σ G,s, such that u n 2 sa, SG,s,. Poincaré s inequality (3.2) asserts that ( u n ) is bounded in H0 1 (), so if lim sup u n 6 = 0 n x R 3 B 1(x) then by Lemma 2.1 in [24] we would have that u n 0 in L 6 (R 3 ) which is impossible because u n 6 = 1. Hence there exist δ > 0 and ξ n R 3 such that, after passing to a subsequence, u n 6 δ. B 1(ξ n)

16 16 Mónica Clapp and Andrzej Szulkin Replacing u n (x) by u n (x + (0, 0, ξ n,3 )) we may assume that (ξ n ) is bounded and hence that u n 6 δ (3.3) B R (0) for R > 0 sufficiently large. Passing to a subsequence, we have that u n u weakly in D 1,2 (, C)G, u n (x) u(x) a.e. in R 3, sa (u n u) 2 µ and u n u 6 ν weakly in M(R 3 ). Arguing as in the proof of Proposition 3.2 we conclude that u 2 sa, + µ + µ = S,s, G ( ) u 6 1/3 6 + ν + ν ( ) S,s, G u ν 1/3 + ν 1/3 u 2 sa, + µ + µ. So one of the quantities u 6, ν, ν is 1 and the other two are 0. Inequality (3.3) implies that ν 1 δ, hence ν = 0. Assume ν = 1. Then u = 0 and S,s, G ν 1/3 = µ, so by Lemma 3.1 ν is concentrated at a finite G-orbit. But does not contain such orbits. Therefore ν = 0 and consequently u 6 = 1 and u 2 sa, = SG,s, Nonexistence of G-invariant ground states Let U be a fixed positive radially symmetric ground state of problem { v = v 5, v D 1,2 (R 3, R). For ε > 0 and x 0 R 3 let U ε,x0 (x) := ε 1/2 U(ε 1 (x x 0 )). Then U ε,x0 ground state. As usual, we denote S := U 2. U 2 6 is also a Proposition 3.4. S,s, = S and it is attained if and only if = 0, and s Z. Proof. Fix x 0 and r > 0 such that B r (x 0 ). Choose ψ Cc (R 3, R) such that 0 ψ 1, ψ(x) = 1 if x x 0 r 2 and ψ(x) = 0 if x x 0 r. Define u ε := ψu ε,x0 D 1,2 (, C). An easy computation shows that u ε 2 sa, = U ε,x o(1) and u ε 6 6 = U ε,x o(1), where o(1) 0 as ε 0. Hence S,s, S. The diamagnetic inequality (2.1) yields S,s, = S. It also implies that if u Σ,s, and u 2 sa, = S,s, then u 2 = S. v But S := inf 2 0 v D 1,2 0 (,R) is not attained if R 3. The remaining part of v 2 6 the argument is the same as in Proposition 2.2. Let G be a closed subgroup of SO(2) and O be G-invariant. For λ > 0 set O λ := {z R 2 : λ 1 z O} and λ := O λ R. Proposition 3.5. If there exists λ (0, ) such that O O λ and O λ O =, then S,s, G is not attained. In particular, if = 0,a or = a, with a (0, ) then S,s, G is not attained.

17 NLS equation with Aharonov-Bohm magnetic field 17 Proof. If u D 1,2 (, C)G, the rescaled function u λ (z) := λ 1/2 u(λ 1 z) satisfies u λ D 1,2 ( λ, C) G, u λ 2 sa, = u 2 sa, and u λ 6 6 = u 6 6. Therefore SG λ,s, = S,s, G. So if u ΣG,s, satisfies u 2 sa, = SG,s,, extending u by 0 outside of we obtain a function ū Σ G λ,s, which satisfies ū 2 sa, = SG,s, = SG λ,s,. Hence (S,s, ) 1 4 ū is a solution of ( λ,s) which vanishes in the nonempty open set λ, contradicting the unique continuation theorem [15]. Here we have used the fact that u Hloc 2 ( λ, C) which follows e.g. from the proof of Proposition 2.2 in [10] Proof of Theorem 1.3 In the remaining part of this section we assume G = SO(2) and either = 0, or = a,b with 0 < a < b <. Then, as in the subcritical case (2.12), one has that u 2 sa, = u 2 + sa 2 u 2 u D 1,2 (, C)G. (3.4) Let D 1,2 (, R) be the subspace of real-valued functions in D1,2 (, C) and Lemma 3.6. One has that Σ G,s,,R := Σ G,s, D 1,2 (, R). inf I s, (u) = inf I s, (u) =: S,s,. G u Σ G,s,,R u Σ G,s, Moreover, if u is a minimizer of I s, on Σ G,s,, then u is a minimizer of I s, on Σ G,s,,R. Proof. The proof is similar to that of Lemma 2.4. Lemma 3.7. S G,s, < SG,t, if s2 < t 2. Proof. The proof is similar to that of Lemma 2.5 (a) using Propositions 3.2 and 3.3. Proof of Theorem 1.3. The proof is analogous to that of Theorem 1.1 using equality (2.6) and Lemmas 3.6 and 3.7. It is not clear whether the solutions given by Theorem 1.3 have the property that lim n u n sa, = Minimizing properties of the solutions As in the subcritical case the solution u m solves a minimization problem. Set X m := {u D 1,2 (, C) : u(g(x 1, x 2 ), x 3 ) = g m u(x 1, x 2, x 3 ) g SO(2), x }. The following holds.

18 18 Mónica Clapp and Andrzej Szulkin Proposition 3.8. The solution u m in Theorem 1.3 satisfies u m X m and u m 2 sa u m 2 6 = inf u X m u 0 u 2 sa u 2. (3.5) 6 Proof. The arguments of Propositions 3.2 and 3.3 go through unchanged to show that u 2 sa inf u X m u 2 (3.6) 6 u 0 is attained, provided an appropriate version of Lemma 3.1 holds. This lemma requires that the sequence (u n ) is invariant with respect to some SO(2)-action on (, C). The appropriate action is obtained by setting D 1,2 u g (x 1, x 2, x 3 ) := g m u(g 1 (x 1, x 2 ), x 3 ). Xm is the fixed point space for this action. Once we have that (3.6) is attained, the rest of the proof goes as that of Proposition 2.6. References [1] L. Abatangelo, S. Terracini, Solutions to a nonlinear Schrödinger equation with singular electro-magnetic potentials, preprint. [2] G. Arioli, A. Szulkin, A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Rational Mech. Anal. 170 (2003), [3] T. Bartsch, E. N. Dancer, S. Peng, On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields, Adv. Differential Equations 11 (2006), [4] H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), [5] J. Chabrowski, A. Szulkin, On the Schrödinger equation involving a critical Sobolev exponent and magnetic field, Topol. Methods Nonlinear Anal. 25 (2005), [6] S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field, J. Differential Equations 188 (2003), [7] S. Cingolani, M. Clapp, Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation, Nonlinearity 22 (2009), [8] S. Cingolani, L. Jeanjean, S. Secchi, Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions, ESAIM Control Optim. Calc. Var. 15 (2009), [9] S. Cingolani, S. Secchi, Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields, J. Math. Anal. Appl. 275 (2002), [10] M. Clapp, R. Iturriaga, A. Szulkin, Periodic and Bloch solutions to a magnetic nonlinear Schrödinger equation, Advanced Nonl. Studies 9 (2009), [11] M. Clapp, A. Szulkin, A positive solution to the pure critical exponent problem in unbounded domains, preprint.

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20 20 Mónica Clapp and Andrzej Szulkin Mónica Clapp Instituto de Matemáticas Universidad Nacional Autónoma de México Circuito Exterior, C.U México D.F. Mexico Andrzej Szulkin Department of Mathematics Stockholm University Stockholm Sweden