Presentation Nodal solutions of a NLS equation concentrating on lower dimensional spheres Marcos T.O. Pimenta and Giovany J. M. Figueiredo Unesp / UFPA Brussels, September 7th, 2015 * Supported by FAPESP - CNPq - Brazil
Presentation Presentation 1 Introduction 2 3 4 5 6
Apresentação 1 Introduction 2 3 4 5 6
Introduction In this work we study some results about existence and concentration of solutions of the following NLS equation { ɛ 2 u + V (x)u = f (u) in R N, u H 1 (R N ), (1) where ɛ > 0, f is a subcritical power-type nonlinearity and the potential V is assumed to be positive and satisfy a local condition. The concentration is going to occur around k dimensional spheres, where 1 k N 1, as ɛ 0.
Introduction We assume the following conditions on de odd nonlinearity f : (f 1 ) There exists ν > 1 such that f ( s ) = o( s ν ) as s 0; (f 2 ) There exist c 1, c 2 > 0 such that f (s) c 1 s + c 2 s p where 0 < p < 2 2; (f 3 ) There exists θ > 2 such that 0 < θf (s) f (s)s, for s 0, where F (s) = s 0 f (t)dt; (f 4 ) s f (s)/s is increasing for s > 0 and decreasing for s < 0.
Introduction Let k N with1 k N 1 and define a subspace H R N such that dim H = N k 1. Hence dim H = k + 1. For x R N, denote x = (x, x ) where x H, x H are such that x = x + x. When u : R N R, we write u(x, x ) = u(x, x ) denoting that u(x, y) = u(x, z) for all y, z H such that y = z. For a R, let u + au = f (u) in R N k. (2) The auxiliar potential M : R N (0, + ] is given by M(x) = x k E(V (x)) where x = (x, x ), x H and x H, where E(V (x)) is the ground-state level of (2) with a = V (x).
Introduction V will be assumed to satisfy the following conditions (V 1 ) 0 < V 0 V (x), x R N and V (x) = V (x, x ) = V (x, x ); (M 1 ) There exists a bounded and open set Ω R N such that (x, x ) Ω implies (x, y ) Ω for all y H such that x = y. Moreover, 0 < M 0 := inf M(x) < inf M(x). x Ω x Ω
Introdução Introduction The main result is the following Teorema 1.1 Let f and V satisfying (f 1 ) (f 4 ) and (V 1 ) e (M 1 ), respectively. Then for all sequence ɛ n 0, up to a subsequence, (1) (with ɛ = ɛ n ) has a nodal solution u n such that, u n (x, x ) = u n (x, x ) and, if ɛ n Pn 1 and ɛ n Pn 2 are respectively, maximum and minimum points of u n, then ɛ n Pn i Ω, i = 1, 2 for n sufficiently large and ɛ n Pn i x 0, as n, where M(x 0 ) = M 0. Moreover ( ) u n (x) C e β ɛn d k(x,ɛ npn 1) + e β ɛn d k(x,ɛ npn 2 ) x R N, where C, β > 0 and d k is a distance defined in (10).
Introdução Introduction Layout of solutions to N = 2 and k = N 1
Introduction The works which more have influenced us are [1], in which Alves and Soares prove the existence and concentration of nodal-peak solutions for (1), and [4] where the authors prove the existence of positive solutions concentrating around lower dimensional spheres. In this work we use the penalization technique developed in [5], which consists in modifying the nonlinearity in such a way to recover the compactness of the functional. However, since we are looking for nodal solutions, we have to carry more thoroughly the penalization and as a consequence, make sharper estimates to prove that the solutions of the modified problem, satisfies the original one.
Apresentação 1 Introduction 2 3 4 5 6
For τ > 2, let r ɛ > 0 be such that f (rɛ) r ɛ = ɛ τ and f ( rɛ) r ɛ = ɛ τ. Since r ɛ 0 as ɛ 0, (f 1 ) implies that ɛ τ = f ( rɛ ) r ɛ r ɛ ν 1. Hence ɛ τ ν 1 rɛ and we can choose an odd function f ɛ C 1 (R) satisfying { f (s) if s f 1 ɛ (s) = 2 ɛ τ ν 1, ɛ τ s if s ɛ τ ν 1, f ɛ (s) ɛ τ s for all s R (3) and 0 f ɛ (s) 2ɛ τ for all s R. (4)
Introduction Penalization
Let g ɛ (x, s) := χ Ω (x)f (s) + (1 χ Ω (x)) f ɛ (s). By (f 1 ) (f 4 ) it follows that g is such that g ɛ (x, x, s) = g ɛ (x, x, s) and satisfies (g 1 ) g ɛ (x, s) = o( s ν ), as s 0, uniformly in compact sets of R N ; (g 2 ) There exist c 1, c 2 > 0 such that g ɛ (x, s) c 1 s + c 2 s p where 1 < p < N+2 N 2 ; (g 3 ) There exists θ > 2 such that: i) 0 < θg ɛ (x, s) g ɛ (x, s)s, para x Ω and s 0, ii) 0 < 2G ɛ (x, s) g ɛ (x, s)s, for x R N \Ω and s 0, where G ɛ (x, s) = s 0 g ɛ(x, t)dt. (g 4 ) s gɛ(x,s) s is nondecreasing for s > 0 and nonincreasing for s < 0, for all x R N.
Let us consider the modified problem or equivalently, ɛ 2 u + V (x)u = g ɛ (x, u) in R N, (5) v + V (ɛx)v = g ɛ (ɛx, u) in R N. (6)
In order to obtain the solutions with the prescribed symmetry, let { } H := v H 1 (R N ); ( v 2 +V (ɛx)v 2 ) < + and v(x, x ) = v(x, x ) which is a Hilbert space when endowed with the inner product u, v ɛ = ( u v + V (ɛx)uv), which gives rise to the norm ( v ɛ = ( v 2 + V (ɛx)v 2 ) 1 2.
Let I ɛ : H 1 (R N ) R be the C 2 (H 1 (R N ), R) functional defined by I ɛ (v) = 1 ( v 2 + V (ɛx)v 2 ) G ɛ (ɛx, v). 2 Note that I ɛ(v)ϕ = ( v ϕ + V (ɛx)vϕ) g ɛ (ɛx, v)ϕ, ϕ H 1 (R N ), and then critical points of I ɛ em H 1 (R N ) are weak solutions of (6). From now on we are going to work with the functional I ɛ restricted to H.
Apresentação 1 Introduction 2 3 4 5 6
Let us consider the Nehari manifold associated to (6) and defined by N ɛ = {v H\{0}; I ɛ(v)v = 0}. Since we are looking for sign-changing solutions, let us consider the Nehari nodal set N ± ɛ = {v H; v ± 0 and I ɛ(v)v ± = 0}, which contains all the sign-changing solutions. Hence, d ɛ := inf N ± ɛ is the nodal ground state level. Let us show that d ɛ is reached by some function in N ± ɛ. I ɛ,
Lemma 3.1 Let v H such that v ± 0. Then there exist t, s > 0 such that tv + + sv N ± ɛ. Let (w n ) be a minimizing sequence for I ɛ in N ± ɛ. One can prove that (w n ) is bounded in H and then w n w ɛ in H, w ± n w ± ɛ in H, where w ± ɛ 0. Let t ɛ, s ɛ > 0 be such that t ɛ w + ɛ + s ɛ w ɛ N ± ɛ. Then and t ɛ w + ɛ Ω ɛ F (t ɛ w + ɛ + s ɛ w ɛ ɛ lim inf n t ɛw + n + s ɛ w n ɛ (7) + s ɛ w ɛ ) = lim inf n Ω ɛ F (t ɛ w + n + s ɛ w n ). (8)
In this point the penalization in fact plays its role. By (3) and (4), one can see that for sufficiently small ɛ > 0, I ɛ,r N \Ω ɛ (v) := 1 ( v 2 + V (ɛx)v 2) F ɛ (v) 2 R N \Ω ɛ R N \Ω ɛ is strictly convex. Then, it follows that I ɛ,r N \Ω ɛ is weakly semicontinuous from below and then I ɛ (t ɛ w + ɛ + s ɛ w ɛ ) lim inf n I ɛ(t ɛ w + n + s ɛ w n ) lim inf (I ɛ(w + n n ) + I ɛ (wn )) = b ɛ.
Hence I ɛ (t ɛ w + ɛ + s ɛ w ɛ ) = b ɛ. Now, arguments similar to those employed in [3] and [2] can be used to, by a contradiction argument, prove that the minimizing of I ɛ on N ± ɛ is a weak solution of (6).
Apresentação 1 Introduction 2 3 4 5 6
Let ɛ n 0 as n and, for each n N, let us denote by v n := v ɛn the solution of the last section, d n := d ɛn, n := ɛn and I n := I ɛn. The following result gives us an estimate from above to the energy levels d n. Lemma 4.1 and lim sup ɛ k nd n 2ω k inf M n Ω ɛ k n v n 2 n C n N.
Next result implies that solutions (v n ) do not present vanishing and neither do (v + n ) and (v n ). Lemma 4.2 Let Pn, 1 Pn 2 local maximum and minimum of v n, respectively. Then Pn i Ω ɛn := ɛ 1 n Ω and v n (P 1 n) a and v n (P 2 n) a, where a > 0 is such that f (a)/a = V 0 /2.
By the last result there exist P 1, P 2 Ω such that, up to a subsequence Lemma 4.3 lim ɛ np i n n = P i, i {1, 2}. lim n P1 n Pn 2 = +.
Lemma 4.4 Let y n = (y n, y n ) R N be a sequence such that ɛ n y n (ȳ, ȳ ) Ω as n. Denoting ṽ n (x, r) := v n (x, x ) where x = r, let us define w n : R N k 1 [ y n, + ) R by w n (x, r) := ṽ n (x, r + y n ). Then there exists w H 1 (R N k ) such that w n w in C 2 loc (RN k ) and w satisfies the limit problem w + V (ȳ, ȳ ) w = g n (x, r, w) in R N k, (9) where g n (x, r, s) := χ(x, r)f (s) + (1 χ(x, r)) f ɛn (s) and χ(x, r) = lim n χ Ω (ɛ n x + ɛ n y n, ɛ n r + ɛ n y n ).
Since the concentration set is expected to be a sphere in R N, it is natural to introduce the distance between two k dimensional spheres in R N, which gives rise to neighborhoods in which we want to estimate the mass of solutions. Then let d k (x, y) = (x y ) 2 + ( x y ) 2, (10) which denotes the distance between k dimensional spheres centered at the origin, parallel to H and of radius x and y, respectively.
Proposition 4.5 Under assumptions of Theorem 1.1, it holds: i) lim n ɛk nd n = 2ω k inf M. Ω ii) lim M(ɛ np i n n) = inf M, i {1, 2}. Ω The proof of this result consists in an analysis of the parts of the integrals of I n (v n ) in B k (P i, R) and in R N \B k (P i, R).
Apresentação 1 Introduction 2 3 4 5 6
First of all we have to prove the following result. Proposição 5.1 lim v n L n (Ω n\(b k (Pn 1,R) B k(pn 2,R))) = 0. By elliptic regularity theory, it follows that v n C 2 (R N ). Then, by continuity, it follows that v n L ((B k (P 1 n,r n) B k (P 2 n,r n))) = o n (1) (11)
Let u n (x) := v n (ɛ 1 n x). In order to prove the exponential decay in ɛ n of u n, we need to consider the following functions W (x) = C(e βd k(x,p 1 n ) + e βd k(x,p 2 n ) ), defined in R N \ ( B k (P 1 n, R) B k (P 2 n, R) ), where C > 0 is a constant to be choosen. For β > 0 sufficiently small, it follows that for all n N and x R N \ ( B k (P 1 n, R) B k (P 1 n, R) ), ( + V (ɛ n x) g ) n(ɛ n, v n ) (W ± v n ) 0. v n
Then, by (11), for x ( B k (P 1 n, R) B k (P 2 n, R) ), W (x) ± v n (x) = C2e βr ± v n 0 for a constant C > 0 sufficiently large and which does not depend on n. Hence by(g 4 ), Maximum Principle applies and v n W (x), em R N \ ( B k (P 1 n, R) B k (P 2 n, R) ), which implies that
Prova do Teorema 1.1 u n (x) C ( e β ɛn d k(x,ɛ np 1 n ) + e β ɛn d k(x,ɛ np 2 n )), (12) for x R N \ ( B k (ɛ n Pn, 1 ɛ n R) B k (ɛ n Pn, 2 ɛ n R) ). In particular u n L (R N \Ω) Ce β ɛn and then u n satisfies the original problem, proving the theorem.
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I C.O. Alves, S.H.M. Soares On the location and profile of spyke-layer nodal solutions to nonlinear Schrödinger equations. Journal of Mathematical Analysis and Applications 296 (2004), 563-577. C.O. Alves, M. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, to appear in Z. Angew. Math. Phys. T. Bartsch, T. Weth, M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math. 96 (2005), 1-18.
II D. Bonheure, J. Di Cosmo, J.V. Schaftingen, Nonlinear Schrödinger equation with unbounded or vanishing potentials: solutions concentrating on lower dimensional spheres, Journal of Differential Equations. 252 (2012), 941-968. M. del Pino, P. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121-137.