Concentrating standing waves for the fractional nonlinear Schrödinger equation

Abstract

Abstract. We consider the semilinear equationis a sufficiently smooth potential with inf R V (x) > 0, and ε > 0 is a small number. Letting w λ be the radial ground state of (−∆) s w λ + λw λ − w p λ = 0 in H 2s (R N ), we build solutions of the formwhere λ i = V (ξ ε i ) and the ξ ε i approach suitable critical points of V . Via a Lyapunov Schmidt variational reduction, we recover various existence results already known for the case s = 1. In particular such a solution exists around k nondegenerate critical points of V . For s = 1 this corresponds to the classical results by Floer-Weinstein [13] and Oh [21,22].

Fractional NLS equations with magnetic field, critical frequency and critical growth

“…By using the Lyapunov-Schmidt reduction method, they showed that (1.6) has a nontrivial solution u ε concentrating to some single point as ε → 0. In [11], assuming that f (x, t) = |t| p−2 t and V is a sufficiently smooth potential with inf R N V > 0, Dávila et al recovered various existence results already known for the case α = 1 and showed the existence of solutions around k nondegenerate critical points of V for (1.6). In [30], Shang and Zhang studied the concentration phenomenon of solutions for (1.6) under the assumptions f (x, t) = K(x)|t| p−2 t, V , K are positive smooth functions and inf R N V > 0.…”

Positive or sign-changing solutions for a critical semilinear nonlocal equation

“…Very recently, in [13], Dávila, del Pino and Wei generalized various existence results known for (1.4) with s = 1 to the case of fractional Laplacian. In [23], the authors gave a result which says that (1.4) with V (x) being radial has solutions with large number of bumps near infinity and the energy of this solutions can be very large when ε is fixed and N ≥ 2.…”

Energy solutions and concentration problem of fractional Schrödinger equation

“…For more related study, the interested reader may consult [24][25][26][27][28][29][30][31][32][33][34] and the references therein. It should also be noted that the concentration phenomena for the fractional Schrödinger equation have been investigated byDávila, del Pino et al [35,36], and Fall, Mahmoudi and Valdinoci [37].…”