No abstract

We consider the equation −ε 2 ∆uwhere Ω is open, smooth and bounded, and we prove concentration of solutions along k-dimensional minimal submanifolds of ∂Ω, for N ≥ 3 and for k ∈ {1, . . . , N − 2}. We impose Neumann boundary conditions, assuming 1 < p < N −k+2 N −k−2 and ε → 0 + . This result settles in full generality a phenomenon previously considered only in the particular case N = 3 and k = 1.

We study the nonlinear Schrödinger equation −ε 2 ψ + V (x)ψ = |ψ| p−1 ψ on a compact manifold or on R n , where V is a positive potential and p > 1. As ε tends to zero, we prove existence of complex-valued solutions which concentrate along closed curves and whose phase is highly oscillatory, carrying quantum-mechanical momentum along the limit set.

We consider the equationwhere Ω is a smooth and bounded domain in R n , ν the outer unit normal to ∂Ω, and a a smooth function satisfying −1 < a(x) < 1 in Ω. We set K, Ω + and Ω − to be respectively the zero-level set of a, {a > 0} and {a < 0}. Assuming ∇a = 0 on K and a = 0 on ∂Ω, we show that there exists a sequence ε j → 0 such that Eq.(1) has a solution u ε j which converges uniformly to ±1 on the compact sets of Ω ± as j → +∞. This result settles in general dimension a conjecture posed in [P. Fife, M.W. Greenlee, Interior transition layers of elliptic boundary value problem with a small parameter, Russian Math. Surveys 29 (4) (1974) 103-131], proved in [M. del Pino, M. Kowalczyk, J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal. 38 (5) (2007) 1542-1564] only for n = 2.

Abstract. We consider here solutions of the nonlinear fractional Schrödinger equationWe show that concentration points must be critical points for V . We also prove that, if the potential V is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point as ε tends to zero. In addition, if the potential V is radial and radially decreasing, then the minimizer is unique provided ε is small.