Concentration on curves for nonlinear Schrödinger Equations

Abstract

We consider the problemwhere p > 1, ε > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let be a closed curve, nondegenerate geodesic relative to the weighted arc length V σ , where σ = ( p + 1)/( p − 1) − 1/2. We prove the existence of a solution u concentrating along the whole of , exponentially small in ε at any positive distance from it, provided that ε is small and away from certain critical numbers. In particular, this establishes the validity of a conjecture raised in [3] in the two-dimensional case.

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

“…and recall that −V x > 0, V > 0 in R. So, condition (78), with L = −M , is satisfied by either one of −V x or V . Nevertheless, observe that one faces a difficulty when proceeding as in [35], namely applying the standard maximum principle in the equation satisfied by…”

“…Actually, the latter case is related to the discussion following Definition 1 below. We refer the interested reader to [20,78], and the references therein.…”

“…To this end, we will follow the general lines set in [78] which dealt with the focusing case (22) + .…”

“…are known in the literature as Fermi coordinates, see for instance [140], and are frequently employed in the study of perturbation problems involving solutions concentrating on manifolds (see for example [78]). Interestingly enough, they owe their name to the physicist E. Fermi in the title of the current paper!…”

“…In contrast, thanks to the (super-) exponentially fast convergence of V to zero, as x → ∞, interpolating directly between u in and zero in R 2 \D ε is rather standard. (This last situation occurs in the construction of spike-layered solutions of the focusing (22) + , see [20,78,98], and transition-layered solutions of Allen-Cahn type equations, see [79,96].) Let δ < δ 0 100(1+max β) be a fixed number.…”