NLS ground states on graphs

Abstract

We investigate the existence of ground states for the subcritical NLS energy on metric graphs. In particular, we find out a topological assumption that guarantees the nonexistence of ground states, and give an example in which the assumption is not fulfilled and ground states actually exist. In order to obtain the result, we introduce a new rearrangement technique, adapted to the graph where it applies. Owing to such a technique, the energy level of the rearranged function is improved by conveniently mixing the symmetric and monotone rearrangement procedures. Mathematics Subject Classification

Variational and Stability Properties of Constant Solutions to the NLS Equation on Compact Metric Graphs

“…We note that for every λ > 0 the function κ = κ(λ) = λ 1 p−2 is a solution of the stationary equation (4). For λ = (µ/ℓ) p/2−1 the function κ(λ) coincides with the constant solution κ µ of mass µ given in (5).…”

“…For i = 1, 2, let ℓ ± i = |{x ∈ G : φ ± i > 0}|, and note that ℓ + i + ℓ − i ≤ ℓ =: |G|. Now take the symmetric rearrangements φ ± i ∈ H 1 (−ℓ ± i /2, ℓ ± i /2) of φ ± i , defined as in [5]. Clearly φ + i (−ℓ + i /2) = φ + i (ℓ + i /2) = 0, and likewise for φ − i .…”

“…The existence of mass-constrained ground states was initiated in the series of works [1,2,3,4], for the special case of a star-graph (a graph obtained by merging N halflines). For generic non-compact metric graphs the problem was addressed in [5,6] in the subcritical regime and in [7] in the critical one (see also [30]). We also note the paper [17], where the stationary solutions for the NLS equation on the tadpole graph (a loop with one half-line attached to it) were characterized and where the existence of the ground state was only conjectured (the conjecture was then confirmed in [6]).…”

Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs

“…The forerunner of the study of nonlinear evolution on metric graphs is considered to be a paper by Ali Mehmeti [15] that dates back to 1984, but it is in the last decade that the study of the NLSE with Kirchhoff's conditions has developed. Several papers [13,14,12] study the existence of ground states for the energy functional under the mass constraint. They analyse the problem for metric graphs with a finite number of vertices and at least one halfline, distinguishing between cases p ∈ (2, 6) (known as subcritical) and p = 6 (called the critical case).…”

Drift of Spectrally Stable Shifted States on Star Graphs

“…The present work is devoted to stability of standing waves in the NLS equation defined on a metric graph, a subject that has seen many recent developments [18]. Existence and variational characterization of standing waves was developed for star graphs [1,2,3,4] and for general Date: February 12, 2019. metric graphs [5,6,7,8,9]. Bifurcations and stability of standing waves were further explored for tadpole graphs [19], dumbbell graphs [12,16], double-bridge graphs [20], and periodic ring graphs [10,11,21,22].…”