On global minimizers for a mass constrained problem

Abstract

In any dimension N ≥ 1, for given mass m > 0 and for the C 1 energy functionalwe revisit the classical problem of finding conditions on F ∈ C 1 (R, R) insuring that I admits global minimizers on the mass constraintUnder assumptions that we believe to be nearly optimal, in particular without assuming that F is even, any such global minimizer, called energy ground state, proves to have constant sign and to be radially symmetric monotone with respect to some point in R N . Moreover, we manage to show that any energy ground state is a least action solution of the associated free functional. This last result settles, under general assumptions, a long standing open problem.

Normalized solutions with positive energies for a coercive problem and application to the cubic-quintic nonlinear Schrödinger equation

“…(iii) Items (iii) and (iv) of Theorem 6.3 demonstrate that for some frequency ω ∈ (0, 3/16) the action ground state U ω is not a minimizer of the global infimum energy E m with m = U ω 2 L 2 (R 3 ) . This is consistent with [5, Theorem 2.5] and in particular we present new counterexamples that disprove the converse statement of [16,Theorem 1.6 (i)].…”

“…This case is commonly called mass subcritical and has been under extensive studies for decades. Among many possible choices, we refer the reader to [6,7,8,12,14,16,20,27,29] and to the references therein. In contrast however, to these previous works, we are herein interested in searching for constrained critical points at positive energy levels.…”

“…To clarify our mass subcritical setting and for convenience of later discussion, we give a brief account of our recent work [16]. Among other issues like the least action characterization and the radial symmetry of global minimizers, the solvability of the global minimization problem (Inf m ) was investigated there under the following assumptions.…”

“…When m ≥ m * , by Theorem 1.1 (ii) and Lemma 2.3, the constrained functional I |Sm admits a global minimizer v ∈ S m with I(v) ≤ 0. In view of [16,Theorem 1.4], this minimizer is radially symmetric up to a translation in R N and hence we may assume that v ∈ S m ∩H 1 r (R N ). Since I(v) ≤ 0, it follows from Lemma 2.2 (iii) that…”

Action versus energy ground states in nonlinear Schrödinger equations

“…After the present work was completed, we became aware of the interesting paper [26], where the authors obtain results strongly related to ours in the case = R N but for a wide class of nonlinearities.…”