The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications
Mathieu Lewin,Simona Rota Nodari +1 more
Reads0
Chats0
TLDR
In this article, the uniqueness and non-degeneracy of positive radial solutions to the double power non-linearity problem were shown to imply the uniqueness of energy minimizers at fixed mass in certain regimes.Abstract:
In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form
$$\Delta u+g(u)=0$$
. Our result applies in particular to the double power non-linearity where
$$g(u)=u^q-u^p-\mu u$$
for
$$p>q>1$$
and
$$\mu >0$$
, which we discuss with more details. In this case, the non-degeneracy of the unique solution
$$u_\mu $$
allows us to derive its behavior in the two limits
$$\mu \rightarrow 0$$
and
$$\mu \rightarrow \mu _*$$
where
$$\mu _*$$
is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the
$$L^2$$
mass of
$$u_\mu $$
in terms of
$$\mu $$
, which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of
$$u_\mu $$
.read more
Citations
More filters
Journal ArticleDOI
Multiple normalized solutions for a Sobolev critical Schr\"odinger equation
Louis Jeanjean,Thanh Trung Le +1 more
TL;DR: In this paper, the existence of standing waves for the Schrodinger equation with mixed power nonlinearities was studied and it was shown that these solutions are unstable by blow-up in finite time.
Posted Content
Orbital stability of ground states for a Sobolev critical Schr\"odinger equation
TL;DR: In this article, the existence of ground state standing waves, of prescribed mass, for the nonlinear Schrodinger equation with mixed power nonlinearities, was studied and the set of ground states is orbitally stable.
Journal ArticleDOI
Multiple normalized solutions for a Sobolev critical Schrödinger equation
Louis Jeanjean,Thanh Trung Le +1 more
TL;DR: In this paper, the existence of standing waves for the non-linear Schrodinger equation with mixed power nonlinearities was studied and it was shown that ground states correspond to local minima of the associated energy functional.
Journal ArticleDOI
General class of optimal Sobolev inequalities and nonlinear scalar field equations
TL;DR: In this paper, a class of optimal Sobolev inequalities up to translations of the nonlinear scalar field equation is presented, and a variational approach based on a new variant of the Lions' lemma in D 1, 2 ( R N ).
Posted Content
General class of optimal Sobolev inequalities and nonlinear scalar field equations
TL;DR: In this paper, a class of optimal Sobolev inequalities for non-zero, positive and infinite mass problems with constant energy functional was presented. But this inequality is not a generalization of the one of Berestycki and Lion's lemma.
References
More filters
Journal ArticleDOI
Pattern formation outside of equilibrium
Michael Cross,P. C. Hohenberg +1 more
TL;DR: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented in this article, with emphasis on comparisons between theory and quantitative experiments, and a classification of patterns in terms of the characteristic wave vector q 0 and frequency ω 0 of the instability.
Journal ArticleDOI
Nonlinear scalar field equations, I existence of a ground state
TL;DR: In this article, a constrained minimization method was proposed for the case of dimension N = 1 (Necessary and sufficient conditions) for the zero mass case, where N is the number of dimensions in the dimension N.
Journal ArticleDOI
The concentration-compactness principle in the calculus of variations. The locally compact case, part 1
TL;DR: In this paper, the equivalence between the compactness of all minimizing sequences and some strict sub-additivity conditions was shown based on a compactness lemma obtained with the help of the notion of concentration function of a measure.
Journal ArticleDOI
Existence of solitary waves in higher dimensions
TL;DR: In this paper, it was shown that Δu=F(u) possesses non-trivial solutions in R n which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear Klein-Gordon equation.