3,418 research outputs found

### On existence and phase separation of solitary waves for nonlinear Schr\"odinger systems modelling simultaneous cooperation and competition

We study the existence of positive bound states for the nonlinear elliptic
system $\begin{cases} - \Delta u_i + \lambda_i u_i = \sum_{j=1}^d \beta_{ij}
u_j^2 u_i & \text{in $\Omega$} \\ u_1 =\cdots = u_d=0 & \text{on $\partial
\Omega$}, \end{cases}$ where $d \ge 2$, $\beta_{ij}= \beta_{ji}$,
$\beta_{ii},\lambda_i >0$, and $\Omega$ is either a bounded domain of
$\mathbb{R}^N$, or $\Omega=\mathbb{R}^N$, with $N \le 3$. In light of its
applicability in several physical contexts, the problem has been intensively
studied in recent years, and several results concerning existence, multiplicity
and qualitative properties of the solutions are available if either
$\beta_{ij}\le 0$ for every $i \neq j$, or $\beta_{ij}>0$ for every $i \neq j$
and some additional assumptions are satisfied. On the other hand, only very
partial results are known in the case of \emph{simultaneous cooperation and
competition}, that is, when there exist two pairs $(i_1,j_1)$ and $(i_2,j_2)$
such that $i_1 \neq j_1$, $i_2 \neq j_2$, $\beta_{i_1 j_1}>0$ and
$\beta_{i_2,j_2}<0$. In this setting, we provide sufficient conditions on the
coupling parameters $\beta_{ij}$ in order to have a positive solution. Our
first main results establishes the existence of solutions with at least $m$
positive components for every $m \le d$. Any such solution is a minimizer of
the energy functional $J$ restricted on a \emph{Nehari-type manifold}
$\mathcal{N}$. By means of level estimates on the constrained second
differential of $J$ on $\mathcal{N}$, we show that, under some additional
assumptions, any such minimizer has all nontrivial components. In order to
prove this second result, we analyse the phase separation phenomena which
involve solutions of the system in a \emph{not completely competitive}
framework.Comment: 27 pages, no figures, published online on Calc. Var. PD

### Symbolic dynamics: from the $N$-centre to the $(N+1)$-body problem, a preliminary study

We consider a restricted $(N+1)$-body problem, with $N \geq 3$ and
homogeneous potentials of degree -\a<0, \a \in [1,2). We prove the
existence of infinitely many collision-free periodic solutions with negative
and small Jacobi constant and small values of the angular velocity, for any
initial configuration of the centres. We will introduce a Maupertuis' type
variational principle in order to apply the broken geodesics technique
developed in the paper "N. Soave and S. Terracini. Symbolic dynamics for the
$N$-centre problem at negative energies. Discrete and Cont. Dynamical Systems
A, 32 (2012)". Major difficulties arise from the fact that, contrary to the
classical Jacobi length, the related functional does not come from a Riemaniann
structure but from a Finslerian one. Our existence result allows us to
characterize the associated dynamical system with a symbolic dynamics, where
the symbols are given partitions of the centres in two non-empty sets.Comment: Revised version, to appear on NoDEA Nonlinear Differential Equations
and Application

### Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case

We study existence and properties of ground states for the nonlinear
Schr\"odinger equation with combined power nonlinearities $-\Delta u= \lambda
u + \mu |u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,}$ having prescribed mass $\int_{\mathbb{R}^N} |u|^2 = a^2,$ in the
\emph{Sobolev critical case}. For a $L^2$-subcritical, $L^2$-critical, of
$L^2$-supercritical perturbation $\mu |u|^{q-2} u$ we prove several
existence/non-existence and stability/instability results.
This study can be considered as a counterpart of the Brezis-Nirenberg problem
in the context of normalized solutions, and seems to be the first contribution
regarding existence of normalized ground states for the Sobolev critical NLSE
in the whole space $\mathbb{R}^N$.Comment: arXiv admin note: text overlap with arXiv:1811.0082

### Liouville theorems and $1$-dimensional symmetry for solutions of an elliptic system modelling phase separation

We consider solutions of the competitive elliptic system $\left\{ \begin{array}{ll} -\Delta u_i = - \sum_{j \neq i} u_i u_j^2 &
\text{in $\mathbb{R}^N$} \\ u_i >0 & \text{in $\mathbb{R}^N$}
\end{array}\right. \qquad i=1,\dots,k.$ We are concerned with the
classification of entire solutions, according with their growth rate. The
prototype of our main results is the following: there exists a function
$\delta=\delta(k,N) \in \mathbb{N}$, increasing in $k$, such that if
$(u_1,\dots,u_k)$ is a solution and $u_1(x)+\cdots+u_k(x) \le C(1+|x|^d)
\qquad \text{for every $x \in \mathbb{R}^N$},$ then $d \ge \delta$. This
means that the number of components $k$ of the solution imposes an increasing
in $k$ minimal growth on the solution itself. If $N=2$, the expression of
$\delta$ is explicit and optimal, while in higher dimension it can be
characterized in terms of an optimal partition problem. We discuss the
sharpness of our results and, as a further step, for every $N \ge 2$ we can
prove the $1$-dimensional symmetry of the solutions satisfying suitable
assumptions, extending known results which are available for $k=2$. The proofs
rest upon a blow-down analysis and on some monotonicity formulae.Comment: 27 page

### Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose-Einstein condensation

We study monotonicity and 1-dimensional symmetry for positive solutions with
algebraic growth of the following elliptic system: $\begin{cases} -\Delta u =
-u v^2 & \text{in $\R^N$}\\ -\Delta v= -u^2 v & \text{in $\R^N$}, \end{cases}$ for every dimension $N \ge 2$. In particular, we prove a Gibbons-type
conjecture proposed by H. Berestycki, T. C. Lin, J. Wei and C. Zhao

### Multidimensional entire solutions for an elliptic system modelling phase separation

For the system of semilinear elliptic equations $\Delta V_i = V_i \sum_{j \neq i} V_j^2, \qquad V_i > 0 \qquad \text{in
$\mathbb{R}^N$}$ we devise a new method to construct entire solutions. The
method extends the existence results already available in the literature, which
are concerned with the 2-dimensional case, also in higher dimensions $N \ge 3$.
In particular, we provide an explicit relation between orthogonal symmetry
subgroups, optimal partition problems of the sphere, the existence of solutions
and their asymptotic growth. This is achieved by means of new asymptotic
estimates for competing system and new sharp versions for monotonicity formulae
of Alt-Caffarelli-Friedman type.Comment: Final version: presentation of the results improved, and several
minor corrections with respect to the first versio

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