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 NN-centre to the (N+1)(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 11-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

Entire solutions with exponential growth for an elliptic system modeling phase-separation

We prove the existence of entire solutions with exponential growth for the semilinear elliptic system [\begin{cases} -\Delta u = -u v^2 & \text{in $\R^N$} -\Delta v= -u^2 v & \text{in $\R^N$} u,v>0, \end{cases}] for every $N \ge 2$. Our construction is based on an approximation procedure, whose convergence is ensured by suitable Almgren-type monotonicity formulae. The construction of \emph{some} solutions is extended to systems with $k$ components, for every $k > 2$