arXiv subjects
Publications and source records attributed to Nicola Soave.
We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity.
For the system of semilinear elliptic equations \[ ΔV_i = V_i \sum_{j \neq i} V_j^2, \qquad V_i > 0 \qquad \text{in $\mathbb{R}^N$} \] 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.
We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincaré-type inequality and classification results for stable solutions, and we apply them to the study of an associated nonlocal problem. We also establish a counterexample in the corresponding framework for the fractional Laplacian.
We consider a family of positive solutions to the system of $k$ components \[ -Δu_{i,β} = f(x, u_{i,β}) - βu_{i,β} \sum_{j \neq i} a_{ij} u_{j,β}^2 \qquad \text{in $Ω$}, \] where $Ω\subset \mathbb{R}^N$ with $N \ge 2$. It is known that uniform bounds in $L^\infty$ of $\{\mathbf{u}_β\}$ imply convergence of the densities to a segregated configuration, as the competition parameter $β$ diverges to $+\infty$. In this paper %we study more closely the asymptotic property of the solutions of the system in this singular limit: we establish sharp quantitative point-wise estimates for the densities around the interface between different components, and we characterize the asymptotic profile of $\mathbf{u}_β$ in terms of entire solutions to the limit system \[ ΔU_i = U_i \sum_{j\neq i} a_{ij} U_j^2. \] Moreover, we develop a uniform-in-$β$ regularity theory for the interfaces.
We consider the system of coupled elliptic equations \[ \begin{cases} -Δu - λ_1 u = μ_1 u^3+ βu v^2 \\ -Δv- λ_2 v = μ_2 v^3 +βu^2 v \end{cases} \text{in $\mathbb{R}^3$}, \] and study the existence of positive solutions satisfying the additional condition \[ \int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a_2^2. \] Assuming that $a_1,a_2,μ_1,μ_2$ are positive fixed quantities, we prove existence results for different ranges of the coupling parameter $β>0$. The extension to systems with an arbitrary number of components is discussed, as well as the orbital stability of the corresponding standing waves for the related Schrödinger systems.
We consider solutions of the competitive elliptic system \[ \left\{ \begin{array}{ll} -Δu_i = - \sum_{j \neq i} u_i u_j^2 & \text{in $\mathbb{R}^N$} \\ u_i >0 & \text{in $\mathbb{R}^N$} \end{array}\right. \qquad i=1,\dots,k. \] 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 $δ=δ(k,N) \in \mathbb{N}$, 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 $d \ge δ$. 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 $δ$ 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.
For a class of systems of semi-linear elliptic equations, including \[ -Δu_i=f_i(x,u_i) - βu_i\sum_{j\neq i}a_{ij}u_j^p,\qquad i=1,\dots,k, \] for $p=2$ (variational-type interaction) or $p = 1$ (symmetric-type interaction), we prove that uniform $L^\infty$ boundedness of the solutions implies uniform boundedness of their Lipschitz norm as $β\to +\infty$, that is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proof rests on monotonicity formulae of Alt-Caffarelli-Friedman and Almgren type in the variational setting and Caffarelli-Jerison-Kenig in the symmetric one.
In this paper we focus on existence and symmetry properties of solutions to the cubic Schrödinger system \[ -Δu_i +λ_i u_i = \sum_{j=1}^d β_{ij} u_j^2 u_i \quad \text{in $Ω\subset \mathbb{R}^N$},\qquad i=1,\dots d \] where $d\geq 2$, $λ_i,β_{ii}>0$, $β_{ij}=β_{ji}\in \mathbb{R}$ for $j\neq i$, $N=2,3$. The underlying domain $Ω$ is either bounded or the whole space, and $u_i\in H^1_0(Ω)$ or $u_i\in H^1_{rad}(\mathbb{R}^N)$ respectively. We establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients, as well as in the purely cooperative case.
We study the existence of positive bound states for the nonlinear elliptic system \[ \begin{cases} - Δu_i + λ_i u_i = \sum_{j=1}^d β_{ij} u_j^2 u_i & \text{in $Ω$} \\ u_1 =\cdots = u_d=0 & \text{on $\partial Ω$}, \end{cases} \] where $d \ge 2$, $β_{ij}= β_{ji}$, $β_{ii},λ_i >0$, and $Ω$ is either a bounded domain of $\mathbb{R}^N$, or $Ω=\mathbb{R}^N$, 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 $β_{ij}\le 0$ for every $i \neq j$, or $β_{ij}>0$ 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 $i_1 \neq j_1$, $i_2 \neq j_2$, $β_{i_1 j_1}>0$ and $β_{i_2,j_2}<0$. In this setting, we provide sufficient conditions on the coupling parameters $β_{ij}$ 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.
In a singular potential setting, we generalize a method which allows to show that minimizers under topological constraints of the action functional (or of the Maupertuis') are collision-free. This methods applies to $3$-dimensional problems of celestial mechanics exhibiting a particular cylindrical symmetry, as well as to planar problems of $N$-centre type, where it gives optimal results.
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.
Under the validity of a Landesman-Lazer type condition, we prove the existence of solutions bounded on the real line, together with their first derivatives, for some second order nonlinear differential equation of the form $\ddot u + g(u) = p(t)$, where the reaction term $g$ is bounded. The proof is variational, and relies on a dual version of the Nehari method for the existence of oscillating solutions to superlinear equations.
We prove the existence of entire solutions with exponential growth for the semilinear elliptic system [\begin{cases} -Δu = -u v^2 & \text{in $\R^N$} -Δv= -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 $k > 2$.
We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: \[ \begin{cases} -Δu = -u v^2 & \text{in $\R^N$}\\ -Δv= -u^2 v & \text{in $\R^N$}, \end{cases} \] 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.
This paper aims at completing and clarifying a delicate step in the proof of Theorem 5.3 of our paper \cite{ST}, where it was used the differentiability of a function $F$, which a priori can appear not necessarily differentiable.
We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\a<0$, $\a \in [1,2)$. We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets.