arXiv subjects

Hugo Tavares

Publications and source records attributed to Hugo Tavares.

48 records · Page 3Linked to original sources

arXiv2015

Existence and symmetry of least energy nodal solutions for Hamiltonian elliptic systems

In this paper we prove existence of least energy nodal solutions for the Hamiltonian elliptic system with Hénon-type weights \[ -Δu = |x|^β |v|^{q-1}v, \quad -Δv =|x|^α|u|^{p-1}u\quad { in } Ω, \qquad u=v=0 { on } \partial Ω, \] where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geq 1$, $α, β\geq 0$ and the nonlinearities are superlinear and subcritical, namely \[ 1> \frac{1}{p+1}+\frac{1}{q+1}> \frac{N-2}{N}. \] When $Ω$ is either a ball or an annulus centred at the origin and $N \geq 2$, we show that these solutions display the so-called foliated Schwarz symmetry. It is natural to conjecture that these solutions are not radially symmetric. We provide such a symmetry breaking in a range of parameters where the solutions of the system behave like the solutions of a single equation. Our results on the above system are new even in the case of the Lane-Emden system (i.e. without weights). As far as we know, this is the first paper that contains results about least energy nodal solutions for strongly coupled elliptic systems and their symmetry properties.

math.AP↗

arXiv2014

New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms

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.

math.AP↗

arXiv2014

Hamiltonian elliptic systems: a guide to variational frameworks

Consider a Hamiltonian system of type \[ -Δu=H_{v}(u,v),\ -Δv=H_{u}(u,v) \ \ \text{ in } Ω, \qquad u,v=0 \text{ on } \partial Ω\] where $H$ is a power-type nonlinearity, for instance $H(u,v)= |u|^p/p+|v|^q/q$, having subcritical growth, and $Ω$ is a bounded domain of $\mathbb{R}^N$, $N\geq 1$. The aim of this paper is to give an overview of the several variational frameworks that can be used to treat such a system. Within each approach, we address existence of solutions, and in particular of ground state solutions. Some of the available frameworks are more adequate to derive certain qualitative properties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentration, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.

math.AP↗

arXiv2014

Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials

For the cubic Schrödinger system with trapping potentials in $\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existence and the orbital stability of standing waves having components with prescribed $L^2$-mass. We provide a variational characterization of such solutions, which gives information on the stability through of a condition of Grillakis-Shatah-Strauss type. As an application, we show existence of conditionally orbitally stable solitary waves when: a) the masses are small, for almost every scattering lengths, and b) in the defocusing, weakly interacting case, for any masses.

math.AP↗

arXiv2013

Existence and orbital stability of the ground states with prescribed mass for the L^2-critical and supercritical NLS on bounded domains

We study solutions of a semilinear elliptic equation with prescribed mass and Dirichlet homogeneous boundary conditions in the unitary ball. Such problem arises in the search of solitary wave solutions for nonlinear Schrödinger equations (NLS) with Sobolev subcritical power nonlinearity on bounded domains. Necessary and sufficient conditions are provided for the existence of such solutions. Moreover, we show that standing waves associated to least energy solutions are always orbitally stable when the nonlinearity is L^2-critical and subcritical, while they are almost always stable in the L^2-supercritical regime. The proofs are obtained in connection with the study of a variational problem with two constraints, of independent interest: to maximize the L^{p+1}-norm among functions having prescribed L^2 and H^1_0-norm.

math.AP↗

arXiv2012

Increasing powers in a degenerate parabolic logistic equation

The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem $$ \partial_t u-Δu=a u-b(x) u^p \text{in} Ω\times \R^+, u(0)=u_0, u(t)|_{\partial Ω}=0 $$ as $p\to +\infty$, where $Ω$ is a bounded domain and $b(x)$ is a nonnegative function. We deduce that the limiting configuration solves a parabolic obstacle problem, and afterwards we fully describe its long time behavior.

math.AP↗

arXiv2012

Existence and symmetry results for competing variational systems

In this paper we consider a class of gradient systems of type $$ -c_i Δu_i + V_i(x)u_i=P_{u_i}(u),\quad u_1,..., u_k>0 \text{in}Ω, \qquad u_1=...=u_k=0 \text{on} \partial Ω, $$ in a bounded domain $Ω\subseteq \R^N$. Under suitable assumptions on $V_i$ and $P$, we prove the existence of ground-state solutions for this problem. Moreover, for $k=2$, assuming that the domain $Ω$ and the potentials $V_i$ are radially symmetric, we prove that the ground state solutions are foliated Schwarz symmetric with respect to antipodal points. We provide several examples for our abstract framework.

math.AP↗

arXiv2011

Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems

In this paper we prove the existence of infinitely many sign-changing solutions for the system of $m$ Schrödinger equations with competition interactions $$ -Δu_i+a_i u_i^3+βu_i \sum_{j\neq i} u_j^2 =λ_{i,β} u_i \quad u_i\in H^1_0(Ω), \quad i=1,...,m $$ where $Ω$ is a bounded domain, $β>0$ and $a_i\geq 0\ \forall i.$ Moreover, for $a_i=0$, we show a relation between critical energies associated with this system and the optimal partition problem $$ \mathop{\inf_{ω_i\subset Ω\text{open}}}_{ω_i\cap ω_j=\emptyset\forall i\neq j} \sum_{i=1}^{m} λ_{k_i}(ω_i), $$ where $λ_{k_i}(ω)$ denotes the $k_i$--th eigenvalue of $-Δ$ in $H^1_0(ω)$. In the case $k_i\leq 2$ we show that the optimal partition problem appears as a limiting critical value, as the competition parameter $β$ diverges to $+\infty$.

math.AP↗

arXiv2010

Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems

In this paper we deal with the cubic Schrödinger system $ -Δu_i = \sum_{j=1}^n β_{ij}u_j^2 u_i$, $u_1,\dots,u_n \geq 0$ in $\mathbb{R}^N (N\leq 3)$, where $β=(β_{i,j})_{ij}$ is a symmetric matrix with real coefficients and $β_{ii}\geq 0$ for every $i=1,\ldots,n$. We analyse the existence and nonexistence of nontrivial solutions in connection with the properties of the matrix $β$, and provide a complete characterization in dimensions $N=1,2$. Extensions to more general power-type nonlinearities are given.

math.AP↗

arXiv2010

Regularity of the nodal set of segregated critical configurations under a weak reflection law

We deal with a class of Lipschitz vector functions $U=(u_1,...,u_h)$ whose components are non negative, disjointly supported and verify an elliptic equation on each support. Under a weak formulation of a reflection law, related to the Pohouzaev identity, we prove that the nodal set is a collection of $C^{1,α}$ hyper-surfaces (for every $0<α<1$), up to a residual set with small Hausdorff dimension. This result applies to the asymptotic limits of reaction-diffusion systems with strong competition interactions, to optimal partition problems involving eigenvalues, as well as to segregated standing waves for Bose-Einstein condensates in multiple hyperfine spin states.

math.AP↗

arXiv2009

Convergence of minimax and continuation of critical points for singularly perturbed systems

We consider a competitive system of two stationary Gross-Pitaevskii equations arising in the theory of Bose-Einstein condensation, and the corresponding scalar equation. We address the question: "Is it true that every bounded family of solutions of the system converges, as the competition parameter goes to infinity, to a pair which difference solves the scalar equation?". We discuss this question in the case when the solutions to the system are obtained as minimax critical points via (weak) L^2 Krasnoselskii genus theory. Our results, though still partial, give a strong indication of a positive answer.

math.AP↗

arXiv2008

Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition

For the positive solutions of the competitive Gross-Pitaevskii system of two equations, we prove that L^\infty boundedness implies uniform Hölder boundedness as the competition parameter goes to infinity. Moreover we prove that the limiting profile is Lipschitz continuous. The proof relies upon the blow-up technique and the monotonicity formulae by Almgren and Alt-Caffarelli-Friedman. This system arises in the Hartree-Fock approximation theory for binary mixtures of Bose-Einstein condensates in different hyperfine states. Extensions to systems with more than two densities are given.

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