Hugo Tavares

SIGN-CHANGING SOLUTIONS OF COMPETITION-DIFFUSION ELLIPTIC SYSTEMS AND OPTIMAL PARTITION PROBLEMS

Abstract In this paper we prove the existence of infinitely many sign-changing solutions for the system of m Schrodinger equations with competition interactions − Δ u i + a i u i 3 + β u i ∑ j ≠ i u j 2 = λ i , β u i , u i ∈ H 0 1 ( Ω ) , i = 1 , … , m where Ω is a bounded domain, β > 0 and a i ⩾ 0 ∀i. Moreover, for a i = 0 , we show a relation between critical energies associated with this system and the optimal partition problem inf ω i ⊂ Ω open ω i ∩ ω j = ∅ ∀ i ≠ j ∑ i = 1 m λ k i ( ω i ) , where λ k i ( ω ) denotes the k i -th eigenvalue of −Δ in H 0 1 ( ω ) . In the case k i ⩽ 2 we show that the optimal partition problem appears as a limiting critical value, as the competition parameter β diverges to +∞.

Existence and orbital stability of the ground states with prescribed mass for the

Given > 0, we study the elliptic problem , N and p) are provided for the existence of solutions. Moreover, we show that standing waves associ- ated to least energy solutions are orbitally stable for every (in the exis- tence range) when p is L 2 -critical and subcritical, i.e. 1 < p 1 + 4=N, while they are stable for almost every in the L 2 -supercritical regime 1 + 4=N < p < 2 1. The proofs are obtained in connection with the study of a variational problem with two constraints, of independent inter- est: to maximize the L p+1 -norm among functions having prescribed L 2 and H 1 0 -norm.

L^2

In this article we deal with the cubic Schrödinger system where β = (β i, j ) ij is a symmetric matrix with real coefficients and β ii ≥ 0 for every i = 1,…, n. We analyze 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.

-critical and supercritical NLS on bounded domains

AbstractLet

Extremality Conditions and Regularity of Solutions to Optimal Partition Problems Involving Laplacian Eigenvalues

{\Omega \subset \mathbb{R}^N}

Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions

Ω⊂RN be an open bounded domain and

Variational Problems with Long-Range Interaction

{m \in \mathbb{N}}