661 research outputs found
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 Schr\"odinger equations with competition
interactions
where is a bounded domain, and
Moreover, for , we show a relation between critical energies associated
with this system and the optimal partition problem
where denotes the --th eigenvalue of
in . In the case we show that the optimal partition
problem appears as a limiting critical value, as the competition parameter
diverges to .Comment: 25 page
Two Positive Normalized Solutions and Phase Separation for Coupled Schr\"odinger Equations on Bounded Domain with L2-Supercritical and Sobolev Critical or Subcritical Exponent
In this paper we study the existence of positive normalized solutions of the
following coupled Schr\"{o}dinger system: \begin{align} \left\{ \begin{aligned}
& -\Delta u = \lambda_u u + \mu_1 u^3 + \beta uv^2, \quad x \in \Omega, \\ &
-\Delta v = \lambda_v v + \mu_2 v^3 + \beta u^2 v, \quad x \in \Omega, \\ & u >
0, v > 0 \quad \text{in } \Omega, \quad u = v = 0 \quad \text{on }
\partial\Omega, \end{aligned} \right. \nonumber \end{align} with the
constraint \begin{align} \int_{\Omega}|u|^2dx = c_1, \quad \quad
\int_{\Omega}|v|^2dx = c_2, \nonumber \end{align} where ,
, , and ()
is smooth, bounded, and star-shaped. Note that the nonlinearities and the
coupling terms are both -supercritical in dimensions 3 and 4, Sobolev
subcritical in dimension 3, Sobolev critical in dimension 4. We show that this
system has a positive normalized solution which is a local minimizer. We
further show that the system has a second positive normalized solution, which
is of M-P type when . This seems to be the first existence result of two
positive normalized solutions for such a Schr\"{o}dinger system, especially in
the Sobolev critical case. We also study the limit behavior of the positive
normalized solutions in the repulsive case , and phase
separation is expected.Comment: 32 page
International Conference on Nonlinear Differential Equations and Applications
Dear Participants, Colleagues and Friends
It is a great honour and a privilege to give you all a warmest welcome to the first Portugal-Italy Conference on Nonlinear Differential Equations and Applications (PICNDEA).
This conference takes place at the Colégio Espírito Santo, University of Évora, located in the beautiful city of Évora, Portugal. The host institution, as well the associated scientific research centres, are committed to the event, hoping that it will be a benchmark for scientific collaboration between the two countries in the area of mathematics.
The main scientific topics of the conference are Ordinary and Partial Differential Equations, with particular regard to non-linear problems originating in applications, and its treatment with the methods of Numerical Analysis. The fundamental main purpose is to bring together Italian and Portuguese researchers in the above fields, to create new, and amplify previous collaboration, and to follow and discuss new topics in the area
Construction of a solution for the two-component radial Gross-Pitaevskii system with a large coupling parameter
We consider strongly coupled competitive elliptic systems that arise in the
study of two-component Bose-Einstein condensates. As the coupling parameter
tends to infinity, solutions that remain uniformly bounded are known to
converge to a segregated limiting profile, with the difference of its
components satisfying a limit scalar PDE. In the case of radial symmetry, under
natural non-degeneracy assumptions on a solution of the limit problem, we
establish by a perturbation argument its persistence as a solution to the
elliptic system
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