54 research outputs found
On the limit configuration of four species strongly competing systems
We analysed some qualitative properties of the limit configuration of the
solutions of a reaction-diffusion system of four competing species as the
competition rate tends to infinity. Large interaction induces the spatial
segregation of the species and only two limit configurations are possible:
either there is a point where four species concur, a 4-point, or there are two
points where only three species concur. We characterized, for a given datum,
the possible 4-point configuration by means of the solution of a Dirichlet
problem for the Laplace equation
On the blow-up threshold for weakly coupled nonlinear Schroedinger equations
We study the Cauchy problem for a system of two coupled nonlinear focusing
Schroedinger equations arising in nonlinear optics. We discuss when the
solutions are global in time or blow-up in finite time. Some results, in
dependence of the data of the problem, are proved; in particular we give a
bound, depending on the coupling parameter, for the blow-up threshold.Comment: 14 page
A Dirichlet problem in the strip
In this paper we investigate a Dirichlet problem in a strip and, using the sliding method, we prove monotonicity for positive and bounded solutions. We obtain uniqueness of the solution and show that this solution is a function of only one variable. From these qualitative properties we deduce existence of a classical solution for this problem
Semiclassical states for weakly coupled nonlinear Schr\"odinger systems
We consider systems of weakly coupled Schr\"odinger equations with
nonconstant potentials and we investigate the existence of nontrivial
nonnegative solutions which concentrate around local minima of the potentials.
We obtain sufficient and necessary conditions for a sequence of least energy
solutions to concentrate.Comment: 23 pages, no figure
Fractional diffusion with Neumann boundary conditions: the logistic equation
Motivated by experimental studies on the anomalous diffusion of biological
populations, we introduce a nonlocal differential operator which can be
interpreted as the spectral square root of the Laplacian in bounded domains
with Neumann homogeneous boundary conditions. Moreover, we study related linear
and nonlinear problems exploiting a local realization of such operator as
performed in [X. Cabre' and J. Tan. Positive solutions of nonlinear problems
involving the square root of the Laplacian. Adv. Math. 2010] for Dirichlet
homogeneous data. In particular we tackle a class of nonautonomous
nonlinearities of logistic type, proving some existence and uniqueness results
for positive solutions by means of variational methods and bifurcation theory.Comment: 36 pages, 1 figur
On the logarithmic Schrodinger equation
In the framework of the nonsmooth critical point theory for lower
semi-continuous functionals, we propose a direct variational approach to
investigate the existence of infinitely many weak solutions for a class of
semi-linear elliptic equations with logarithmic nonlinearity arising in
physically relevant situations. Furthermore, we prove that there exists a
unique positive solution which is radially symmetric and nondegenerate.Comment: 10 page
Geometry of the limiting solution of a strongly competing system
We report on known results on the geometry of the limiting solutions of a reaction-diffusion system in any number of competing species k as the competition rate m tends to infinity. The case k=8 is studied in detail. We provide numerical simulations of solutions of the system for k=4,6,8 and large competition rate. Thanks to FreeFEM++ software, we obtain nodal partitions showing the predicted limiting configurations
Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schr\"odinger systems
We study the spectral structure of the complex linearized operator for a
class of nonlinear Schr\"odinger systems, obtaining as byproduct some
interesting properties of non-degenerate ground state of the associated
elliptic system, such as being isolated and orbitally stable.Comment: 18 pages, 1 figur
Oscillating solutions for nonlinear nonlinear Helmholtz equations
Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behavior at infinity is established. Some generalizations to nonautonomous radial equations as well as existence results for nonradial solutions are found. Our theorems prove the existence of standing waves solutions of nonlinear Klein-Gordon or Schrödinger equations with large frequencies
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