201,652 research outputs found
Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains
We provide new results on the existence, non-existence, localization and
multiplicity of nontrivial solutions for systems of Hammerstein integral
equations. Some of the criteria involve a comparison with the spectral radii of
some associated linear operators. We apply our results to prove the existence
of multiple nonzero radial solutions for some systems of elliptic boundary
value problems subject to nonlocal boundary conditions. Our approach is
topological and relies on the classical fixed point index. We present an
example to illustrate our theory.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1404.139
Boundary-layers for a Neumann problem at higher critical exponents
We consider the Neumann problem where is an open bounded
domain in is the unit inner normal at the boundary and
For any integer, we show that, in some suitable domains
problem has a solution which blows-up along a
dimensional minimal submanifold of the boundary as
approaches from either below or above the higher critical Sobolev exponent
Comment: 13 page
Mean Field Games models of segregation
This paper introduces and analyses some models in the framework of Mean Field
Games describing interactions between two populations motivated by the studies
on urban settlements and residential choice by Thomas Schelling. For static
games, a large population limit is proved. For differential games with noise,
the existence of solutions is established for the systems of partial
differential equations of Mean Field Game theory, in the stationary and in the
evolutive case. Numerical methods are proposed, with several simulations. In
the examples and in the numerical results, particular emphasis is put on the
phenomenon of segregation between the populations.Comment: 35 pages, 10 figure
Multi-Peak Solutions for a Wide Class of Singular Perturbation Problems
In this paper we are
concerned with a wide class of singular perturbation problems arising
from such diverse fields as phase transitions,
chemotaxis, pattern formation,
population dynamics and chemical reaction theory.
We study the corresponding elliptic
equations in a bounded domain without any symmetry
assumptions. We assume that the
mean curvature of the boundary
has \overline{M} isolated, non-degenerate critical points.
Then we show that for any positive integer m\leq \overline{M}
there exists a stationary
solution with M local peaks which are attained on the boundary and
which lie close to these critical points.
Our method is based on Liapunov-Schmidt reduction
A positive fixed point theorem with applications to systems of Hammerstein integral equations
We present new criteria on the existence of fixed points that combine some
monotonicity assumptions with the classical fixed point index theory. As an
illustrative application, we use our theoretical results to prove the existence
of positive solutions for systems of nonlinear Hammerstein integral equations.
An example is also presented to show the applicability of our results.Comment: 11 page
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