1,013 research outputs found
Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions
Assuming is a ball in , we analyze the positive
solutions of the problem that branch out from the constant solution as grows from to
. The non-zero constant positive solution is the unique positive
solution for close to . We show that there exist arbitrarily many
positive solutions as (in particular, for supercritical exponents)
or as for any fixed value of , answering partially a
conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for
and so that a given number of solutions exist. The geometrical properties
of those solutions are studied and illustrated numerically. Our simulations
motivate additional conjectures. The structure of the least energy solutions
(among all or only among radial solutions) and other related problems are also
discussed.Comment: 37 pages, 24 figure
Global bifurcation for monotone fronts of elliptic equations
In this paper, we present two results on global continuation of monotone
front-type solutions to elliptic PDEs posed on infinite cylinders. This is done
under quite general assumptions, and in particular applies even to fully
nonlinear equations as well as quasilinear problems with transmission boundary
conditions. Our approach is rooted in the analytic global bifurcation theory of
Dancer and Buffoni--Toland, but extending it to unbounded domains requires
contending with new potential limiting behavior relating to loss of
compactness. We obtain an exhaustive set of alternatives for the global
behavior of the solution curve that is sharp, with each possibility having a
direct analogue in the bifurcation theory of second-order ODEs.
As a major application of the general theory, we construct global families of
internal hydrodynamic bores. These are traveling front solutions of the full
two-phase Euler equation in two dimensions. The fluids are confined to a
channel that is bounded above and below by rigid walls, with incompressible and
irrotational flow in each layer. Small-amplitude fronts for this system have
been obtained by several authors. We give the first large-amplitude result in
the form of continuous curves of elevation and depression bores. Following the
elevation curve to its extreme, we find waves whose interfaces either overturn
(develop a vertical tangent) or become exceptionally singular in that the flow
in both layers degenerates at a single point on the boundary. For the curve of
depression waves, we prove that either the interface overturns or it comes into
contact with the upper wall.Comment: 60 pages, 6 figure
- …