697 research outputs found
Spacelike radial graphs of prescribed mean curvature in the Lorentz-Minkowski space
In this paper we investigate the existence and uniqueness of spacelike radial
graphs of prescribed mean curvature in the Lorentz-Minkowski space
, for , spanning a given boundary datum lying on the
hyperbolic space
On a fourth order nonlinear Helmholtz equation
In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz
equation in for positive, bounded and -periodic functions . Using
the dual method of Evequoz and Weth, we find solutions to this equation and
establish some of their qualitative properties
Qualitative properties of solutions to mixed-diffusion bistable equations
We consider a fourth-order extension of the Allen-Cahn model with
mixed-diffusion and Navier boundary conditions. Using variational and
bifurcation methods, we prove results on existence, uniqueness, positivity,
stability, a priori estimates, and symmetry of solutions. As an application, we
construct a nontrivial bounded saddle solution in the plane.Comment: New version with minor change
Nonlinear Schr{\"o}dinger equation: concentration on circles driven by an external magnetic field
In this paper, we study the semiclassical limit for the stationary magnetic
nonlinear Schr\"odinger equation \begin{align}\label{eq:initialabstract}\left(
i \hbar \nabla + A(x) \right)^2 u + V(x) u = |u|^{p-2} u, \quad x\in
\mathbb{R}^{3},\end{align}where p\textgreater{}2, is a vector potential
associated to a given magnetic field , i.e and is a
nonnegative, scalar (electric) potential which can be singular at the origin
and vanish at infinity or outside a compact set.We assume that and
satisfy a cylindrical symmetry. By a refined penalization argument, we prove
the existence of semiclassical cylindrically symmetric solutions of upper
equation whose moduli concentrate, as , around a circle. We
emphasize that the concentration is driven by the magnetic and the electric
potentials. Our result thus shows that in the semiclassical limit, the magnetic
field also influences the location of the solutions of
(\ref{eq:initialabstract}) if their concentration occurs around a locus, not
a single point
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
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