46 research outputs found
H\'enon type equations and concentration on spheres
In this paper we study the concentration profile of various kind of symmetric
solutions of some semilinear elliptic problems arising in astrophysics and in
diffusion phenomena. Using a reduction method we prove that doubly symmetric
positive solutions in a -dimensional ball must concentrate and blow up on
-spheres as the concentration parameter tends to infinity. We also
consider axially symmetric positive solutions in a ball in , , and show that concentration and blow up occur on two antipodal points,
as the concentration parameter tends to infinity
On the finite space blow up of the solutions of the Swift-Hohenberg equation
The aim of this paper is to study the finite space blow up of the solutions
for a class of fourth order differential equations. Our results answer a
conjecture in [F. Gazzola and R. Pavani. Wide oscillation finite time blow up
for solutions to nonlinear fourth order differential equations. Arch. Ration.
Mech. Anal., 207(2):717--752, 2013] and they have implications on the
nonexistence of beam oscillation given by traveling wave profile at low speed
propagation.Comment: 24 pages, 2 figure
Hamiltonian elliptic systems: a guide to variational frameworks
Consider a Hamiltonian system of type where is a power-type nonlinearity, for instance , having subcritical growth, and is a bounded domain
of , . The aim of this paper is to give an overview of
the several variational frameworks that can be used to treat such a system.
Within each approach, we address existence of solutions, and in particular of
ground state solutions. Some of the available frameworks are more adequate to
derive certain qualitative properties; we illustrate this in the second half of
this survey, where we also review some of the most recent literature dealing
mainly with symmetry, concentration, and multiplicity results. This paper
contains some original results as well as new proofs and approaches to known
facts.Comment: 78 pages, 7 figures. This corresponds to the second version of this
paper. With respect to the original version, this one contains additional
references, and some misprints were correcte
Local minimizers in spaces of symmetric functions and applications
We study versus local minimizers for functionals defined on
spaces of symmetric functions, namely functions that are invariant by the
action of some subgroups of . These functionals, in many cases,
are associated with some elliptic partial differential equations that may have
supercritical growth. So we also prove some results on classical regularity for
symmetric weak solutions for a general class of semilinear elliptic equations
with possibly supercritical growth. We then apply these results to prove the
existence of a large number of classical positive symmetric solutions to some
concave-convex elliptic equations of H\'enon type
Periodic solutions and torsional instability in a nonlinear nonlocal plate equation
A thin and narrow rectangular plate having the two short edges hinged and the
two long edges free is considered. A nonlinear nonlocal evolution equation
describing the deformation of the plate is introduced: well-posedness and
existence of periodic solutions are proved. The natural phase space is a
particular second order Sobolev space that can be orthogonally split into two
subspaces containing, respectively, the longitudinal and the torsional
movements of the plate. Sufficient conditions for the stability of periodic
solutions and of solutions having only a longitudinal component are given. A
stability analysis of the so-called prevailing mode is also performed. Some
numerical experiments show that instabilities may occur. This plate can be seen
as a simplified and qualitative model for the deck of a suspension bridge,
which does not take into account the complex interactions between all the
components of a real bridge.Comment: 34 pages, 4 figures. The result of Theorem 6 is correct, but the
proof was not correct. We slightly changed the proof in this updated versio
On Hamiltonian systems with critical Sobolev exponents
In this paper we consider lower order perturbations of the critical
Lane-Emden system posed on a bounded smooth domain , with , inspired by the classical results of Brezis and
Nirenberg \cite{BrezisNirenberg1983}. We solve the problem of finding a
positive solution for all dimensions . For the critical dimension
we show a new phenomenon, not observed for scalar problems. Namely, there
are parts on the critical hyperbola where solutions exist for all
-homogeneous or subcritical superlinear perturbations and parts where there
are no solutions for some of those perturbations.Comment: 26 page
Orbitally stable standing waves of a mixed dispersion nonlinear Schr\"odinger equation
We study the mixed dispersion fourth order nonlinear Schr\"odinger equation
\begin{equation*} %\tag{\protect{4NLS}}\label{4nls} i \partial_t \psi -\gamma
\Delta^2 \psi +\beta \Delta \psi +|\psi|^{2\sigma} \psi =0\ \text{in}\ \R
\times\R^N, \end{equation*} where and . We
focus on standing wave solutions, namely solutions of the form , for some . This ansatz yields the
fourth-order elliptic equation \begin{equation*}
%\tag{\protect{*}}\label{4nlsstar} \gamma \Delta^2 u -\beta \Delta u +\alpha u
=|u|^{2\sigma} u. \end{equation*} We consider two associated constrained
minimization problems: one with a constraint on the -norm and the other on
the -norm. Under suitable conditions, we establish existence of
minimizers and we investigate their qualitative properties, namely their sign,
symmetry and decay at infinity as well as their uniqueness, nondegeneracy and
orbital stability.Comment: 37 pages. To appear in SIAM J. Math. Ana
On unique continuation principles for some elliptic systems
In this paper we prove unique continuation principles for some systems of
elliptic partial differential equations satisfying a suitable superlinearity
condition. As an application, we obtain nonexistence of nontrivial (not
necessarily positive) radial solutions for the Lane-Emden system posed in a
ball, in the critical and supercritical regimes. Some of our results also apply
to general fully nonlinear operators, such as Pucci's extremal operators, being
new even for scalar equations.Comment: 18 page