1,721,005 research outputs found
Bifurcation diagrams for singularly perturbed system: the multi-dimensional case.
Full text linkWe consider a singularly perturbed system where the fast dynamics of the unperturbed problem exhibits a trajectory homoclinic to a critical point. We assume that the slow time system admits a unique critical point, which undergoes a bifurcation as a second parameter varies: transcritical, saddle-node, or pitchfork. We generalize to the multi-dimensional case the results obtained in a previous paper where the slow-time system is -dimensional. We prove the existence of a unique trajectory homoclinic to a centre manifold of the slow manifold. Then we construct curves in the -dimensional parameters space, dividing it in different areas where is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples
Bifurcation diagrams for singularly perturbed system
Full text linkWe consider a singularly perturbed system where the fast dynamic of the unperturbed problem exhibits a trajectory homoclinic to a critical point. We assume that the slow time system is -dimensional and it admits a unique critical point, which undergoes to a bifurcation as a second parameter varies: transcritical, saddle-node, or pitchfork. In this setting Battelli and Palmer proved the existence of a unique trajectory homoclinic to the slow manifold. The purpose of this paper is to construct curves which divide the -dimensional parameters space in different areas where is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples
Ground states and singular ground states for quasilinear elliptic equations in the subcritical case
Full text linkWe consider radial solution u(vertical bar x vertical bar), x is an element of R-n, of a p-Laplace equation with non-linear potential depending also on the space variable x. We assume that the potential is polynomial and it is negative for u small and positive and subcritical for u large.
We prove the existence of radial Ground States under suitable Hypotheses on the potential f(u, vertical bar x vertical bar). Furthermore we prove the existence of uncountably many radial Singular Ground States; this last result seems to be new even for the spatial independent case and even for p = 2.
The proofs combine an energy analysis and the dynamical systems approach developed by Johnson, Pan, Yi and Battelli for the p = 2 case
Fowler transformation and radial solutions for quasilinear elliptic equations. Part 1: the subcritical and the supercritical case.
Full text linkWe illustrate a method, based on a generalized Fowler transformation,
to discuss the existence and the asymptotic behavior of positive
radial solutions for the following equation:
where , , .
This approach proves to be particularly
useful in the spatial dependent case. Moreover it is a good tool
to detect singular and fast decay solutions.
We apply it to the case in which is either subcritical or supercritical, obtaining structure
results for positive solutions and refining the estimates on the asymptotic behavior.
The equation has been proposed as a reaction diffusion model for a non-Newtonian fluid and
can also be regarded as the constitutive law for a problem in elasticity theory
Phase plane analysis for radial solutions to supercritical quasilinear elliptic equations in a ball
No full textWe consider the following problem
u>0 ; extrm{ in $B$, } quad extrm{ and } quad u=0 extrm{ on $; partial B$.}
where is the unitary ball in .
Merle and Peletier considered the classical Laplace case , and proved the existence of a unique value for which
a radial singular positive solution exists, assuming
and q>2^*:=rac{2n}{n-2}.
Then Dolbeault and Flores proved that, if q>2^* but is smaller than the Joseph-Lundgren exponent
, then there is an unbounded sequence of radial positive classical solutions for (1), which accumulate at ,
again for
On a non-homogeneous and non-linear heat equation
No full textWe consider the Cauchy-problem for a parabolic equation of the
following type:
where , n >2, is supercritical. We supplement this equation by the initial
condition , and we allow to be either bounded or
unbounded in the origin but smaller than stationary singular
solutions. We discuss local existence and long time behaviour for
the solutions for a wide class of non-homogeneous
non-linearities . We show that in the supercritical case, ground
states with slow decay lie on the threshold between
initial data corresponding to blow-up solutions, and the basin of attraction of the null solution. Our
results extend previous ones in that we allow to be a Matukuma-type potential and in that we allow
it to depend on in a more general way.
We explore such a threshold in the subcritical case
too, and we obtain a result which is new even for the model case .
We find a family of initial data
which have fast decay (i.e. ), are
arbitrarily small in - norm, but which correspond to blow-up solutions
On a diffusion model with absorption and production
No full textWe discuss the structure of radial solutions of some superlinear elliptic equations which model
diffusion phenomena when both absorption and production are present.
We focus our attention on solutions defined in (regular) or in
(singular) which are infinitesimal at infinity, discussing also their asymptotic behavior.
The phenomena we find are present only if absorption and production coexist, i.e., if
the reaction term changes sign. Our results are then generalized to include the case where Hardy potentials are considered
Non-Autonomous Quasilinear Elliptic Equations and Wazewski's principle
Full text linkIn this paper we investigate positive radial solutions of the
following equation
Δpu + K(r)u|u|σ−2 = 0
where r = |x|, x ∈ Rn, n > p > 1, σ = np/(n − p) is the Sobolev critical
exponent and K(r) is a function strictly positive and bounded.
This paper can be seen as a completion of the work started in [9],
where structure theorems for positive solutions are obtained for potentials
K(r) making a finite number of oscillations. Just as in [9], the starting
point is to introduce a dynamical system using a Fowler transform. In [9]
the results are obtained using invariant manifold theory and a dynamical
interpretation of the Pohozaev identity; but the restriction 2n/(n + 2) ≤
p ≤ 2 is necessary in order to ensure local uniqueness of the trajectories of
the system. In this paper we remove this restriction, repeating the proof
using a modification of Ważewski’s principle; we prove for the cases p > 2
and 1 < p < 2n/(n + 2) results similar to the ones obtained in the case
2n/(n + 2) ≤ p ≤ 2.
We also introduce a method to prove the existence of Ground States
with fast decay for potentials K(r) which oscillates indefinitely. This new
tool also shed some light on the role played by regular and singular perturbations
in this problem, see [10]
Asymptotic expansion of solutions of an elliptic equation related to the non-linear Schr"{o}dinger equation
No full textWe study the radially symmetric blow-up solutions of the nonlinear Schrödinger
equation. We give a method for developing such a solution in a series which
represents it asymptotically
Positive solutions of semilinear elliptic equations: a dynamical approach
No full textThis paper is devoted to the study of the structure of positive radial solutions
for the following semi-linear equation:
We require to be nonnegative and to exhibit
both subcritical and supercritical behavior with respect to the Sobolev
critical exponent. More precisely we assume
that is subcritical for small and large and supercritical for
large and small, and we give existence and non-existence results for ground states
regular and singular, with either fast or slow decay.
We find a surprisingly rich structure, which is
characterized by two different patterns of bifurcations.
We perform a Fowler transformation and we use a dynamical approach, exploiting
some ideas borrowed from Bamon, Del Pino, Flores, combining them with the use of the translation
of the Pohozaev function for this dynamical context
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