68 research outputs found
On Regions of Existence and Nonexistence of solutions for a System of --Laplacians
We give a new region of existence of solutions to the superhomogeneous
Dirichlet problem \quad \begin{array}{l} -\Delta_{p} u= v^\delta\quad
v>0\quad {in}\quad B,\cr -\Delta_{q} v = u^{\mu}\quad u>0\quad {in}\quad B, \cr
u=v=0 \quad {on}\quad \partial B, \end{array}\leqno{(S_R)} where is the
ball of radius centered at the origin in \RR^N. Here
and is the Laplacian
operator for .Comment: 17 pages, accepted in Asymptotic Analysi
A new dynamical approach of Emden-Fowler equations and systems
We give a new approach on general systems of the form (G){[c]{c}%
-\Delta_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=\epsilon_{1}|x|
^{a}u^{s}v^{\delta}, -\Delta_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla
u)=\epsilon_{2}|x|^{b}u^{\mu}v^{m}, where are
real parameters, and In
the radial case we reduce the problem to a quadratic system of order 4, of
Kolmogorov type. Then we obtain new local and global existence or nonexistence
results. In the case we also describe the
behaviour of the ground states in two cases where the system is variational. We
give an important result on existence of ground states for a nonvariational
system with and In the nonradial case we solve a conjecture of
nonexistence of ground states for the system with and and
Comment: 43 page
Trapped surfaces in prolate collapse in the Gibbons-Penrose construction
We investigate existence and properties of trapped surfaces in two models of
collapsing null dust shells within the Gibbons-Penrose construction. In the
first model, the shell is initially a prolate spheroid, and the resulting
singularity forms at the ends first (relative to a natural time slicing by flat
hyperplanes), in analogy with behavior found in certain prolate collapse
examples considered by Shapiro and Teukolsky. We give an explicit example in
which trapped surfaces are present on the shell, but none exist prior to the
last flat slice, thereby explicitly showing that the absence of trapped
surfaces on a particular, natural slicing does not imply an absence of trapped
surfaces in the spacetime. We then examine a model considered by Barrabes,
Israel and Letelier (BIL) of a cylindrical shell of mass M and length L, with
hemispherical endcaps of mass m. We obtain a "phase diagram" for the presence
of trapped surfaces on the shell with respect to essential parameters and . It is found that no trapped surfaces are
present on the shell when or are sufficiently small. (We are
able only to search for trapped surfaces lying on the shell itself.) In the
limit , the existence or nonexistence of trapped surfaces lying
within the shell is seen to be in remarkably good accord with the hoop
conjecture.Comment: 22 pages, 6 figure
Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks
This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems
Sign-changing bubble-tower solutions to fractional semilinear elliptic problems
We study the asymptotic and qualitative properties of least energy radial
sign-changing solutions to fractional semilinear elliptic problems of the form
where , is the s-Laplacian, is a ball of ,
is the critical Sobolev exponent and
is a small parameter. We prove that such solutions have the limit profile of a
"tower of bubbles", as , i.e. the positive and negative
parts concentrate at the same point with different concentration speeds.
Moreover, we provide information about the nodal set of these solutions
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