658 research outputs found
Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems
We prove the existence of positive periodic solutions for the second order
nonlinear equation , where has superlinear growth at
zero and at infinity. The weight function is allowed to change its sign.
Necessary and sufficient conditions for the existence of nontrivial solutions
are obtained. The proof is based on Mawhin's coincidence degree and applies
also to Neumann boundary conditions. Applications are given to the search of
positive solutions for a nonlinear PDE in annular domains and for a periodic
problem associated to a non-Hamiltonian equation.Comment: 41 page
Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case
We study the periodic and the Neumann boundary value problems associated with
the second order nonlinear differential equation \begin{equation*} u'' + c u' +
\lambda a(t) g(u) = 0, \end{equation*} where is a
sublinear function at infinity having superlinear growth at zero. We prove the
existence of two positive solutions when and
is sufficiently large. Our approach is based on Mawhin's
coincidence degree theory and index computations.Comment: 26 page
A constructive approach to regularity of Lagrangian trajectories for incompressible Euler flow in a bounded domain
The 3D incompressible Euler equation is an important research topic in the
mathematical study of fluid dynamics. Not only is the global regularity for
smooth initial data an open issue, but the behaviour may also depend on the
presence or absence of boundaries.
For a good understanding, it is crucial to carry out, besides mathematical
studies, high-accuracy and well-resolved numerical exploration. Such studies
can be very demanding in computational resources, but recently it has been
shown that very substantial gains can be achieved first, by using Cauchy's
Lagrangian formulation of the Euler equations and second, by taking advantages
of analyticity results of the Lagrangian trajectories for flows whose initial
vorticity is H\"older-continuous. The latter has been known for about twenty
years (Serfati, 1995), but the combination of the two, which makes use of
recursion relations among time-Taylor coefficients to obtain constructively the
time-Taylor series of the Lagrangian map, has been achieved only recently
(Frisch and Zheligovsky, 2014; Podvigina {\em et al.}, 2016 and references
therein).
Here we extend this methodology to incompressible Euler flow in an
impermeable bounded domain whose boundary may be either analytic or have a
regularity between indefinite differentiability and analyticity.
Non-constructive regularity results for these cases have already been obtained
by Glass {\em et al.} (2012). Using the invariance of the boundary under the
Lagrangian flow, we establish novel recursion relations that include
contributions from the boundary. This leads to a constructive proof of
time-analyticity of the Lagrangian trajectories with analytic boundaries, which
can then be used subsequently for the design of a very high-order
Cauchy--Lagrangian method.Comment: 18 pages, no figure
Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem
We study the second-order nonlinear differential equation u\u2032\u2032+a(t)g(u)=0 , where g is a continuously differentiable function of constant sign defined on an open interval I 86R and a(t) is a sign-changing weight function. We look for solutions u(t) of the differential equation such that u(t) 08I, satisfying the Neumann boundary conditions. Special examples, considered in our model, are the equations with singularity, for I=R+0 and g(u) 3c 12u 12\u3c3, as well as the case of exponential nonlinearities, for I=R and g(u) 3cexp(u) . The proofs are obtained by passing to an equivalent equation of the form x\u2032\u2032=f(x)(x\u2032)2+a(t)
Periodic solutions for critical fractional problems
We deal with the existence of -periodic solutions to the following
non-local critical problem \begin{equation*} \left\{\begin{array}{ll}
[(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in}
(-\pi,\pi)^{N} \\ u(x+2\pi e_{i})=u(x) &\mbox{for all} x \in \mathbb{R}^{N},
\quad i=1, \dots, N, \end{array} \right. \end{equation*} where , , , is the fractional critical
Sobolev exponent, is a positive continuous function, and is a
superlinear -periodic (in ) continuous function with subcritical
growth. When , the existence of a nonconstant periodic solution is
obtained by applying the Linking Theorem, after transforming the above
non-local problem into a degenerate elliptic problem in the half-cylinder
, with a nonlinear Neumann boundary
condition, through a suitable variant of the extension method in periodic
setting. We also consider the case by using a careful procedure of limit.
As far as we know, all these results are new.Comment: Calculus of Variations and Partial Differential Equations (2018
- …