273,025 research outputs found
Existence of Nontrivial Solutions for p-Laplacian Equations in {R}^{N}
In this paper, we consider a p-Laplacian equation in {R}^{N}with
sign-changing potential and subcritical p-superlinear nonlinearity. By using
the cohomological linking method for cones developed by Degiovanni and
Lancelotti in 2007, an existence result is obtained. We also give a result on
the existence of periodic solutions for one-dimensional -Laplacian equations
which can be proved by the same method.Comment: 19 pages, submitte
Periodic orbits in Hamiltonian systems with involutory symmetries
We study the existence of families of periodic solutions in a neighbourhood
of a symmetric equilibrium point in two classes of Hamiltonian systems with
involutory symmetries. In both classes, involutions reverse the sign of the
Hamiltonian function. In the first class we study a Hamiltonian system with a
reversing involution R acting symplectically. We first recover a result of
Buzzi and Lamb showing that the equilibrium point is contained in a three
dimensional conical subspace which consists of a two parameter family of
periodic solutions with symmetry R and there may or may not exist two families
of non-symmetric periodic solutions, depending on the coefficients of the
Hamiltonian. In the second problem we study an equivariant Hamiltonian system
with a symmetry S that acts anti-symplectically. Generically, there is no
S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we
prove the existence of at least 2 and at most 12 families of non-symmetric
periodic solutions. We conclude with a brief study of systems with both forms
of symmetry, showing they have very similar structure to the system with
symmetry R
A mechanism for detecting normally hyperbolic invariant tori in differential equations
Determining the existence of compact invariant manifolds is a central quest
in the qualitative theory of differential equations. Singularities, periodic
solutions, and invariant tori are examples of such invariant manifolds. A
classical and useful result from the averaging theory relates the existence of
isolated periodic solutions of non-autonomous periodic differential equations,
given in a specific standard form, with the existence of simple singularities
of the so-called guiding system, which is an autonomous differential equation
given in terms of the first non-vanishing higher order averaged function. In
this paper, we provide an analogous result for the existence of invariant tori.
Namely, we show that a non-autonomous periodic differential equation, given in
the standard form, has a normally hyperbolic invariant torus in the extended
phase space provided that the guiding system has a hyperbolic limit cycle. We
apply this result to show the existence of normally hyperbolic invariant tori
in a family of jerk differential equations
Almost periodic linear differential equations with non-separated solutions
AbstractA celebrated result by Favard states that, for certain almost periodic linear differential systems, the existence of a bounded solution implies the existence of an almost periodic solution. A key assumption in this result is the separation among bounded solutions. Here we prove a theorem of anti-Favard type: if there are bounded solutions which are non-separated (in a strong sense) sometimes almost periodic solutions do not exist. Strongly non-separated solutions appear when the associated homogeneous system has homoclinic solutions. This point of view unifies two fascinating examples by Zhikov–Levitan and Johnson for the scalar case. Our construction uses the ideas of Zhikov–Levitan together with the theory of characters in topological groups
Singularity \& Regularity Issues for Simplified Models of Turbulence
We consider a family of Leray- models with periodic boundary
conditions in three space dimensions. Such models are a regularization, with
respect to a parameter , of the Navier-Stokes equations. In particular,
they share with the original equation (NS) the property of existence of global
weak solutions. We establish an upper bound on the Hausdorff dimension of the
time singular set of those weak solutions when is subcritical. The
result is an interpolation between the bound proved by Scheffer for the
Navier-Stokes equations and the regularity result proved in \cite{A01}
Mathematical study of a parabolic system describing the evolution of the solar magnetic field
We study a system of two strongly coupled parabolic
equations describing a solar dynamo wave. We investigate the
existence and uniqueness of a classical solution and the existence
of a periodic in time solution. In the Appendix, an existence
result of periodic solutions for an auxiliary quasilinear parabolic
equation is provided, together with a C0 estimate of such solution
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