1,114 research outputs found
Multiple positive solutions to elliptic boundary blow-up problems
We prove the existence of multiple positive radial solutions to the
sign-indefinite elliptic boundary blow-up problem where is a function superlinear at zero and at infinity,
and are the positive/negative part, respectively, of a sign-changing
function and is a large parameter. In particular, we show how the
number of solutions is affected by the nodal behavior of the weight function
. The proof is based on a careful shooting-type argument for the equivalent
singular ODE problem. As a further application of this technique, the existence
of multiple positive radial homoclinic solutions to is also considered
Exact meromorphic stationary solutions of the real cubic Swift-Hohenberg equation
We show that all meromorphic solutions of the stationary reduction of the
real cubic Swift-Hohenberg equation are elliptic or degenerate elliptic. We
then obtain them all explicitly by the subequation method, and one of them
appears to be a new elliptic solution.Comment: 15 pages, 3 figures, to appear, Studies in Applied Mathematic
A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations on and those
of the parabolic equations on a bounded
domain . We give details on the similarities of these dynamics in the
cases , and and in the corresponding cases ,
and dim() respectively. In addition to
the beauty of such a correspondence, this could serve as a guideline for future
research on the dynamics of parabolic equations
Global Saddles for Planar Maps
We study the dynamics of planar diffeomorphisms having a unique fixed point
that is a hyperbolic local saddle. We obtain sufficient conditions under which
the fixed point is a global saddle. We also address the special case of
-symmetric maps, for which we obtain a similar result for
homeomorphisms. Some applications to differential equations are also given
Singular solutions to a semilinear biharmonic equation with a general critical nonlinearity
We consider positive solutions of the semilinear biharmonic equation
in with non-removable singularities at the origin. Under natural
assumptions on the nonlinearity , we show that is a
periodic function of and we classify all such solutions.Comment: To V. Maz'ya on the occasion of his 80th birthday; references adde
- …