For a class of semilinear elliptic systems, the existence of a broader class of multibump solutions is established under considerably weaker conditions than in earlier works. The key tool is the use ...of variational methods.
This paper studies a Hamiltonian system possessing a double well potential for which the existence of multitransition heteroclinic and homoclinic solutions that are local minimizers of an associated ...functional is known. Under an additional mild non-degeneracy condition on the set of all homoclinic and heteroclinic solutions, the existence of further heteroclinic and homoclinic solutions that are of mountain pass type is established. A key tool for the existence arguments is a variant of the Mountain Pass Theorem that is of independent interest.
This monograph presents extensions of the Moser-Bangert approach that include solutions of a family of nonlinear elliptic PDEs on Rn and an Allen-Cahn PDE model of phase transitions. The text may be ...used as a text for a graduate course in PDEs.
On a phase transition model Byeon, Jaeyoung; Rabinowitz, Paul H.
Calculus of variations and partial differential equations,
05/2013, Letnik:
47, Številka:
1-2
Journal Article
Recenzirano
An Allen–Cahn phase transition model with a periodic nonautonomous term is presented for which an infinite number of transition states is shown to exist. A constrained minimization argument and the ...analysis of a limit problem are employed to get states having a finite number of transitions. A priori bounds and an approximation procedure give the general case. Decay properties are also studied and a sharp transition result with an arbitrary interface is proved.
For about 25 years, global methods from the calculus of variations have been used to establish the existence of chaotic behavior for some classes of dynamical systems. Like the analytical approaches ...that were used earlier, these methods require nondegeneracy conditions, but of a weaker nature than their predecessors. Our goal here is study such a nondegeneracy condition that has proved useful in several contexts including some involving partial differential equations, and to show this condition has an equivalent formulation involving stable and unstable manifolds.
This paper surveys some recent work on a variant of the Mountain Pass Theorem that is applicable to some classes of differential equations involving unbounded spatial or temporal domains. In ...particular its application to a system of semilinear elliptic PDEs on
R
n
and to a family of Hamiltonian systems involving double well potentials will also be discussed.
The existence of solutions undergoing multiple spatial transitions between isolated periodic solutions is studied for a class of systems of semilinear elliptic partial differential equations. A key ...tool is a new result on the possible behavior of the set of single transition solutions.
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