We study the existence of solution for a boundary value problem at resonance where the nonlinearity depends only on the derivative. In a sense, we can say that the problem considered is strongly resonant. Our proofs make use of the Lyapunov–Schmidt reduction; in so doing, we are led with the asymptotic estimate of the corresponding bifurcation equation.  2004 Elsevier Inc. All rights reserved.

We show that there exists multi-vortex, non-radial, finite energy solutions to the magnetic Ginzburg-Landau equations on all of R2. We use Lyapunov-Schmidt reduction to construct solutions which are invariant under rotations by 2π k (but not by rotations in O(2) in general) and reflections in the x− axis for some k ≥ 7. 1 Ginzburg-Landau equations 1.

We prove existence of vibrations of a nonhomogeneous string under a nonlinear time periodic forcing term in the case in which the forcing frequency avoids resonances with the vibration modes of the string (nonresonant case). The proof relies on a Lyapunov–Schmidt reduction and a Nash–Moser iteration scheme.

In this paper we employ the Lyapunov–Schmidt procedure to set up equivariant Hopf bifurcation theory of functional differential equations of mixed type. In the process we derive criteria for the existence and direction of branches of bifurcating periodic solutions in terms of the original system, avoiding the process of center manifold reduction. © 2010 Elsevier Ltd. All rights reserved.

The paper extends the applicability of our freely accessible Matlab package Chebpack to calculate the eigenvalues and eigenfunctions of some higher order differential problems as well as to semidiscretize evolution problems, directly or by using the Lyapunov-Schmidt reduction method. The numerical examples illustrate the accuracy and the simplicity of the algorithms and prove the importance of ...

We examine semilinear Neumann problems driven by the Laplacian plus an unbounded and indefinite potential. The reaction is a Carathéodory function which exhibits linear growth near ±∞. We allow for resonance to occur with respect to a nonprincipal nonnegative eigenvalue, and we prove several multiplicity results. Our approach uses critical point theory, Morse theory and the reduction method (th...

[reaction: see text] A highly enantioselective Meerwein-Schmidt-Ponndorf-Verley (MSPV) reduction of N-phosphinoyl ketimines by (BINOL)Al(III)/2-propanol is reported. Yields ranging between 79 and 85% with high enantiomeric excesses (93-98%) are observed for a wide range of structurally diverse ketimines. A [2.0.4] bicyclic chelation model is proposed to account for this high selectivity.

From minimal surfaces such as Simons’ cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin’s theorem [43] is optimal.

Using the Lyapunov-Schmidt reduction method, the authors discuss the existence and multiplicity of periodic solutions for a piecewise linear beam equation with damping and nonconstant load when the nonlinearities cross the eigenvalues. The result answers partially an open question possed by Lazer and McKenna [12].

‎‎for any group $g$‎, ‎we define an equivalence relation $thicksim$ as below‎: ‎[forall g‎, ‎h in g gthicksim h longleftrightarrow |g|=|h|]‎ ‎the set of sizes of equivalence classes with respect to this relation is called the same-order type of $g$ and denote by $alpha{(g)}$‎. ‎in this paper‎, ‎we give a partial answer to a conjecture raised by shen‎. ‎in fact‎, ‎we show that if $g$ is a nilpot...