: The eigenvalue of minimum modulus of the Frechet derivative of a nonlinear operator is estimated along a bifurcating curve of zeroes of the operator. This result is applied to the study of a number of differential equations. Parallel results are developed for a class of nonlinear eigenvalue problems of positive type. (Author)

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Bifurcation, perturbation of simple eigenvalues, itand linearized stability

Studies a nonlinear heat equation in a finite interval of space subject to a white noise forcing term. The equation without the forcing term exhibits several equilibrium configurations, two of which are stable. The solution of the complete forced equation is a stochastic process in space and time that has a unique stochastic equilibrium. The authors study this process in the limit of small noise, and obtain lower and upper bounds for the probability of large fluctuations. They then apply these estimates to calculate the transition probability between the stable configurations (tunnelling). This model problem can be interpreted as a rigorous version of some recent attempts to describe Euclidean quantum systems in terms of stochastic equilibrium states of a nonlinear stochastic differential equation in infinite dimensions. However, its significance goes beyond this situation and the authors' methods may be applicable to models in other areas of natural science.

https://iopscience.iop.org/article/10.1088/0305-4470/15/10/011/pdf

Large fluctuations for a nonlinear heat equation with noise

The effect of domain shape on the number of positive solutions of certain nonlinear equations, II

On the indices of fixed points of mappings in cones and applications

Le nombre de solutions de certains problèmes semi-linéaires elliptiques

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The Number of Solutions of a Nonlinear Two Point Boundary Value Problem

/pdf/positive-solutions-of-convex-nonlinear-eigenvalue-problems-22pcrz7ojg.pdf

Positive solutions of convex nonlinear eigenvalue problems.

/pdf/critical-solutions-of-autonomous-nonlinear-boundary-value-1g4uax1hol.pdf

Theodore Laetsch

Papers

The Number of Solutions of a Nonlinear Two Point Boundary Value Problem

Positive solutions of convex nonlinear eigenvalue problems.

Critical solutions of autonomous nonlinear boundary value problems.