Global Leray-Schauder continuation for Fredholm operators

TL;DR

This paper explores the global solution structure of Fredholm maps using Leray-Schauder continuation, revealing conditions under which solution branches are unbounded, reach domain boundaries, or accumulate at other solutions, with applications to boundary value problems.

Contribution

It extends the Leray-Schauder continuation theorem to analyze the global behavior of solution branches for Fredholm operators, including real-analytic maps and applications to quasilinear boundary value problems.

Findings

Solution branches are either unbounded, reach domain boundary, or accumulate at other solutions.

Bounded components in the interior satisfy a degree balance with an even number of contact points.

Constructs local parameterizations for real-analytic maps showing blow-up or boundary approach.

Abstract

This paper ascertains the global behavior of the forward and backward branches of solutions provided by the Leray-Schauder continuation theorem for orientable $\mathcal{C}^1$ Fredholm maps, as developed by the authors in [54]. Under properness on bounded sets and a nonzero local index at the given base solution, each branch satisfies the following alternative: either it is unbounded, or it reaches the boundary of the domain, or it accumulates at a different solution on the base parameter level. When the component is bounded and stays in the interior, there is a degree balance on the base slice entailing a vanishing sum of local indices and, in particular, the existence of an even number of non-degenerate contact points. For real-analytic maps we construct locally injective parameterizations that exhibit blow-up, approach to the boundary, or return to the base level. An application to a quasilinear boundary value problem driven by the mean-curvature and Minkowski operators illustrates the global results.