Topological results on Fredholm maps and application to a superlinear differential equation

Abstract

Consider the equation F(u) = h, where F: X → Y is a smooth map between Banach spaces. By the Inverse Image Theorem regular solutions are isolated, so that only singular solutions may accumulate. Call S={u ∈ X: F′(u) is not surjective} the singular set of F; then the problem of analyzing the structure of the solution set consists in trying to give a description of F ⁻¹ (h) ∩ S. The point of view we adopt in this paper is to consider real-analytic Fredholm maps of index 0. In this case F ⁻¹ (h) ∩ S is a real-analytic set A such that dim _u A = dimKer F ¹ (^u). This implies in particular that pure one-dimensional components of A are real-analytic curves, whose tangent line at every point ^u is exactly Ker F′(u). If moreover F is proper then these components are circles. One of the first questions one can raise is whether there are general conditions under which even singular solutions are isolated. Or, in other words, are there detectable obstructions to the accumulation of solutions ? It turns out that there actually is a variety of situations in which the solution set cannot contain circles. In this paper we give an exemple of such a situation: the two-point boundary value problem

$$ \left\{ {\begin{array}{*{20}{c}}{u'' + {u^3} = h in \left( {0,1} \right)} \\ {u(0) = u(1) = 0.} \end{array}} \right\} $$