Non-degenerate solutions of boundary-value problems

Abstract

(Received 7 September 1978) together with homogeneous Dirichlet or Neumann boundary conditions. Suppose that f is cubic-like, i.e., f( - A) = f(0) = f(B) = 0, A, B > 0, and f'(0) > 0. Clearly u = 0 is a solution, and it is also well-known that as s -+ 0, non-constant solutions bifurcate out of the zero solution& Our aim is to determine whether these non-constant solutions undergo bifurcations as s + 0. Solutions which do not bifurcate are termed A solution us(x) of (l), together with the given boundary conditions, is called if and only if the equation su + f'(u@))u = 0 together with the same boundary conditions implies u E 0. A straightforward application of the implicit function theorem shows that a strongly non- degenerate solution is non-degenerate. Our object in this paper is to derive an easily checked non-degeneracy condition and to show that when our condition holds non-degenerate solutions are also strongly non-degenerate. We consider general systems of equations in the plane, subject to linear homogeneous bound- ary conditions. Our technique is to rephrase the problem in a qualitative geometric way where our non-degeneracy condition is easily derivable. We shall also show that if our condition is ever violated, then we can find a boundary-value problem which admits a degenerate solution. In the second part of the paper, we shall attack the problem by topological methods; we feel that these techniques should prove useful for other types of problems. Thus for equations of the form (1) we shall consider first the homogeneous Neumann problem and we shall explicitly construct local conjugations of the associated one-parameter family of vector fields Xs =