Genericity of hyperbolicity and saddle-node bifurcations in reaction-diffusion equations depending on a parameter

Semilinear elliptic equations of the form $$\begin{array}{*{20}c} {\sum\limits_{i,j = 1}^n {(a_{ij} (x)u_{xi} (x))_{x_j } + } f(\lambda ,x,u(x)) = 0,} & {x \in \Omega ,} \\ {u(x) = 0,} & x \\ \end{array} $$ are considered, where λ ε ? is a parameter, Ω ? ? ⁿ is a bounded domain andf is a smooth non-linear function. It is shown that for ‘generic’ functionsf, the set of non-trivial solutions (λ,u) consists of a finite, or countable, collection of smooth, 1-dimensional curves and any such solution is either hyperbolic or is a saddle-node bifurcation point of the curve.