We extend bifurcation results of nonlinear eigenvalue problems from real Banach spaces to any neighbourhood of a given point. For points of odd multiplicity on these restricted domains, we establish that the component of solutions through the bifurcation point either is unbounded, admits an accumulation point on the boundary, or contains an even number of odd-multiplicity points. In the simple-multiplicity case, we show that branches of solutions in the directions of corresponding eigenvectors satisfy similar conditions on such restricted domains.

Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax + ε B(x) =δ x, where A: H → H is a bounded self-adjoint (linear) operator with nontrivial kernel Ker A, and B: H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x0 ∈ S∩ Ker A of A (thus corresponding to the eigenvalue δ=0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S∩ Ker A), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ ∈ ℝ and ε ≠ 0. In this paper, we prove that if B is a C1 gradient mapping and the eigenvalue δ=0 has finite multiplicity, then the sphere S∩ Ker A contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.

the degree defined by p. benevieri and m. furi (see ‘a simple notion of orientability for fredholm maps of index zero between banach manifolds and degree theory’, ann. sci. math. québec 22 (1998) 131–148) is used here to obtain some sharp lower bounds for the number of solutions of the $\lambda$-sections of the compact components of the set of non-trivial solutions of $\mf{f}(\lambda,x)=0$, where $\mf{f}$ is a $\cal{c}^1$ fredholm map of index 1 such that $\mf{f}(\lambda,0)=0$ for all $\lambda\in\mathbb{r}$. these bounds are given in terms of the parity of the linearized fredholm family $d_2\mf{f}(\cdot,0)$. the parity is a local invariant measuring the change of the orientation of $d_2\mf{f}(\lambda,0)$ as $\lambda$ crosses an interval. the set of eigenvalues of $d_2\mf{f}(\cdot,0)$ is not assumed to be discrete. therefore, even in the classical situation when $\mf{f}$ is a compact perturbation of the identity, the results presented here are completely new.

We develop results for bifurcation from the principal eigenvalue for certain operators based on the $p$-Laplacian and containing a superlinear nonlinearity with a critical Sobolev exponent. The main result concerns an asymptotic estimate of the rate at which the solution branch departs from the eigenspace. The method can also be applied for nonpotential operators.