Bifurcation from the essential spectrum.

*(English)*Zbl 0888.47045
Matzeu, Michele (ed.) et al., Topological nonlinear analysis II: degree, singularity and variations. Papers presented in part at the 2nd topological analysis workshop on degree, singularity and variations: developments of the last 25 years, Frascati, Italy, June 1995. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 27, 397-443 (1997).

A survey of some of the results obtained during the past twenty years concerning bifurcation from a point of the essential spectrum of the linearization of a nonlinear equation is given. First, bifurcation problems
\[
Au - N(u) = \lambda Lu \tag{1}
\]
are considered where \(A\), \(L\) are bounded linear selfadjoint operators in a Hilbert space, \(L\) is positive, \(N= \nabla \varphi\), \(\varphi\) is a differentiable functional, \(\lim_{|u|\to 0}\frac{\varphi(u)}{|u|^2} = 0\). General basic properties and relations of the discrete and essential spectrum to bifurcation points are recalled. Eigenvalue problems of the type
\[
Su + R(u) = \lambda u
\]
with an unbounded selfadjoint operator \(S\) in a real Hilbert space and a nonlinear perturbation \(R\) are simultaneously studied. It is shown how such problems can be cast in the form (1) under certain assumptions. Two types of results concerning bifurcation from the infimum \(\Lambda\) of the spectrum are explained. First, under certain assumptions, for any \(\lambda < \Lambda\), a nontrivial \(u_\lambda\) satisfying (1) is found as a critical point of the functional \(F(\lambda,u)=J(u)+\frac12 \lambda (Lu,u) \) where \(J(u)=\frac12 (Au,u)-\varphi(u)\) and it is proved that \(|u_\lambda|\to 0\) for \(\lambda \to \Lambda\). Second, for any \(r>0\), a couple \(\lambda_r, u_r\) is found such that \(u_r\) is a stationary point of \(J\) subject to the constraint \((Lu,u) =r\), \(\lambda_r\) is a Lagrange multiplier. Further, general situations are described when there is a maximal interval \((a,b)\) lying in the resolvent set of \(A,L\) and \(b\) is a bifurcation point. Moreover, there is a bifurcation to the left at \(b\) (bifurcation into spectral gaps). In some cases, also the order of the bifurcation is described. The last section is devoted to the problem
\[
-\Delta u(x) + V(x)u(x) - r(x)|u(x)|^{p-2}u(x) = \lambda u(x) \text{ for } x \in \mathbf R^N
\]
where \(V \in \mathbf L^\infty (R^N)\), \(r\in \mathbf L^\infty (R^N)\), \(r\geq 0\) and \(p<2\) or \(2<p<\frac{2N}{N-2}\) in the case \(N=1,2\) or \(N\geq 3\), respectively. Known results about this problem are summarized. Particularly, general theory from previous sections is applied, i.e. bifurcations obtained by fixed \(\lambda\) approach, fixed norm approach and bifurcations into spectral gaps are described in various cases.

For the entire collection see [Zbl 0866.00047].

For the entire collection see [Zbl 0866.00047].

Reviewer: M.Kučera (Praha)

##### MSC:

47J10 | Nonlinear spectral theory, nonlinear eigenvalue problems |

58C30 | Fixed-point theorems on manifolds |

47A10 | Spectrum, resolvent |

58E07 | Variational problems in abstract bifurcation theory in infinite-dimensional spaces |

35B32 | Bifurcations in context of PDEs |