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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].
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