The index of a critical point for nonlinear elliptic operators with strong coefficient growth. (English) Zbl 0953.47042

Kartsatos, Athanassios G.; Skrypnik, Igor V.

J. Math. Soc. Japan 52, No. 1, 109-137 (2000).

Nonlinear operators with strong coefficient growth associated with boundary value problems of the type \[ \sum_{|\alpha|} D^{\alpha} \{ \rho^{2}(u) D^{\alpha} u +a_{\alpha}(x, D^{1}u)\} =\lambda a_{0}(x,u,D^{1} u),\;x \in \Omega; u(x)=0,x\in \partial \Omega \] (\(\Omega\subset R^{n} \) is bounded domain with \(\partial\Omega\in C^{2}, \rho:R\rightarrow R_{+}\) can have exponential growth) are considered.
The authors give an index formula for such densely defined operators acting from Sobolev spase \(W^{1,m}_{0}(\Omega)\) into its dual. This formula is very important for various nonlinear problems containing such kind operators. An abstract variant (to operators of \(S_{+}\)-type) is suggested with applications to the relevant bifurcation problems.

Reviewer: Boris V.Loginov (Ul’yanovsk)

MSC:

47H11	Degree theory for nonlinear operators
47J05	Equations involving nonlinear operators (general)
47J10	Nonlinear spectral theory, nonlinear eigenvalue problems
35B32	Bifurcations in context of PDEs
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47J15	Abstract bifurcation theory involving nonlinear operators

Keywords:

\((S_{+})\)-operators; degree theory; index of an isolated critical point; nonlinear elliptic PDEs; nonlinear operators with strong coefficient growth; boundary value problems; index formula; Sobolev space; bifurcation problems