Bifurcation of obstacle problems with inclusions follow from degree results for variational inequalities. (English) Zbl 1313.35020

Väth, Martin

Differ. Integral Equ. 26, No. 11-12, 1235-1262 (2013).

The main result of the paper states that the Leray-Schauder degree corresponding to a variational inequality on a cone \(K\) in a Hilbert space (i.e., the degree of the projection onto \(K\) of the map under consideration) coincides with the local degree corresponding to a wide class of inclusions having the same homogenization as that variational inequality. As a consequence, bifurcation results for variational inequalities based on degree theory can be automatically generalized for such inclusions. Unilateral obstacles given by inclusions can be more natural from the point of view of applications than those described by variational inequalities. The unilateral obstacles considered can describe, e.g., sources which are active only if the concentration is less than a given threshold. The abstract result is applied to reaction-diffusion systems with Neumann boundary conditions exhibiting Turing’s diffusion-driven instability. Using recent surprising results concerning variational inequalities of reaction-diffusion type, it is shown that there are global bifurcations of stationary spatially non-homogeneous solutions to such systems with unilateral sources in the region of parameters where the system without unilateral conditions can have no bifurcation, and where bifurcations for unilateral problems were not expected. Also, a single elliptic non-linear equation with unilateral conditions given by inclusions is studied. The existence of bifurcation points for such problems lying strictly between certain eigenvalues of the linearized equation is given, which is again a consequence of the former results for variational inequalities. Furthermore, bifurcation at some eigenvalues without any multiplicity assumption is proved.

Reviewer: Milan Kučera (Praha)

MSC:

35B32	Bifurcations in context of PDEs
35K57	Reaction-diffusion equations
35J60	Nonlinear elliptic equations
35J88	Unilateral problems for elliptic systems and systems of variational inequalities with elliptic operators
47J20	Variational and other types of inequalities involving nonlinear operators (general)

Keywords:

inclusion; variational inequality; obstacle problem; local degree; global bifurcation; reaction-diffusion system