Eigenvalues and bifurcation for problems with positively homogeneous operators and reaction-diffusion systems with unilateral terms

Reaction-diffusion systems satisfying assumptions guaranteeing Turing's instability and supplemented by unilateral terms of type v− and v+ are studied. Existence of critical points and sometimes also bifurcation of stationary spatially non-homogeneous solutions are proved for rates of diffusions for which it is excluded without any unilateral term. The main tool is a general result giving a variational characterization of the largest eigenvalue for positively homogeneous operators in a Hilbert space satisfying a condition related to potentiality, and existence of bifurcation for equations with such operators. The originally non-variational (non-symmetric) system is reduced to a single equation with a positively homogeneous potential operator and the abstract results mentioned are used.