Abstract: Consider the following general nonlinear system
Au = N(u) (1)
where H is a Hilbert space, A is a self-adjoint operator, and N is a (nonlinear) gradient operator. Typical example are Dirac equations and reaction-diffusion systems where \sigma(A) (the spectrum) is unbounded from below and above, and particularly, \sigma_e(A)\cap\mathbb R^{\pm}\not=\empty. The talk focus on
1) to establish general variational setting for (1) by using the operator interpolation theory;
2) certain critical point theory;
3) the existence, concentration and exponential decay for semi-classical solutions of Dirac equation and the reaction-diffusion systems, etc.;
4) bifurcation of Dirac equation on spin manifolds.
