For partial differential equations of mixed elliptic-hyperbolic
and degenerate types which are the Euler-Lagrange equations for an
associated Lagrangian, invariance with respect to changes in
independent and dependent variables is investigated, as are
results in the classification of continuous one-parameter symmetry
groups. For the variational and divergence symmetries,
conservation laws are derived via the method of multipliers. The
conservation laws resulting from anisotropic dilations are applied
to prove uniqueness theorems for linear and nonlinear problems,
and the invariance under dilations of the linear part is used to
derive critical exponent phenomena and to obtain localized energy
estimates for supercritical problems.