  This paper addresses some fundamental issues in nonconvex analysis. By using
pure complementary energy principle proposed by the author, a class of fully
nonlinear partial diforerential equations in nonlinear elasticity is able to
converted a unified algebraic equation, a complete set of analytical solutions
are obtained for 3-D finite deformation problems governed by generalized
neo-Hookean model. Both global and local extremal solutions to the nonconvex
variational problem are identifored by a triality theory. Connection between
challenges in nonlinear analysis and NP-hard problems in computational science
is revealed. Results show that Legendre-Hadamard condition can only guarantee
ellipticity for convex problems. For nonconvex systems, the ellipticity depends
not only on the stored energy, but also on the external force field. Uniqueness
is proved based on a quasiconvexity and a generalized ellipticity condition.
Application is illustrated for logarithm stored energy.
