  This paper mainly addresses the extrema of a nonconvex functional with
double-well potential in higher dimensions through the approach of nonlinear
partial differential equations. Based on the canonical duality method, the
corresponding Euler--Lagrange equation with Neumann boundary condition can be
converted into a cubic dual algebraic equation, which will help find the local
extrema for the primal problem. In comparison with the 1D case discussed by D.
Gao and R. Ogden, there exists huge difference in higher dimensions, which will
be explained in the theorem.
