Finiteness and Positivity Results for Global Minimizers of a Semilinear Elliptic Problem

We consider the Landau–Ginzburg free energy functional whereW(t)=(t2−1)2is a double-well potential and prove that it has only finitely many global minimizers. We investigate the stability properties of these minimizers and show that the second variationF″ is positive definite in one dimension for nondegenerateu.