Critical points of nonconvex and noncoercive functionals (Q1871852)

In the paper one-dimensional variational integrals of the form \[ E(u)= \int^1_0 [W(u')+ G(u)] dx \] are considered, where \(W\) and \(G\) are smooth functions which do not satisfy the usual conditions of coercivity and convexity necessary to apply the direct methods of the calculus of variations. In fact, \(W\) is assumed to be a double well potential and \(G\) is not bounded from below. The elliptic \(\varepsilon\)-regularization \[ E_\varepsilon(u)= \int^1_0 \Biggl[{\varepsilon\over 2} |u''|^2+ W(u')+ G(u)\Biggr] dx \] is considered, as well as the larger family of functionals \[ E_\varepsilon(\lambda, u)= \int^1_0 \Biggl[{\varepsilon\over 2}|u''|^2+ W(\lambda+ u')+ G(u)\Biggr] dx, \] where the parameter \(\lambda\) varies in \(\mathbb{R}\). The existence of minimizers and of critical points of the functional \(E\) is studied through the analysis of the critical points \(u_\varepsilon\) of \(E_\varepsilon\). This issue is in turn studied through a global continuation property of the critical points of \(E_\varepsilon(\lambda, u)\) bifurcating from the trivial solution line \(\{\lambda, 0\}\) at some \(\lambda\neq 0\).