\(S\)-shaped bifurcation of a singularly perturbed boundary value problem (Q1265088)

Consider the singularly perturbed boundary value problem \[ \varepsilon^2 u'' = h(u) - \mu x,\quad u(0)=u'(1)=0,\tag{*} \] with \(\mu > 0\), \(0 < \varepsilon \ll 1\). \(h:[0,\infty) \rightarrow [0,\infty)\) is a \(C^4\)-function, positive for \(u > 0,\;h(0)=0,\) having a unique maximum \(\gamma_M\) at \(\tilde{u}\) and a unique minimum \(\gamma_m\) at \(\hat{u}\), \(0 < \tilde{u} < \hat{u}\). Let \(\gamma_0, \gamma_m < \gamma_0 < \gamma_M\), be the ``Maxwell'' value of the parameter \(\mu\) satisfying some ``equal area rule''. The author proves that to given \(\overline{\mu} > \gamma_M\) there is an \(\varepsilon_0 > 0\) such that for \(0 < \varepsilon < \varepsilon_0\) there exist \(\mu_0 (\varepsilon), \mu_M (\varepsilon)\) satisfying \(\mu_0 (\varepsilon) \rightarrow \gamma_0, \mu_M (\varepsilon) \rightarrow \gamma_M\) for \( \varepsilon \rightarrow 0\) such that (i) For \(0 < \mu < \mu_0 (\varepsilon)\) there is a unique nondecreasing solution \(u_1\) to \((*)\). This solution has no layers. (ii) For \(\mu_0 (\varepsilon) < \mu < \mu_M (\varepsilon)\) there are three nondecreasing solutions \(u_1 ,u_2 , u_3\) to \((*)\). \(u_1\) has no layers, \(u_2\) has an internal and \(u_3\) a boundary layer. (iii) For \(\bar{\mu} > \mu > \mu_M (\varepsilon)\) there is a unique nondecreasing solution \(u_2\) to \( (*). u_2\) has an internal layer. (iv) \(u_1\) merges with \(u_3\) at \(\mu_M (\varepsilon)\), and \(u_2\) merges with \(u_3\) at \(\mu_0 (\varepsilon)\) in a generic fold bifurcation. The author develops new techniques to study the bifurcations near \(\gamma_0\) and \(\gamma_M\), especially he proves a \(C^2\)-extension of the exchange lemma.