Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications
Consider positive solutions of the eigenvalue problem \[ -\Biggl({u'\over \sqrt{1-u^{\prime 2}}}\Biggr)'=\lambda f(u),\quad u(-L)= u(L)= 0,\tag{1} \] where \(\lambda\), \(L\) are positive, \(f\) is sufficiently smooth and satisfies \(f(u)>0\) for \(u>0\). For any fixed \(L\), the bifurcation curve \(S_L\) of (1) is defined by \[ S_L:= \{(\lambda,\|u_\lambda\|_\infty):\lambda>0,\;u_\lambda>0\text{ and smooth}\}. \] The author introduces a classification of the bifurcation curves (e.g. monotone increasing or decreasing, \(S\)-shaped) and describes the dependence of the shape of the bifurcation curve \(S_L\) on the parameter \(L\) and on the function \(f\). He compares the eigenvalue problem (1) with the eigenvalue problems \[ -u''=\lambda f(u),\quad u(-L)= u(L)= 0, \] \[ -\Biggl({u'\over \sqrt{1+ u^{\prime 2}}}\Biggr)'=\lambda f(u),\quad u(-L)= u(L)= 0. \] The proofs are based on the time-map \(T_\lambda(\alpha)\) such that \[ S_L:= \{(\lambda,\alpha): T_\lambda(\alpha)= L\text{ for some }\alpha,\lambda>0\}. \]
scientific article; zbMATH DE number 6844168
scientific article