Remarks on time map for quasilinear equations
This paper deals with one-dimensional quasilinear equations \[ \varphi (u')' + f(u) = 0, \;u(0) = 0, \;u'(0) = r > 0 \] including the case of the \(p\)-Laplacian, where \(\varphi (s) = | s | ^{p-2}s, \;p > 1.\) As the authors explain using a ballistic analogy, the previous initial value problem can be interpreted as shooting from the ground level at an angle \(r > 0.\) Then, the time map \(T = T(r)\) gives the time when the projectile falls back to the ground. In the paper, two different generalizations of Schaaf's time formula [\textit{R. Schaaf}, Global solution branches of two point boundary value problems. Lecture Notes in Mathematics. 1458. Berlin etc.: Springer-Verlag (1990; Zbl 0780.34010)] are presented. The first is obtained by globally \(p\)-linearizing the equation. The second involves parametric integration of the equation. These two formulas can be used to prove uniqueness or multiplicity results for the corresponding Dirichlet boundary value problem and for the computation of bifurcation diagrams.
scientific article; zbMATH DE number 5848912
scientific article