Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions
The authors study the existence of \(T\)-periodic solutions of the capillarity equation \[ -(u'/\sqrt{1+(u')^2})'=f(t,u), \] with \(f\) a Carathéodory function. To this end, they develop the method of lower and upper solutions both in the case in which such functions are well ordered (the lower solution is under the upper one) and in the case in which they can cross. These functions are assumed to be of bounded variation and their definition is in a variational setting. By assuming some monotonicity and regularity conditions on \(f\), it is proved that the solutions of the problem are in \(C^3\). In the last section, some stability properties are deduced.
quasilinear ordinary differential equation
scientific article; zbMATH DE number 6118822
scientific article