Singular Dirichlet problem for ordinary differential equations with \(\phi \)-Laplacian. (English) Zbl 1114.34017

Polášek, Vladimír; Rachůnková, Irena

Math. Bohem. 130, No. 4, 409-425 (2005).

Summary: We provide sufficient conditions for the solvability of a singular Dirichlet boundary value problem with \(\phi \)-Laplacian \[ (\phi (u'))' = f(t, u, u'),\quad u(0) = A, \;u(T) = B, \]
where \(\phi \) is an increasing homeomorphism, \(\phi (\mathbb R )=\mathbb R \), \(\phi (0)=0\), \(f\) satisfies the Carathéodory conditions on each set \([a, b]\times \mathbb R ^{2}\) with \([a, b]\subset (0, T)\) and \(f\) is not integrable on \([0, T]\) for some fixed values of its phase variables. We prove the existence of a solution which has continuous first derivative on \([0, T]\).

MSC:

34B16	Singular nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

existence of smooth solution; lower function; upper function