Boundary value problems for singular second order equations

• Main Authors: Alessandro Calamai, Cristina Marcelli, Francesca Papalini
• Format: Article
• Language: English
• Published: SpringerOpen 2018-09-01
• Series: Fixed Point Theory and Applications
• Subjects: Boundary value problems; Nonlinear differential operators; Φ-Laplacian operator; Singular equation; Nagumo condition
• Online Access: http://link.springer.com/article/10.1186/s13663-018-0645-0

Abstract We investigate strongly nonlinear differential equations of the type (Φ(k(t)u′(t)))′=f(t,u(t),u′(t)),a.e. on [0,T], $$\bigl(\Phi \bigl(k(t) u'(t) \bigr) \bigr)'= f \bigl(t,u(t),u'(t) \bigr), \quad\text{a.e. on } [0,T], $$ where Φ is a strictly increasing homeomorphism and the nonnegative function k may vanish on a set of measure zero. By using the upper and lower solutions method, we prove existence results for the Dirichlet problem associated with the above equation, as well as for different boundary conditions involving the function k. Our existence results require a weak form of a Wintner–Nagumo growth condition.