Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian

Abstract

Using Leray–Schauder degree theory we obtain various existence and multiplicity results for nonlinear boundary value problems ( ϕ ( u ′ ) ) ′ = f ( t , u , u ′ ) , l ( u , u ′ ) = 0 where l ( u , u ′ ) = 0 denotes the Dirichlet, periodic or Neumann boundary conditions on [ 0 , T ] , ϕ : ] − a , a [ → R is an increasing homeomorphism, ϕ ( 0 ) = 0 . The Dirichlet problem is always solvable. For Neumann or periodic boundary conditions, we obtain in particular existence conditions for nonlinearities which satisfy some sign conditions, upper and lower solutions theorems, Ambrosetti–Prodi type results. We prove Lazer–Solimini type results for singular nonlinearities and periodic boundary conditions.