Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations

Abstract

In this paper, we consider the second order discontinuous differential equation in the real line, a t , u ϕ u ′ ′ = f t , u , u ′ , a . e . t ∈ R , u ( − ∞ ) = ν − , u ( + ∞ ) = ν + , with ϕ an increasing homeomorphism such that ϕ ( 0 ) = 0 and ϕ ( R ) = R , a ∈ C ( R 2 , R ) with a ( t , x ) > 0 for ( t , x ) ∈ R 2 , f : R 3 → R a L 1 -Carathéodory function and ν − , ν + ∈ R such that ν − < ν + . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities ϕ and f . To the best of our knowledge, this result is even new when ϕ ( y ) = y , that is for equation a t , u ( t ) u ′ ( t ) ′ = f t , u ( t ) , u ′ ( t ) , a . e . t ∈ R . Moreover, these results can be applied to classical and singular ϕ -Laplacian equations and to the mean curvature operator.