Year-end Sale % OFF 
Abstract
Abstract We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations ( a ( x ) Φ ( x ′ ( t ) ) ) ′ = f ( t , x ( t ) , x ′ ( t ) ) , a . e . t ∈ ℝ governed by a nonlinear differential operator Φ extending the classical p- Laplacian, with right-hand side f having the critical rate of decay -1 as |t| → +∞, that is f ( t , ⋅ , ⋅ ) ≈ 1 t . We prove general existence and non-existence results, as well as some simple criteria useful for right-hand side having the product structure f(t, x, x') = b(t, x)c(x, x'). Mathematical subject classification: Primary: 34B40; 34C37; Secondary: 34B15; 34L30.
Highlights
We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations (a(x) (x (t))) = f (t, x(t), x (t)), a.e. t ∈ R
Differential equations governed by nonlinear differential operators have been extensively studied in the last decade, due to their several applications in various sciences
In [11] we studied boundary value problems on the whole real line
Summary
Differential equations governed by nonlinear differential operators have been extensively studied in the last decade, due to their several applications in various sciences. In many applications the dynamic is described by a differential operator depending on the state variable, like (a(x)x’)’ for some sufficiently regular function a(x), which can be everywhere positive [non-negative] (as in the diffusion [degenerate] processes), or a changing sign function, as in the diffusion-aggregation models (see [7], [11,12,13]) It naturally arises the interest for mixed nonlinear differential operators of the type (a(x)F(x’))’. In the present article we focus our attention on right-hand sides having the critical rate of decay δ = -1 and show that, contrary to the situation studied in [11], the solvability of the boundary value problem is influenced by the behavior of the righthand side and of the differential operator with respect to the state variable x. Assume that there exists a pair of lower and upper solutions a, b Î C1 (R) of the equation (2.1), satisfying a(t) ≤ b(t), for every t Î R, with a increasing in (-∞, -L), b increasing in (L, +∞), for some L >0
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
