On the solvability of a boundary value problem on the real line

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.

Keywords

• boundary value problems
• unbounded domains
• heteroclinic solutions
• nonlinear differential operators
• <it>p</it>-Laplacian operator
• &#934;-Laplacian operator