Homoclinic solution to zero of a non-autonomous, nonlinear, second order differential equation with quadratic growth on the derivative

Abstract

The aim of this work is to obtain a positive, smooth, even and homoclinic solution to the problem −(A(u)u 0 ) 0 (t) + u(t) = λa1(t)|u(t)| q−1 + |u(t)| p−1 + g(|u 0 (t)|), in R. Considering 1 < q < 2 < p < +∞ and a1 ∈ L s (R) ∩ C(R), s = 2 2−q , a positive even function. Also A : R → R a Lipschitz, smooth (at least C 1 (R)), nondecreasing function satisfying ∃γ ∈ (0, 1) such that 0 < γ ≤ A(t) ∀t ∈ R, and g : R −→ R a continuous function satisfying 0 ≤ sg(s) ≤ |s| θ for all s ∈ R, where 2 < θ ≤ 3. By homoclinic we mean “homoclinic to the origin” or “homoclinic to zero” , i.e, the solution must verify limx→±∞ u(x) = 0.