Homoclinics for singular strong force Lagrangian systems

We study the existence of homoclinic solutions for a class of Lagrangian systems ddt$\begin{array}{} \frac{d}{dt} \end{array} $(∇Φ(u̇(t))) + ∇uV(t, u(t)) = 0, where t ∈ ℝ, Φ: ℝ2 → [0, ∞) is a G-function in the sense of Trudinger, V: ℝ × (ℝ2 ∖ {ξ}) → ℝ is a C1-smooth potential with a single well of infinite depth at a point ξ ∈ ℝ2 ∖ {0} and a unique strict global maximum 0 at the origin. Under a strong force condition around the singular point ξ, via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions u±: ℝ → ℝ2 ∖ {ξ}.

Abstract licence: CC BY

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