Homoclinic Solution to Zero of a Non-autonomous, Nonlinear, Second Order Differential Equation with Quadratic Growth on the Derivative

TL;DR

This paper establishes the existence of positive, smooth, even, homoclinic solutions to a class of non-autonomous, nonlinear second-order differential equations with quadratic growth, using Galerkin's method and Strauss' approximation.

Contribution

It introduces a novel combination of Galerkin's method and Strauss' approximation to prove the existence of homoclinic solutions with specific properties.

Findings

Existence of positive, smooth, even, homoclinic solutions.

Solutions depend continuously on parameters.

Regularity results for the solutions.

Abstract

This work aims to obtain a positive, smooth, even, and homoclinic to zero (i.e zero at infinity) solution to a non-autonomous, second-order, nonlinear differential equation involving quadratic growth on the derivative. We apply Galerkin's method combined with Strauss' approximation on the term involving the first derivative to obtain weak solutions. We also study the regularity of the solutions and the dependence on their existence with a parameter