Existence of positive quasi-homoclinic solutions for damped \(p\)-Laplacian differential equations (Q2113819)

Summary: In this paper we prove the existence of nontrivial quasi-homoclinic solutions for the damped \(p\)-Laplacian differential equation \[ (|u'|^{p -2}u')' + c|u'|^{p -2}u'-a(t)|u|^{p -2}u+f(t, u) = 0,\ t\in\mathbb{R}, \] where \(p\geq 2\), \(c\geq 0\) is a constant and the functions \(a\) and \(f\) are continuous and not necessarily periodic in \(t\). Using the Mountain-Pass Theorem, we obtain the existence of positive homoclinic solution in both cases sub-quadratic and super-quadratic.