Multiple homoclinic solutions for fourth-order p-Laplacian differential equations

Abstract

This article concerns the existence of infinitely many homoclinic solutions for the following fourth-order $p-$Laplacian differential equation
$$\Big(\left|u''(t)\right|^{p-2}u''(t)\Big)''-\omega\Big(\left|u'(t)\right|^{p-2}u'(t)\Big)'+V(t)\left|u(t)\right|^{p-2}u(t)=f(t,u(t))\leqno(1)$$
where $p\geq 2$, $\omega$ is a constant, $V\in C(\mathbb{R},\mathbb{R})$ is a positive function bounded from below and $f\in C(\mathbb{R}^{2},\mathbb{R})$. Applying Fountain Theorem and Dual Fountain Theorem, we prove that equation (1) possesses two different sequences of homoclinic solutions when $V$ satisfies a new coercive condition and the potential $f(t,x)$ is a combination of a superquadratic and a subquadratic functions.