Some multiplicity results of homoclinic solutions for second order Hamiltonian systems

Abstract

We look for homoclinic solutions $q:\mathbb{R} \rightarrow \mathbb{R}^N$ to the class of second order Hamiltonian systems $-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}$ where $L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}$ and $a,b: \mathbb{R}\rightarrow \mathbb{R}$ are positive bounded functions, $G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}$ are positive homogeneous functions and $f:\mathbb{R}\rightarrow\mathbb{R}^N$. Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if $f\equiv 0$ and the existence of at least three solutions if $f$ is not trivial but small enough.