Note on homoclinic solutions to nonautonomous Hamiltonian systems with sign-changing nonlinear part

TL;DR

The paper addresses the existence of nontrivial homoclinic solutions to a nonautonomous Hamiltonian system with sign-changing superquadratic nonlinearity, formulated as $\dot{z}= J D_z H(z,t)$ with $H(z,t)=\frac{1}{2} A z\cdot z + \Gamma(t)(F(z)-\lambda G(z))$. It develops a variational framework on the fractional Sobolev space $X=H^{1/2}(\mathbb{R};\mathbb{R}^{2N})$, employing the KS $\tau$-topology and an abstract theorem from BB to obtain a Cerami sequence despite indefiniteness and sign changes. The authors verify the geometry (A1)–(A3) for small $\lambda$, establish boundedness of Cerami sequences, and use concentration-compactness in $H^{1/2}$ to extract a nontrivial limit that is a critical point of the energy functional, hence a homoclinic solution. This work extends linking-type variational methods to Hamiltonian systems with sign-changing nonlinearities and provides a rigorous existence result via modern topological-variational techniques.

Abstract

In the paper, we utilize the recent variational, abstract theorem to show the existence of homoclinic solutions to the Hamiltonian system $$ \dot{z} = J D_z H(z, t), \quad t \in \mathbb{R}, $$ where the Hamiltonian $H : \mathbb{R}^{2N} \times \mathbb{R} \rightarrow \mathbb{R}$ is of the form $$ H(z, t) = \frac12 Az \cdot z + Γ(t) \left( F(z) - λG(z) \right) $$ for some symmetric matrix $A$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 3.1: BB
• Proposition 4.1
• Remark 4.2
• Lemma 4.3
• Lemma 4.4
• proof
• Remark 4.5
• Remark 5.1
• Lemma 5.2