Non-collision Orbits for a Class of Singular Hamiltonian Systems on the Plane with Weak Force Potentials

We study the existence of non-collision orbits for a class of singular Hamiltonian systems \(\begin{aligned} \ddot{q}+ V'(q)=0 \end{aligned}\) $$\begin{array}{c}\hfill \ddot{q}+V^{\prime }(q)=0\end{array}$$ where \(q:{\mathbb {R}} \longrightarrow {\mathbb {R}}^2\) $q:\mathbb{R}⟶{\mathbb{R}}^2$ and \(V\in C^2({\mathbb {R}}^2 {\setminus } \{e\},\, {\mathbb {R}})\) $V\in C^2({\mathbb{R}}^2\backslash \{e\},\mathbb{R})$ is a potential with a singularity at a point \(e\not =0\) $e\ne 0$ . We consider V which behaves like \(\displaystyle -1/|q-e|^\alpha \) ${\displaystyle {-1/|q-e|}^{\alpha }}$ as \( q\rightarrow e \) $q\to e$ with \(\alpha \in ]0,2[.\) $\alpha \in ]0,2[.$ Under the assumption that 0 is a strict global maximum for V, we establish the existence of a homoclinic orbit emanating from 0. Moreover, in case \(\displaystyle V(q) \longrightarrow 0\) ${\displaystyle V(q)⟶0}$ as \(|q|\rightarrow +\infty \) $|q|\to +\infty$ , we prove the existence of a heteroclinic orbit “at infinity" i.e. a solution q such that \(\begin{aligned} \lim _{t\rightarrow -\infty } q(t)=0,\,\, \lim _{t \rightarrow +\infty }|q(t)|=+\infty \,\, \hbox {and} \, \lim _{t \rightarrow \pm \infty }\dot{q}(t)=0. \end{aligned}\) $$\begin{array}{c}\hfill \underset{t\to -\infty }{lim}q(t)=0,\underset{t\to +\infty }{lim}|q(t)|=+\infty \text{and}\underset{t\to \pm \infty }{lim}\dot{q}(t)=0.\end{array}$$