Existence and multiplicity of homoclinic orbits for a class of asymptotically periodic second order Hamiltonian systems
The paper deals with the system \(d^2 q/dt^2 = L(t) q - \nabla V(t,q)\), \(q(t) \to 0\), \(dq (t)/dt \to 0\) as \(|t |\to \infty\), where \(q \in \mathbb{C}^2 (\mathbb{R}, \mathbb{R}^m)/ \{0\}\) and \(L \in \mathbb{C} (\mathbb{R}, \mathbb{R}^{m^2})\), \(V \in \mathbb{C}^1 (\mathbb{R}, \mathbb{R}^m)\) are asymptotic to certain periodic functions \(L_-\), \(V_-\) as \(t \to - \infty\). (More in detail: \(L(t)\), \(L_- (t)\) are symmetrical; \(ax^2 \geq L(t)x \cdot x\) and \(L_- (t)x \cdot x \geq bx^2\) for appropriate \(a,b > 0\); \(L_- (t + 1) = L_- (t)\); \(|L(t) - L_- (t) |\to 0\) as \(t \to - \infty\); \(V\) and \(V_-\) are \(o(x)\) as \(x \to 0\); \(0 < \mu V < \nabla V \cdot x\) and \(0 < \mu V_- < \nabla V_- \cdot x\) for appropriate \(\mu > 0\) and all \(x \neq 0\); \(V_- (t + 1,x) = V_- (t,x)\); \(|\nabla V(t,x) - \nabla V_- (t,x) |\to 0\) as \(t \to - \infty\) uniformly on compacts.) The solutions are found as the critical points of the functional \(\varphi (u) = {1 \over 2} |u |^2 - \int V(t,u) dt\) on the space \(W^{1,2} (\mathbb{R}, \mathbb{R}^m)\) with the norm \(|u |^2 = \int (|du/dt |^2 + L(t)u \cdot u) dt\) under the following additional hypothesis: there exists \(c^* > c\) such that the set \(K(c^*)/ \mathbb{Z}\) is finite, where \(c\) is the mountain pass level, and \(K(c^*)\) is the set of critical values which are less or equal to \(c^*\) of the functional \(\varphi_- (u) = {1 \over 2} \int (|du/dt |^2 + L_- (t)u \cdot u - V(t,u)) dt\). A rather thorough analysis results in the existence of even infinitely many homoclinic solutions \(u \neq 0\) with the following \((k\)-bump) property: for every \(r > 0\) and each finite sequence \(p_1, \dots, p_k \in \mathbb{Z}\) that verifies \(p_j < n_0 (r)\), \(p_j - p_{j + 1} > M(r) > 0\) (with appropriate \(n_0 (r)\) and \(M(r) > 0)\) there exists a solution \(\overline q\) such that \(\sup |\overline q(t) - q(t) |< r({1 \over 2} (p_j + p_{j + 1}) < t < {1 \over 2} (p_{j - 1} + p_j))\) where \(p_0 = \infty\), \(p_{k + 1} = - \infty\).
scientific article; zbMATH DE number 878453