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Existence of homoclinic solution for the second order Hamiltonian systems. (English) Zbl 1057.34038
The authors apply a generalized version of the mountain pass theorem [P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)] for establishing the existence of a nontrivial homoclinic solution for the second-order Hamiltonian systems \[ \Ddot{u}(t)- L(t)u(t)+\nabla W(t,u(t))= 0, \qquad t\in \mathbb{R},\tag{\(*\)} \] where the symmetric-matrix-valued function \(L \in C(\mathbb{R},\mathbb{R}^{N^2})\) and \(W\in C^1(\mathbb{R}\times \mathbb{R}^N,\mathbb{R})\) satisfy some specific conditions. These conditions are weaker that those considered by other authors in proving the same type of results, namely \(L(t)\) is not supposed to be uniformly positive definite, and \(W\) satisfies a superquadratic condition. The authors consider the following standard functional \[ f(u)=\frac{1}{2}\int_{\mathbb{R}}(| \Dot{u}| ^2+ L(t)u,u))\,dt- \int_{\mathbb{R}}W(t,u)\,dt. \] The generalized mountain pass theorem gives the existence of a critical point \(u\) of \(f\) such that \(f(u)\geq \alpha_0 >0\). Furthermore, it is shown that \(u\in D(A)\), where \(A=-d^2/dt^2+L(t)\), and therefore \(| u(t)| \to 0\), \(| \Dot{u}(t)| \to 0\) as \(| t| \to \infty\). So, \(u\) is a homoclinic solution of \((*)\).

MSC:
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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