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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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