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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 [{\it 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 $$\Ddot{u}(t)- L(t)u(t)+\nabla W(t,u(t))= 0, \qquad t\in \Bbb{R},\tag$*$ $$ where the symmetric-matrix-valued function $L \in C(\Bbb{R},\Bbb{R}^{N^2})$ and $W\in C^1(\Bbb{R}\times \Bbb{R}^N,\Bbb{R})$ 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 $$f(u)=\frac{1}{2}\int_{\Bbb{R}}(\vert \Dot{u}\vert ^2+ L(t)u,u))\,dt- \int_{\Bbb{R}}W(t,u)\,dt.$$ 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 $\vert u(t)\vert \to 0$, $\vert \Dot{u}(t)\vert \to 0$ as $\vert t\vert \to \infty$. So, $u$ is a homoclinic solution of $(*)$.

MSC:
34C37Homoclinic and heteroclinic solutions of ODE
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
47J30Variational methods (nonlinear operator equations)
58E05Abstract critical point theory
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References:
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