Existence and multiplicity of solutions for a class of Hamiltonian systems

Khaled Khachnaoui

doi:10.3842/umzh.v76i5.7497

Khaled Khachnaoui Department of Mathematics, University of Kairouan, Preparatory Institute for Engineering Studies, Tunisia

DOI: https://doi.org/10.3842/umzh.v76i5.7497

Keywords: Nehari manifold methods; Hamiltonian systems; Variational methods; Critical point

Abstract

UDC 517.9

We investigate a class of Hamiltonian systems \begin{gather*} {-q''}(t)+(L(t)-\xi)q(t)= a(t)|q(t)|^{p-2}q(t)+\eta f(t),\\ q\in H^1(\mathbb{R},\mathbb{R}^N),\end{gather*} where $(t,q)\in \mathbb{R}\times \mathbb{R}^N$, $p>2,$ $a\in C(\mathbb{R},(0,+\infty)),$ $f\in C(\mathbb{R},\mathbb{R}^N),$ $\xi, \eta$ are real parameters, and $L\in C(\mathbb{R},\mathbb{R}^{N^2})$ is a positive definite symmetric matrix for all $t\in \mathbb{R}.$ The main technical approach is based on the Nehari manifold method combined with variational and topological methods. The obtained results extend and complement the results available in the literature.

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