Multiple Normalized Solutions for First Order Hamiltonian Systems

.In this paper, we study the following first order Hamiltonian systems: \(\mathcal{J}\dot{u}+M(t)u=|K(t)u|^{p-2}K(t)^TK(t)u+\lambda u\; \text{in}\;\mathbb{R},\) where \(u:\mathbb{R}\to \mathbb{R}^{2N}\), \(p\in (2,4)\), \(\lambda \in \mathbb{R}\) arises as the Lagrange multiplier, and \(\mathcal{J},M, K(t)\) are \(2N\times 2N\) real matrices with \(\mathcal{J}^{-1}=\mathcal{J}^T=-\mathcal{J}, M=M^T\). Using the multiplicity theorem of Ljusternik–Schnirelmann together with variational methods, we show the existence of multiple normalized homoclinic solutions for this problem. We deal with not only the case det\(K(t)\neq 0\) for all \(t\) in a set of nonzero measure, but also the case det\(K(t)=0\) for all \(t\in \mathbb{R}\). In particular, we also obtain bifurcation results of this problem.Keywordsmultiple normalized solutionsHamiltonian systemsvariational methodsMSC codes70H0535J50