In this paper, we consider the existence of periodic solutions for a class of nonautonomous second-order discrete Hamiltonian systems in case the sum on the time variable of potential is periodic. The tools used in our paper are the direct variational minimizing method and Rabinowitz's saddle point theorem.
MSC: 34C25; 58E50
This paper is devoted to study the following systems of coupled elliptic equations with quadratic nonlinearity −ε 2 ∆v + P (x)v = µvw, x ∈ R N , −ε 2 ∆w + Q(x)w = µ 2 v 2 + γw 2 , x ∈ R N , which arises from second-harmonic generation in quadratic optical media. We assume that the potential functions P (x) and Q(x) are positive functions and have a strict local maxima at x 0. Applying the finite dimensional reduction method, for any integer 1 ≤ k ≤ N + 1, we prove the existence of positive solutions which have k local maximum points that concentrate at x 0 simultaneously when ε is small.
This paper is devoted to a class of singularly perturbed nonlinear Schrödinger systems defined on a smooth bounded domain in RN(N=2,3). We use the Lyapunov–Schmidt reduction method to construct synchronized vector solutions with multiple spikes both on the boundary and in the interior of the domain. For each vector solution that has been constructed, we point out that the interior spikes locate near sphere packing points in the domain, and the boundary spikes locate near the critical points of the mean curvature function related to the boundary of the domain.
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