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The purpose of this article is twofold. First, we study the existence and asymptotic behavior of ground states of a fractional Schrodinger system with quadratic interaction. Second, we give some conditions, in terms of the mass and energy of the ground states, under which the solutions of the associated initial value problem have the uniform bound or may blow up in finite time. As a corollary, we show the strong instability of the ground states.

This paper focuses on the scalar field equation Dx2αu−νu−uyy=f(u), where α ∈ (0, 1), (x,y)∈RN×(−L,L)⊂RN+1, with N ≥ 1 and Dx2α stands for the fractional Laplacian. By using several variational methods, we establish the existence, long behavior, and multiplicity of solutions of this equation under the Dirichlet and Neumann boundary conditions.