We study the mixed dispersion fourth order nonlinear Schrödinger equationwhere γ, σ > 0 and β ∈ R. We focus on standing wave solutions, namely solutions of the form ψ(x, t) = e iαt u(x), for some α ∈ R. This ansatz yields the fourth-order elliptic equationWe consider two associated constrained minimization problems: one with a constraint on the L 2 -norm and the other on the L 2σ+2 -norm. Under suitable conditions, we establish existence of minimizers and we investigate their qualitative properties, namely their sign, symmetry and decay at infinity as well as their uniqueness, nondegeneracy and orbital stability.

Abstract. We study an elliptic system of the form Lu = |v| p−1 v and Lv = |u| q−1 u in Ω with homogeneous Dirichlet boundary condition, where Lu := −Δu in the case of a bounded domain and Lu := −Δu + u in the cases of an exterior domain or the whole space R N . We analyze the existence, uniqueness, sign and radial symmetry of ground state solutions and also look for sign changing solutions of the system. More general non-linearities are also considered.

We consider a fourth-order quasilinear nonhomogeneous equation which is equivalent to a nonhomogeneous Hamiltonian system. The purpose of this work is to prove the existence of at least two solutions for such equation when a certain parameter is small enough. Furthermore, under an additional hypothesis on positiveness of the nonhomogeneous part we prove that our solutions are positive.

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