We construct solutions to a class of Schrödinger equations involving the fractional laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.Keywords: Fractional laplacian, Nehari manifold.

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Abstract. We consider the stationary nonlinear magnetic Choquard equationwhere A is a real valued vector potential, V is a real valued scalar potential, N ≥ 3, α ∈ (0, N ) and 2 − (α/N ) < p < (2N − α)/(N − 2). We assume that both A and V are compatible with the action of some group G of linear isometries of R N . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry conditionwhere τ : G → S 1 is a given group homomorphism into the unit complex numbers.MSC2010: 35Q55, 35Q40, 35J20, 35B06.

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The semi-classical regime of standing wave solutions of a Schrödinger equation in the presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is shown that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.