This paper focuses on the following scalar field equation involving a fractional Laplacian: where N ⩾ 2, α ∈ (0, 1), (−Δ)α stands for the fractional Laplacian. Using some minimax arguments, we obtain a positive ground state under the general Berestycki–Lions type assumptions.

For a class of quasilinear Schrödinger equations we establish the existence of ground states of soliton type solutions by a minimization argument.