We consider the existence of multiple positive solutions to the nonlinear Schrödinger systems set onHere a 1 , a 2 > 0 are prescribed, µ 1 , µ 2 , β > 0, and the frequencies λ 1 , λ 2 are unknown and will appear as Lagrange multipliers. Two cases are studied, the first when N ≥ 1, 2 < p 1 , p 2 < 2 + 4 N , r 1 , r 2 > 1, 2 + 4 N < r 1 + r 2 < 2 * , the second when N ≥ 1, 2 + 4 N < p 1 , p 2 < 2 * , r 1 , r 2 > 1, r 1 + r 2 < 2 + 4 N . In both cases, assuming that β > 0 is sufficiently small, we prove the existence of two positive solutions. The first one is a local minimizer for which we establish the compactness of the minimizing sequences and also discuss the orbital stability of the associated standing waves. The second solution is obtained through a constrained mountain pass and a constrained linking respectively.

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equationWe assume γ > 0, N ≥ 1, 4 ≤ σN < 4N (N−4) + , whereas the parameter α ∈ R will appear as a Lagrange multiplier. Given c ∈ R + , we consider several questions including the existence of ground states, of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.