NORMALIZED SOLUTIONS TO THE MIXED DISPERSION NONLINEAR SCHRODINGER EQUATION IN THE MASS CRITICAL AND SUPERCRITICAL REGIME

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrodinger equation gamma Delta(2)u - Delta u + alpha u = vertical bar u vertical bar(2 sigma) u, u is an element of H-2(R-N), under the constraint integral(RN) vertical bar u vertical bar(2) dx - c > 0. We assume that gamma > 0,N >= 1,4 >= sigma N < 4N/(N-4), whereas the parameter alpha is an element of E R will appear as a Lagrange multiplier. Given c is an element of R+, we consider several questions including the existence of ground states and of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.