Minimal mass blow up solutions for a double power nonlinear Schrodinger equation

We consider a nonlinear Schrodinger equation with double power nonlinearity i partial derivative(t)u + Delta u + vertical bar u vertical bar(4/d)u + epsilon vertical bar u vertical bar(p-1)u = 0, epsilon is an element of {-1, 0, 1}, 1 < p < 1 + 4/d in R-d (d = 1, 2, 3). Classical variational arguments ensure that H-1(R-d) data with parallel to u(0)parallel to(2) < parallel to Q parallel to(2) lead to global in time solutions, where Q is the ground state of the mass critical problem (epsilon = 0). We are interested by the threshold dynamic parallel to u(0)parallel to(2) = parallel to Q parallel to(2) and in particular by the existence of finite time blow up minimal solutions. For epsilon = 0, such an object exists thanks to the explicit conformal symmetry, and is in fact unique from the seminal work [22]. For epsilon = -1, simple variational arguments ensure that minimal mass data lead to global in time solutions. We investigate in this paper the case epsilon = 1, exhibiting a new class of minimal blow up solutions with blow up rates deeply affected by the double power nonlinearity. The analysis adapts the recent approach [31] for the construction of minimal blow up elements.