The double-power nonlinear Schrodinger equation and its generalizations: uniqueness, non-degeneracy and applications

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form Delta u + g(u) = 0. Our result applies in particular to the double power non-linearity where g(u) = u(q) - u(p) - mu u for p > q > 1 and mu > 0, which we discuss with more details. In this case, the non-degeneracy of the unique solution u mu allows us to derive its behavior in the two limits mu -> 0 and mu -> mu* where mu* is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the L-2 mass of u(mu) in terms of mu, which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of u(mu).