Non-homogeneous Gagliardo-Nirenberg inequalities in RN and application to a biharmonic non-linear Schrödinger equation

Abstract

We develop a new method, based on the Tomas-Stein inequality, to establish non-homogeneous Gagliardo-Nirenberg-type inequalities in RN. Then we use these inequalities to study standing waves minimizing the energy when the L2-norm (the mass) is kept fixed for a fourth-order Schrödinger equation with mixed dispersion. We prove optimal results on the existence of minimizers in the mass-subcritical and mass-critical cases. In the mass-supercritical case global minimizers do not exist. However, if the Laplacian and the bi-Laplacian in the equation have the same sign, we are able to show the existence of local minimizers. The existence of those local minimizers is significantly more difficult than the study of global minimizers in the mass-subcritical and mass-critical cases. They are global in time solutions with small H2-norm that do not scatter. Such special solutions do not exist if the Laplacian and the bi-Laplacian have opposite sign. If the mass does not exceed some threshold μ0∈(0,+∞), our results on “best” local minimizers are optimal.