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

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 study standing waves minimizing energy when L2-norm (the mass) is kept fixed for fourth-order Schrödinger equation with mixed dispersion. prove optimal results existence of minimizers mass-subcritical and mass-critical cases. In mass-supercritical case global do not exist. However, if Laplacian bi-Laplacian have same sign, are able show local minimizers. The those significantly more difficult than They time solutions small H2-norm that scatter. Such special exist opposite sign. If mass does exceed some threshold ?0?(0,+?), our “best” optimal.