Ground States to the Generalized Nonlinear Schrödinger Equations with Bernstein Symbols

Abstract

This paper concerns with existence and qualitative properties of ground states to generalized nonlinear Schrödinger equations (gNLS) with abstract symbols. Under some structural assumptions on the symbol, we prove a ground state exists and it satisfies several fundamental properties that the ground state to the standard NLS enjoys. Furthermore, by imposing additional assumptions, we construct, in small mass case, a nontrivial radially symmetric solution to gNLS with $H^1$-subcritical nonlinearity, even if the natural energy space does not control the $H^1$-subcritical nonlinearity.