Extremals to new Gagliardo–Nirenberg inequality and ground states

Abstract

In this paper, we consider extremals of a new Gagliardo–Nirenberg type inequality with symmetry, which may be useful for the study of the existence result of ground states to nonlinear fractional Schrödinger equations. We prove the existence of ground states of the fractional Schrödinger equation iut+(−Δ)su+Qu=λV|u|p−1u (where s∈(0,1), 1<p,(n−2s)p<n+2s) on the space Rn. Here Q is a non-negative function on Rn and V is a nontrivial nonnegative function on Rn. Under some suitable restriction on the smooth functions V and Q, we prove existence results about first eigen-functions of non-local problems on the whole space.