A new deduce of the strict binding inequality and its application: Ground state normalized solution to Schrödinger equations with potential

Abstract

In the present paper, we prove the existence of solutions $(\lambda ,u)\in \mathbb{R}\times H^1({\mathbb{R}}^N)$ to the following elliptic equations with potential $-\Delta u+(V(x)+\lambda )u=g(u)in{\mathbb{R}}^N,$ satisfying the normalization constraint ${\int }_{{\mathbb{R}}^N}{u}^2=a>0,$ which is deduced by searching for solitary wave solution to the time-dependent nonlinear Schrödinger equations. Besides the importance in the applications, not negligible reasons of our interest for such problems with potential $V(x)$ are their stimulating and challenging mathematical difficulties. We develop an interesting way basing on iteration and give a new proof of the so-called "sub-additive inequality", which can simply the standard process in the traditional sense. Under some very relax assumption on the potential $V(x)$ and some other suitable assumptions on $g$, we can obtain the existence of ground state solution for prescribed $a>0$.

20 pages