A new deduction of the strict sub-additive inequality and its application: Ground state normalized solution to Schrödinger equations with potential

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 $$\displaystyle -\Delta u+(V(x)+\lambda)u=g(u)\;\hbox{in}\;\mathbb R^N, $$ 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 based on iteration and give a new proof of the so-called ``sub-additive inequality", which can simplify the standard process in the traditional sense. Under some mild assumption on the potential $V(x)$ and some other suitable assumptions on $g$, we can obtain the existence of ground state solution for prescribed $a > 0$.