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In the present paper, we prove the existence of solutions $(\lambda, u)\in\R\times H^1(\R^N)$ to the following elliptic equations with potential$\displaystyle -\Delta u+(V(x)+\lambda)u=g(u)\;\hbox{in}\;\R^N, $ satisfyingthe normalization constraint $\displaystyle \int_{\R^N}u^2=a>0,$ which isdeduced by searching for solitary wave solution to the time-dependent nonlinearSchr\"odinger equations. Besides the importance in the applications, notnegligible reasons of our interest for such problems with potential $V(x)$ aretheir stimulating and challenging mathematical difficulties. We develop aninteresting way basing on iteration and give a new proof of the so-called"sub-additive inequality", which can simply the standard process in thetraditional sense. Under some very relax assumption on the potential $V(x)$ andsome other suitable assumptions on $g$, we can obtain the existence of groundstate solution for prescribed $a>0$.

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