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

In the present paper, we prove the existence of solutions (λ, u) ∈ R×H(R ) to the following elliptic equations with potential −∆u+(V (x) + λ)u = g(u) in R , satisfying the normalization constraint ∫ RN u = 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 “subadditive 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.