Berestycki-Lions conditions on ground state solutions for a Nonlinear Schr\"odinger equation with variable potentials

TL;DR

This paper establishes the existence of ground state solutions for nonlinear Schrödinger equations with variable potentials under Berestycki-Lions conditions, using new techniques that extend previous results and handle more general cases.

Contribution

The paper introduces novel methods to prove ground state solutions for Schrödinger equations with variable potentials, generalizing and improving prior results under Berestycki-Lions assumptions.

Findings

Existence of ground state solutions of Pohozaev type.

Existence of least energy solutions.

Results applicable to non-radial and sign-indefinite cases.

Abstract

This paper is dedicated to studying the nonlinear Schr\"odinger equations of the form \begin{equation*}\label{KE} \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(u), & x\in \R^N; u\in H^1(\R^N), \end{array} \right. \end{equation*} where $V\in \mathcal{C}^1(\R^N, [0, \infty))$ satisfies some weak assumptions, and $f\in \mathcal{C}(\R, \R)$ satisfies the general Berestycki-Lions assumptions. By introducing some new tricks, we prove that the above problem admits a ground state solution of Poho\u{z}aev type and a least energy solution. These results generalize and improve some ones in [L. Jeanjean, K. Tanka,Indiana Univ. Math. J. 54 (2005), 443-464], [L. Jeanjean, K. Tanka, Proc. Amer. Math. Soc. 131 (2003) 2399-2408], [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, our assumptions are "almost" necessary when $V(x)\equiv V_{\infty}>0$, moreover, our approach could be useful for the study of other problems where radial symmetry of bounded sequence either fails or is not readily available, or where the ground state solutions of the problem at infinity are not sign definite.