Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials

By Sitong Chen, Xianhua Tang. Advances in Nonlinear Analysis, 2018

66 citations recorded by the source corpus.

Abstract This paper is dedicated to studying the nonlinear Schrödinger equations of the form −△u+V(x)u=f(u),x∈RN;u∈H1(RN), $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(u), & x\in \mathbb{R}^N; \\ u\in H^1(\mathbb{R}^N), \end{array} \right. \end{array}$$ where V ∈ 𝓒1(ℝN, [0, ∞)) satisfies some weak assumptions, and f ∈ 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 Pohožaev 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) ≡ V∞ > 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3a5fa21c9b2ab0744dfd46eae5ae7b97c9020e5a