Ground state for relativistic nonlinear Schrödinger equations involving general nonlinear term

Abstract

In this paper, the existence and nonexistence of positive solutions for a relativistic nonlinear Schrödinger equation with vanish potential and general nonlinear term were obtained. First, we reduce this equation to a semilinear elliptic equation by a change. Then, using mountain pass theorem and Strauss compactness lemma, we get the existence of positive radial ground state solutions for this equation. Moreover, the nonexistence of nontrivial solutions for this equation was gotten by Pohozaev identity.

Keywords:

• Ground state
• relativistic nonlinear Schrödinger equations
• mountain pass theorem
• strauss compactness lemma

2020 Mathematics Subject Classifications:

• 35J60
• 35J20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 11771127] and the Fundamental Research Funds for the Central Universities [grant numbers WUT:2020IB011, 2020IB017, 2020IB019].