Existence and multiplicity results for the fractional Schrödinger equations with indefinite potentials

ABSTRACT

In this paper, we consider the fractional Schrödinger equations $(-\mathrm{\Delta }{)}^{s}u+V(x)u=f(x,u)\text{in}{\mathbb{R}}^N,$ where s ∈ (0,1) and N>2s. Firstly, we prove that the problem admits a nontrivial solution and infinitely many nontrivial solutions under the assumptions that V is allowed to be indefinite potential and f is superlinear and subcritical. In addition, we establish an existence criterion of infinitely many nontrivial solutions for the aforementioned problem with concave and critical nonlinearities as well as indefinite potential, which improves the result of Du and Tian [Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete Contin Dyn Syst B. 2016;21(10):3407–3428 (Theorem 1.6)].

KEYWORDS:

• Fractional Schrödinger equations
• indefinite potential
• eigenvalue problem

AMS SUBJECT CLASSIFICATION (2010):

• 35R11
• 35A15
• 35S05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research is supported in part by the National Natural Science Foundation of China (NSF) of China (11671181).