Existence of positive ground state solutions for fractional Schrödinger equations with a general nonlinearity

ABSTRACT

In this paper, we study the existence of positive ground state solutions for the nonlinear fractional Schrödinger equation: $${(-▵)}^{\mathit{\alpha }}u+V(x)u=f(u),\text{in}{\mathbb{R}}^N,$$

where $N\ge 2$, $\mathit{\alpha }\in (0,1)$, $f\in C^1(\mathbb{R},\mathbb{R})$ is subcritical near infinity and superlinear near zero and satisfies the Berestycki–Lions condition. Using the monotonic trick established by Struwe–Jeanjean, the method of Pohozaev manifold and establishing a global compactness lemma, we show that the above problem has at least a positive ground state solution.

KEYWORDS:

• Fractional Schrödinger equation
• ground state solution
• variational method
• global compactness lemma

AMS SUBJECT CLASSIFICATIONS:

• 35J50
• 35B38
• 35J65

Acknowledgements

The authors would like to express their sincere gratitude to the anonymous referee for carefully reading the manuscript, giving valuable comments and suggestions to improve the results as well as the exposition of the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Z. Liu is supported by the NSFC [grant number 11626127]; Science and Technology Development Project of Hengyang City [grant number 2016KJ59].