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# Dynamics of Partial Differential Equations

## Volume 18 (2021)

### Number 3

### Lions-type theorem of the fractional Laplacian and applications

Pages: 211 – 230

DOI: https://dx.doi.org/10.4310/DPDE.2021.v18.n3.a3

#### Authors

#### Abstract

In this paper, our goal is to establish a generalized version of Lions-type theorem for the fractional Laplacian. As an application of this theorem, we consider the existence of ground state solutions of a fractional equation:\[(-\Delta)^s u + V (\lvert x \rvert) u = f(u), \; x \in \mathbb{R}^N ,\]where $N \geqslant 3, s \in (\frac{1}{2}, 1), V$ is a singular potential with $\alpha \in (0, 2s) \cup (2s, 2N - 2s)$, and the nonlinearity $f$ has the critical growth, discussed without any boundary value condition.

#### Keywords

Lions theorem, fractional equation, singular potential, critical exponent, Nehari manifold, ground state solution

#### 2010 Mathematics Subject Classification

Primary 35A15, 35R11. Secondary 35Q55.

This work is supported by China Postdoctoral Science Foundation 2020M671835 and Key Projects of Anhui University of Science and Technology QN2019101. It is also partially supported by National Science Foundation of China 12001160.

Received 16 May 2020

Published 22 July 2021