## Advances in Differential Equations

- Adv. Differential Equations
- Volume 23, Number 5/6 (2018), 455-488.

### Mountain pass solutions for the fractional Berestycki-Lions problem

#### Abstract

We investigate the existence of least energy solutions and infinitely many solutions for the following nonlinear fractional equation \begin{align*} (-\Delta)^{s} u = g(u) \mbox{ in } \mathbb R^{N}, \end{align*} where $s\in (0,1)$, $N\geq 2$, $(-\Delta)^{s}$ is the fractional Laplacian and $g: \mathbb R \rightarrow \mathbb R $ is an odd $\mathcal{C}^{1, \alpha}$ function satisfying Berestycki-Lions type assumptions. The proof is based on the symmetric mountain pass approach developed by Hirata, Ikoma and Tanaka in [33]. Moreover, by combining the mountain pass approach and an approximation argument, we also prove the existence of a positive radially symmetric solution for the above problem when $g$ satisfies suitable growth conditions which make our problem fall in the so called “zero mass” case.

#### Article information

**Source**

Adv. Differential Equations Volume 23, Number 5/6 (2018), 455-488.

**Dates**

First available in Project Euclid: 23 January 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1516676484

**Subjects**

Primary: 35A15: Variational methods 35J60: Nonlinear elliptic equations 35R11: Fractional partial differential equations 45G05: Singular nonlinear integral equations 49J35: Minimax problems

#### Citation

Ambrosio, Vincenzo. Mountain pass solutions for the fractional Berestycki-Lions problem. Adv. Differential Equations 23 (2018), no. 5/6, 455--488. https://projecteuclid.org/euclid.ade/1516676484