Advances in Differential Equations

Mountain pass solutions for the fractional Berestycki-Lions problem

Vincenzo Ambrosio

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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

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

First available in Project Euclid: 23 January 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Ambrosio, Vincenzo. Mountain pass solutions for the fractional Berestycki-Lions problem. Adv. Differential Equations 23 (2018), no. 5/6, 455--488.

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