Zero mass case for a fractional Berestycki–Lions-type problem.

In this work we study the following fractional scalar field equation: { ( - Δ ) s ⁢ u = g ′ ⁢ ( u ) in ⁢ ℝ N , u > 0 , <graphic></graphic> \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=g^{\prime}(u)% \quad\mbox{in }\mathbb{R}^{N},\\ \displaystyle u&\displaystyle>0,\end{aligned}\right. where N ≥ 2 {N\geq 2} , s ∈ ( 0 , 1 ) {s\in(0,1)} , ( - Δ ) s {(-\Delta)^{s}} is the fractional Laplacian and the nonlinearity g ∈ C 2 ⁢ ( ℝ ) {g\in C^{2}(\mathbb{R})} is such that g ′′ ⁢ ( 0 ) = 0 {g^{\prime\prime}(0)=0}. By using variational methods, we prove the existence of a positive solution which is spherically symmetric and decreasing in r = | x | {r=|x|}. [ABSTRACT FROM AUTHOR]