(Super)critical nonlocal equations with periodic boundary conditions
In this paper the authors discuss the existence and multiplicity of periodic solutions for a class of parametric nonlocal equations with critical and supercritical growth. More specifically, given \(s\in (0,1)\), \(N>2s\), \(m>0\), the authors investigate the existence and multiplicity of periodic solutions for the class of problems \[ ((-\Delta+m^2)^s)u=\lambda f(x,u)+\mu g(x,u) \text{ in } (0,T)^N, \] \[ u(x+Te_i)=u(x) \text{ for all } x\in \mathbb{R}^N, \;i=1,\ldots, N. \] Here \(f\) has subcritical growth and \(g\) behaves as \(|u|^{p-2}u\), with \(p\geq 2N/(N-2s)\), besides satisfying additional conditions. The first key ingredient in the proof is an extension technique in the periodic setting, inspired by the one of Caffarelli-Silvestre, which realizes the equations as local degenerate elliptic problems in a half-cylinder of \(\mathbb{R}^{N+1}_+\) with nonlinear Neumann boundary conditions. The authors then truncate the critical term \(g\) and prove that the critical points of the truncated functional are bounded. The Moser iteration scheme adapted to the fractional setting is also instrumental. Finally, a multiplicity result valid for differentiable functionals due to Ricceri is used, and the authors show that the problem under consideration admits at least three period solutions with the property that their Sobolev norms are bounded by a suitable constant. Additionally, the authors also provide a concrete estimate on the range of these parameters by using properties of fractional calculus.
scientific article; zbMATH DE number 6941791
scientific article