Periodic solutions for critical fractional problems

Abstract

We deal with the existence of $2\pi$-periodic solutions to the following non-local critical problem \begin{equation*} \left\{\begin{array}{ll} [(-Δ_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in} (-π,π)^{N} \\ u(x+2πe_{i})=u(x) &\mbox{for all} x \in \mathbb{R}^{N}, \quad i=1, \dots, N, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N\ge 4s$, $m\ge 0$, $2_s^{\ast }=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $W(x)$ is a positive continuous function, and $f(x,u)$ is a superlinear $2\pi$-periodic (in $x$) continuous function with subcritical growth. When $m>0$, the existence of a nonconstant periodic solution is obtained by applying the Linking Theorem, after transforming the above non-local problem into a degenerate elliptic problem in the half-cylinder $(-\pi ,\pi {)}^N\times (0,\infty )$, with a nonlinear Neumann boundary condition, through a suitable variant of the extension method in periodic setting. We also consider the case $m=0$ by using a careful procedure of limit. As far as we know, all these results are new.

Calculus of Variations and Partial Differential Equations (2018)

References in corpus (2)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications

Cited by in corpus (2)

• Lamellar phase solutions for diblock copolymers with nonlocal diffusions
• Multiple solutions for superlinear fractional problems via theorems of mixed type