Periodic solutions for critical fractional problems

Abstract

We deal with the existence of 2 π-periodic solutions to the following non-local critical problem[(-Δx+m2)s-m2s]u=W(x)|u|2s∗-2u+f(x,u)in(-Nu(x+2πei)=u(x)forallx∈RN,i=1,⋯,N,where s∈ (0 , 1) , N≥ 4 s, m≥ 0 , 2s∗=2NN-2s is the fractional critical Sobolev exponent, W(x) is a positive continuous function, and f(x, u) is a superlinear 2 π-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 (-N×(0,∞), 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