Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian

Abstract

We establish sharp energy estimates for some solutions, such asglobal minimizers, monotone solutions and saddle-shaped solutions,of the fractional nonlinear equation $(-\Delta)$1/2 $u=f(u)$ in R n. Our energyestimates hold for every nonlinearity $f$ and are sharp since they areoptimal for one-dimensional solutions, that is, for solutionsdepending only on one Euclidian variable. As a consequence, in dimension $n=3$, we deduce theone-dimensional symmetry of every global minimizer and of everymonotone solution. This result is the analog of a conjecture of DeGiorgi on one-dimensional symmetry for the classical equation$-\Delta u=f(u)$ in R n.