The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970s. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). ...

We study the nonlinear fractional equation (−Δ)su=f(u) in Rn, for all fractions 0<s<1 and all nonlinearities f . For every fractional power s∈(0,1) , we obtain sharp energy estimates for bounded global minimizers and for ...

We consider the monomial weight |x1|A1⋯|xn|An in Rn, where Ai⩾0 is a real number for each i=1,…,n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of ...

In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension n¿9. This result, that was only known to be true for n¿4 , is ...

We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including ...

We study a natural question that, apparently, has not been well addressed in the literature. Given functions \upsilon with support in the unit ball B1 \subset \mathbb{R}n and with gradient in the Morrey space Mp,\lambda(B1), ...

We study the regularity with respect to parameters of the invariant manifolds associated to non-resonant subspaces obtained in the previous article [CFdlL00].