Existence of ground states for semilinear elliptic equations with decayingmass: a parabolic approach

Consider the elliptic problem: Deltau - Vu + u(p) = 0 in R-n, with 1 < p < (n + 2)/(n - 2), n greater than or equal to 2, and non-negative bounded pot ential V(x) which may decay to 0 at infinity. We prove that if V satisfies a(1)/(1+ \x \ (b)) less than or equal to V(x) less than or equal to a(2) wi th 0 less than or equal to b < 2(n - 1)(p - 1)/(p +3) and V is radial, then it admits a (ground state) positive solution. The result relies on the stu dy of global solutions of the associated parabolic problem. Indeed, we will show that, under suitable conditions on V (not necessarily radial), this p roblem admits global positive solutions and that when V and u (.,0) are rad ial some global solutions have <omega>-limit sets containing a positive equ ilibrium. (C) 2001 Academie des sciences/Editions scientifiques et medicale s Elsevier SAS.