a1 Dipartimento di Matematica, Università degli Studi ‘Roma Tre’, Largo San Leonardo Murialdo 1, 00146 Roma, Italy (email@example.com).
We study the effect of the potential |y|α on the stability of entire solutions for elliptic equations on N, N ≥ 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet nonlinear eigenvalue problem on bounded domains. As a by-product of our results, we will improve the known results about the regularity of such solutions.
(Received February 19 2007)
(Accepted July 20 2007)