We provide a sufficient condition for the existence of a positive solution to -Delta u + V(vertical bar x vertical bar)u = u(p) in B-1, when p is large enough. Here B-1 is the unit ball of R-n, n >= 2, and we deal with both Neumann and Dirichlet homogeneous boundary conditions. The solution turns out to be a constrained minimum of the associated energy functional. As an application we show that in case V(vertical bar x vertical bar) >= 0, V not equivalent to 0 is smooth and p is sufficiently large, and the Neumann problem always admits a solution.
Positive constrained minimizers for supercritical problems in the ball / Grossi, Massimo; Benedetta, Noris. - In: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9939. - 140:6(2012), pp. 2141-2154. [10.1090/s0002-9939-2011-11133-x]
Positive constrained minimizers for supercritical problems in the ball
GROSSI, Massimo;
2012
Abstract
We provide a sufficient condition for the existence of a positive solution to -Delta u + V(vertical bar x vertical bar)u = u(p) in B-1, when p is large enough. Here B-1 is the unit ball of R-n, n >= 2, and we deal with both Neumann and Dirichlet homogeneous boundary conditions. The solution turns out to be a constrained minimum of the associated energy functional. As an application we show that in case V(vertical bar x vertical bar) >= 0, V not equivalent to 0 is smooth and p is sufficiently large, and the Neumann problem always admits a solution.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
