Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8904428 | Acta Mathematica Scientia | 2018 | 12 Pages |
Abstract
In this article, we study constrained minimizers of the following variational problem
e(Ï):=inf{uâH1(R3),âuâ22=Ï}E(u),Ï>0,where E(u) is the Schrödinger-Poisson-Slater (SPS) energy functional
E(u):=12â«R3|âu(x)|2dx-14â«R3â«R3u2(y)u2(x)|x-y|dydx-1pâ«R3|u(x)|pdxinR3,and pâ (2, 6). We prove the existence of minimizers for the cases 2 < p <
103, Ï > 0, and p =
103, 0 < Ï < Ï *, and show that e(Ï)= â â for the other cases, where Ï* =
âÏâ22 and Ï(x) is the unique (up to translations) positive radially symmetric solution of
-Îu+u=u73 in
R3. Moreover, when
e(Ï*)=-â, the blow-up behavior of minimizers as
ÏâÏ* is also analyzed rigorously.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Xincai ZHU,