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SUMMARY:Estimates on the $L^2$ norm of the positive solutions of a two par
 ameter family of nonlinear PDE´s of the TFW type. - Rafael Benguria (Pont
 ificia Universidad Católica de Chile)
DTSTART:20260304T110000Z
DTEND:20260304T120000Z
UID:TALK245092@talks.cam.ac.uk
DESCRIPTION:In this talk I consider the two parameter family of PDE&acute\
 ;s (generalized TFWequation):\\[-\\Delta u + (\\gamma u^{2p-2} - \\phi) u=
 0\,\\]on $\\mathbb{R}^3$\, where\\[\\phi(x)= \\frac{Z}{|x|}-\\int_{\\mathb
 b{R}^3} \\frac{u^2(y)}{|x-y|} \\\, dy.\\]Here\, $Z>0$ is fixed and $\\gamm
 a \\ge 0$\, and $1<p$. The case $p=5/3$corresponds to the Thomas-Fermi-von
  Weizs&auml\;cker equation in the atomiccase\, which was first studied by 
 R. Benguria\, H. Brezis\, and E.H. Lieb in1981. The case $\\gamma =0$ corr
 esponds to the Hartree equation.Here we are interested in estimating the s
 o called ``excess charge&acute\;&acute\;\,given by $Q=\\int_{\\mathbb{R}^3
 }\\|u\\|_2^2 - Z$\, for different values of $p$ and $\\gamma$\, where $u$ 
 is thepositive solution of the generalized TFW equation. This is joint wor
 k withHeinz Siedentop (Ludwig Maximilians University).
LOCATION:Seminar Room 2\, Newton Institute
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