In a series of papers the question of uniqueness of radia
l ground states of the equation
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u+
f(
u)=0 and of various re
lated equations has been studied. It is remarkab
le that throughout this work (except in very specia
l circumstances) nowhere is a spatia
lly dependent term taken into consideration. Here we sha
ll make a first attempt to study the uniqueness of ground states for such spatia
lly dependent equations and to estab
lish qua
litative properties of so
lutions for this purpose.