Under lim t->0 f(t)/t=0 and lim t->inf f(t)/t^p=0 for p<(N+1)/(N-1), solutions to the extension problem and Hartree extension problem are radially symmetric, with added regularity, exponential decay, and ground state existence under AR condition.
Tartar, An introduction to Sobolev spaces and interp olation spaces, Springer, Berlin, 2007
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Existence, regularity, asymptotic decay and radiality of solutions to some extension problems
Under lim t->0 f(t)/t=0 and lim t->inf f(t)/t^p=0 for p<(N+1)/(N-1), solutions to the extension problem and Hartree extension problem are radially symmetric, with added regularity, exponential decay, and ground state existence under AR condition.