We consider a constrained variational problem where the energy functional includes a logarithmic convolution term and an external potential . There is a threshold that we establish existence and nonexist-ence results for constraint minimizers: for , minimizers exist for any ; for with and with , no minimizer exists. Furthermore, for and , we analyze the limiting behavior of positive minimizers, showing that after suitable scaling, they converge to the standard ground state solution of in . We also de-rive asymptotic estimates for the location of the maximum points of mini-mizers.
Cite this paper
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