In this article, our focus lies on a Schrödinger equation incorporating a Hardy term. To identify the global minimizers of the functional Iunder a mass constraint, we utilize the Hardy inequality. In addition, we reveal that every energy ground state is directly linked to the least action solution of the associated action functional. This finding affirmatively addresses the question of whether, under broad assumptions, The functional Iis characterized by a mountain pass structure and satisfies the (PS)C condition. Subsequently, this guarantees the existence of a nontrivial critical point for the energy functional I.
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