ABSTRACT Let , , and be an open bounded domain in with smooth boundary. We consider the minimum problem over a certain class , where and are constants, and . The corresponding Euler–Lagrange equation is related to the Ginzburg–Landau equation and involves a subcritical exponent when . For and , we prove the existence, non‐negativity, and uniform boundedness of minimizers of . Then, we show that any minimizer is locally ‐continuous with some and admits the optimal growth near the free boundary. Finally, under the additional assumption that , we establish non‐degeneracy for minimizers near the free boundary and show that there exists at least one minimizer for which the corresponding free boundary has finite ()‐dimensional Hausdorff measure.
Hu et al. (Tue,) studied this question.
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