Let N>2, p (2NN+2, +), and be an open bounded domain in RN. We consider the minimum problem J (u): = (1p| u| ᵖ+₁ (1- (u^+) ²) ²+₂u^+) dx min over a certain class K, where ₁ 0 and ₂ R are constants, and u^+: =\u, 0\. The corresponding Euler-Lagrange equation is related to the Ginzburg-Landau equation and involves a subcritical exponent when ₁>0. For ₁ 0 and ₂ R, we prove the existence, non-negativity, and uniform boundedness of minimizers of J (u). Then, we show that any minimizer is locally C^1, -continuous with some (0, 1) and admits the optimal growth pp-1 near the free boundary. Finally, under the additional assumption that ₂>0, 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 (N-1) -dimensional Hausdorff measure.
Hu et al. (Wed,) studied this question.