A minimum problem associated with scalar Ginzburg-Landau equation and free boundary

Abstract: Let $N>2$, $p\in \left(\frac{2N}{N+2},+\infty\right)$, and $\Omega$ be an open bounded domain in $\mathbb{R}^N$. We consider the minimum problem $$ \mathcal{J} (u) := \displaystyle\int_{\Omega } \left(\frac{1}{p}| \nabla u| ^p+\lambda_1\left(1-(u^+)^2\right)^2+\lambda_2u^+\right)\text{d}x\rightarrow \text{min} $$ over a certain class $\mathcal{K}$, where $\lambda_1\geq 0$ and $ \lambda_2\in \mathbb{R}$ are constants, and $u^+:=\max{u,0}$. The corresponding Euler-Lagrange equation is related to the Ginzburg-Landau equation and involves a subcritical exponent when $\lambda_1>0$. For $\lambda_1\geq 0$ and $ \lambda_2\in \mathbb{R}$, we prove the existence, non-negativity, and uniform boundedness of minimizers of $\mathcal{J} (u) $. Then, we show that any minimizer is locally $C^{1,\alpha}$-continuous with some $\alpha\in (0,1)$ and admits the optimal growth $\frac{p}{p-1}$ near the free boundary. Finally, under the additional assumption that $\lambda_2>0$, 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.