A Minimum Problem Associated With Scalar Ginzburg–Landau Equation and Free Boundary

Let N≥2$N\ge 2$, p∈2NN+2,+∞$p\in \left(\frac{2N}{N+2},+\infty \right)$, and Ω$\Omega$ be an open bounded domain in RN$\mathbb {R}^N$ with smooth boundary. We consider the minimum problem J(u):=∫Ω1p|∇u|p+λ11−(u+)22+λ2u+dx→min$\mathcal {J} (u):= \int _{\Omega } \left(\frac{1}{p}| \nabla u| ^p+\lambda _1{\left(1-(u^+)^2\right)}^2+\lambda _2u^+\right)\text{d}x\rightarrow \text{min}$ over a certain class K$\mathcal {K}$, where λ1≥0$\lambda _1\ge 0$ and λ2∈R$ \lambda _2\in \mathbb {R}$ are constants, and u+:=max{u,0}$u^+:=\max \lbrace u,0\rbrace$. The corresponding Euler–Lagrange equation is related to the Ginzburg–Landau equation and involves a subcritical exponent when λ1>0$\lambda _1>0$. For λ1≥0$\lambda _1\ge 0$ and λ2∈R$ \lambda _2\in \mathbb {R}$, we prove the existence, non‐negativity, and uniform boundedness of minimizers of J(u)$\mathcal {J} (u)$. Then, we show that any minimizer is locally C1,α$C^{1,\alpha }$‐continuous with some α∈(0,1)$\alpha \in (0,1)$ and admits the optimal growth pp−1$\frac{p}{p-1}$ near the free boundary. Finally, under the additional assumption that λ2>0$\lambda _2>0$, 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$N-1$)‐dimensional Hausdorff measure.