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

Let $N>2$, p∈ (2N/N+2,+∞), and Ω be an open bounded domain in RN. We consider the minimum problem J (u) := ∫Ω (1/p| ∇ u| ᵖ+λ₁(1-(u⁺)²)²+λ₂u⁺)dx→ min over a certain class K, where λ₁≥ 0 and λ₂∈ R are constants, and u⁺:=max,0\. The corresponding Euler-Lagrange equation is related to the Ginzburg-Landau equation and involves a subcritical exponent when λ₁>0. For λ₁≥ 0 and λ₂∈ R, we prove the existence, non-negativity, and uniform boundedness of minimizers of J (u). Then, we show that any minimizer is locally C1,α-continuous with some α∈ (0,1) and admits the optimal growth p/p-1 near the free boundary. Finally, under the additional assumption that λ₂>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.