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<p>We study a two-phase problem for an elliptic equation with a singular term plus a critical Sobolev power. We obtain the existence of a one-parameter family of solutions. Furthermore, we study the asymptotic behavior of the family with respect to the parameter. The strategy to obtain a solution relies on a limit process of the approximated solutions of an equation without the singular term. We analyze the regularity of the solutions as well as the regularity of the free boundaries, which are smooth hypersurfaces outside a set of Hausdorff co-dimension at most 3.</p>

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Existence of Solution for a Two-Phase Elliptic Equation

  • 期刊论文

  • 发表日期：

    2025年5月9日

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  • Eduardo Hitomi,
  • Marcelo Montenegro

摘要

We study a two-phase problem for an elliptic equation with a singular term plus a critical Sobolev power. We obtain the existence of a one-parameter family of solutions. Furthermore, we study the asymptotic behavior of the family with respect to the parameter. The strategy to obtain a solution relies on a limit process of the approximated solutions of an equation without the singular term. We analyze the regularity of the solutions as well as the regularity of the free boundaries, which are smooth hypersurfaces outside a set of Hausdorff co-dimension at most 3.