The Bifurcation Phenomenon for the Regularized Two-Phase Problem Associated with the $p$-Laplacian

Alaa Haj Ali; Catherine Lebiedzik; Peiyong Wang

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arXiv · 2609.28227

The Bifurcation Phenomenon for the Regularized Two-Phase Problem Associated with the $p$-Laplacian

Alaa Haj Ali·Catherine Lebiedzik·Peiyong Wang

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Abstract

In this paper, we verify a bifurcation phenomenon regarding the multiplicity of weak solutions, subject to the Dirichlet boundary condition, of a regularized two-phase free boundary problem associated with the $p$-Laplacian. In fact, we prove the existence of a mountain pass solution when the boundary data is small. In this case, three weak solutions of the regularized problem exist: the $p$-harmonic function, a minimizer of the corresponding functional, and the mountain pass solution. Finally, we consider the special case of radially symmetric solutions. By solving the associated nonlinear ordinary differential equation in the unregularized case, we show explicitly how the Bernoulli condition at the free boundary gives rise to the bifurcation of solutions.

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Alaa Haj Ali, Catherine Lebiedzik, Peiyong Wang. 2026-09-23. The Bifurcation Phenomenon for the Regularized Two-Phase Problem Associated with the $p$-Laplacian. https://arxiv.org/abs/2609.28227

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