The Bifurcation Phenomenon for the Regularized Two-Phase Problem Associated with the p-Laplacian
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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BibTeX
@misc{ali2026bifurcation,
title = {{The Bifurcation Phenomenon for the Regularized Two-Phase Problem Associated with the $p$-Laplacian}},
author = {Alaa Haj Ali and Catherine Lebiedzik and Peiyong Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28227},
eprint = {2609.28227},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28227},
}
RIS
TY - UNPB
TI - The Bifurcation Phenomenon for the Regularized Two-Phase Problem Associated with the p-Laplacian
AU - Alaa Haj Ali
AU - Catherine Lebiedzik
AU - Peiyong Wang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28227
UR - https://arxiv.org/abs/2609.28227
AB - 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.
ER -