The Bifurcation Phenomenon for the Regularized Two-Phase Problem Associated with the $p$-Laplacian
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Institutions: University of Michigan, Wayne State University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214240124