A bifurcation phenomenon in a singularly perturbed one-phase free boundary problem of phase transition

In this paper, we prove that a bifurcation phenomenon exists in a one-phase singularly perturbed free boundary problem of phase transition. Namely, the uniqueness of a solution of the one-phase problem breaks down as the boundary data decreases through a threshold value. The minimizer of the functional in consideration separates from the trivial harmonic solution. Moreover, we prove a third solution, a critical point of the functional being minimized, exists in this case by using the Mountain Pass Lemma. We prove convergence of the evolution with initial data near the minimizer or the trivial harmonic solution to the minimizer or to the trivial solution respectively, which means both the minimizer and the trivial harmonic solution are stable, while a saddle point solution of the free boundary problem is unstable in this sense.