Singular perturbation method for inhomogeneous nonlinear free boundary problems

In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type: F(D(2)u, x) = beta(epsilon)(u) + f(epsilon)(x) and Delta(p)u = beta(e)(u) + f(epsilon)(x), where beta(epsilon) approaches Dirac delta(0) as epsilon -> 0 and f(epsilon) has a uniform control in L-q, q > N. Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the epsilon-level surfaces are established for these variational and nonvaritional solutions. Finally, letting epsilon -> 0 basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained.