Nonlinear free boundary problems with singular source terms (Q1885254)

The authors consider nonlinear free boundary problems of the kind: \(\mathcal{ L}u=-\sum_{1}^{k}c_{j}\delta _{x^{j}}\) in \(\Omega \), \(u=0\) on \(\partial \Omega \), \(| \nabla u| =1\) on \(\partial \Omega \), where \( \mathcal{L}\) is a quasilinear uniformly elliptic operator written in divergence form: \(\text{div}( f_{p}( \nabla u)) \), with \(f_{p}=\nabla f( p) \) and \(f( p) =F(t| p| ^{2}) \), for some convex and increasing function \(F\) satisfying \(F(0) =0\). The case of the linear Laplace operator (obtained when \(F(p) =p\)) has already been studied by \textit{H. Shahgholian} in [Potential Anal. 3, No. 2, 245--255 (1994; Zbl 0796.31004)]. The case of the \(p\)-Laplace operator \(\;\Delta _{p}\) has been studied by \textit{D. Danielli} and \textit{A. Petrosyan} in [Calc. Var. Partial Differ. Equ. 23, No. 1, 97--124 (2005; Zbl 1068.35187)]. In the above problem \(c_{j}\) is a positive real and \(\delta _{x^{j}}\) is the Dirac measure with support at \(x^{j}\in \mathbb{R}^{n}\), \(n\geq 3\). The authors prove that, for given \(( c_{j}) _{j}\) and \(( x^{j}) _{j}\), there exists an open subset \(\Omega \) of \(\mathbb{R}^{n}\) containing the points \(x^{j}\) and a solution in \(W_{0}^{1,2}( \Omega) \) of the free boundary problem. Defining subsets \(D_{m,j}\) of \( \mathbb{R}^{n}\) verifying \(\cap _{m}D_{m,j}=\{ x^{j}\} \), the authors first introduce the auxiliary free boundary problem \(\mathcal{L} u_{m}=0\) in \(\Omega _{m}\setminus \cup _{j=1}^{k}D_{m,j}\), \(u_{m}=0\) on \( \partial \Omega _{m}\), \(u_{m}=a_{j}m^{n-2}\) in \(\overline{D_{m,j}}\), \( \lim_{y\rightarrow x}| \nabla u_{m}| \leq 1\) on \(\partial \Omega _{m}\), for arbitrary chosen positive \(a_{j}\). Then, proving uniform estimates and \(W^{1,2}\)-bounds on the solution \(( \Omega _{m},u_{m}) \) of this auxiliary problem, the authors establish the main existence result.