A limit case in non-isotropic two-phase minimization problems driven by $p$-Laplacians

In this work we study a minimization problem with two-phases where in each phase region the problem is ruled by a quasi-linear elliptic operator of p−Laplacian type. The problem in its variational form is as follows: min v  ∫ Ω∩{v>0} ( 1 p |∇v|p +λ p +(x)+ f+(x)v ) dx+ ∫ Ω∩{v≤0} ( 1 q |∇v|q +λ q −(x)+ f−(x)v ) dx  . Here we minimize among all admissible functions v in an appropriate Sobolev space with a prescribed boundary datum v = g on ∂Ω. First, we show existence of a minimizer, prove some properties, and provide an example for non-uniqueness. Moreover, we analyze the limit case where p and q go to infinity, obtaining a limiting free boundary problem governed by the ∞−Laplacian operator. Consequently, Lipschitz regularity for any limiting solution is obtained. Finally, we establish some weak geometric properties for solutions.