A free boundary problem for quasilinear elliptic equations in exterior domains
This paper studies the quasilinear elliptic equations of the type \(\operatorname {div} a(x,u,\nabla u) =f(x,u,\nabla u)\) in an exterior domain in \(\mathbb{R}^n\). The fact that the function \(f\) is allowed to be discontinuous converts the problem into one of the free-boundary type. The main result of the paper is that, assuming an ordered pair of appropriately defined upper and lower solutions, there exist solutions of the problem within the interval specified by the upper and lower solution. Moreover, the provided solution is an extremal one. The proof of the main result is based on the abstract fixed point theorem. Since no decay of the solution is specified as \(|x |\to \infty \), suitable local Sobolev spaces are the solution spaces. An example with the \(p\)-Laplacian is given at the end of the paper. The paper is clearly written and concise. The result is interesting.
scientific article; zbMATH DE number 1453641