The authors study the existence and regularity results of positive solutions for nonlinear problem containing the square root of the Laplacian stands for the square root of the operator in a bounded domain ) with zero Dirichlet boundary conditions.
Among other results they prove:
Theorem 1.1 (The existence result). Let be an integer and when . Suppose that is a smooth bounded domain in and . Assume that if , or that if .
Then, problem admits at least one solution. This solution (as well as every weak solution) belongs to for some .
Theorem 1.3 (A priori estimates of Gidas-Spruck type). Let and . Assume that is a smooth bounded domain and , .
Then there exists a constant , which depends only on and , such that every weak solution of the problem satisfies .
Theorem 1.6 (Symmetry results of Gidas-Ni-Nirenberg type). Assume that is a bounded smooth domain of in which is convex in the direction and symmetric with respect to the hyperplane . Let be Lipschitz continuous and be a solution of the problem.
Then is symmetric with respect to , i.e., for all . In addition, for .
In particular, if is a ball, then is radially symmetric, for , and it is decreasing, i.e., for .
The introduction contains a detailed review of earlier results and a comparison with the results obtained in the reviewed article.