Boundary blow-up for nonlinear elliptic equations with general growth in the gradient: an approach via symmetrisation

Abstract

In this paper we give a survey of some recent results obtained via symmetrization methods for solutions of elliptic equations in the form A(u) = H(x, u, Du), where the principal term is a laplacian-type operator and H(x, u, Du) grows with respect to Du at most like |Du|q , 1 ≤ q ≤ 2. In particular, it is considered the case where the solution blows up on the boundary and some comparison results are illustrated. Also an isoperimetric inequality for the so-called “ergodic constant” is given and the connections with the homogeneous Dirichlet problem for the quoted equations are discussed.

Keywords

• Rearrangements
• Blow-up solutions.