Behavior of solutions of semilinear elliptic equations in domains with complicated boundary

Abstract

In the paper, boundary value problems in a bounded domain for semilinear elliptic equations are considered. It is claimed that, for small exponent of nonlinearity, the support of a bounded solution is concentrated near the part of the boundary of the domain where the boundary condition is inhomogeneous. Estimates for the size of a neighborhood containing the support of the solution are established. For a supercritical exponent of nonlinearity, the convergence of solutions to a limit solution in the unperforated domain is established for some family of perforated domains. The rate of convergence is polynomial, and it depends on the exponent of the nonlinearity and on the rate at which the sizes of the cavities decrease simultaneously with the growth of the number of cavities and does not depend on the boundary conditions at the boundaries of the cavities. No restrictions on the displacement of the cavities are imposed.