A RELATION BETWEEN THE SIZE OF A DOMAIN AND THE SOLVABILITY OF THE DIRICHLET PROBLEMS FOR SEMILINEAR ELLIPTIC-EQUATIONS ON THE DOMAIN

We establish a relationship between the solvability of Dirichlet probl ems for semilinear elliptic equations on a domain and the measure of t he set where the nonlinear term is positive. Given a nonlinear term, a n upper bound for volumes of domains on which we can solve Dirichlet p roblems for the semilinear equation is obtained. The upper bound can b e estimated explicitly by data depending only on the nonlinear term. C onsequently, various kinds of existence results for Dirichlet problems for semilinear elliptic equations are proved. In our results, the cor responding functionals to Dirichlet problems may not satisfy the Palai s-Smale condition.