Some Weighted Inequalities on Polyharmonic Operators and Applications

This work is devoted to weighted inequalities on polyharmonic operators and their applications to the critical exponent problems of semilinear polyharmonic operator with Dirichlet boundary value conditions. Some newer weighted inequalities on polyharmonic operator are obtained. By applying these new weighted inequalities to the critical exponent problems, it obtains a preferable necessary condition for nontrivial radial solutions of the critical exponent problems of semilinear polyharmonic operator with Dirichlet boundary conditions existing. This preferable necessary condition extends the scope in which the critical exponent problems has nontrivial radial solutions to two times of that in Bernis' paper.