The Sharp Apriori Estimates for Some Superlinear Degenerate Elliptic Problems

Abstract

The principal objective of this paper (talk) is the systematic development of the general integral identities and their applications to nonlinear elliptic problems.

Here we shall restrict our consideration to specific nonlinear elliptic problems of the “simple” form

$$\left\{ {_{u = 0 on \partial \Omega }^{\Delta u + f(x,u) = h(x) in \Omega \subset {\mathbb{R}^N}, N \geqslant 3}} \right.$$

containing such equations as

$$\Delta u + \frac{1}{{{{\left| x \right|}^\gamma }}}{\left| u \right|^{p - 2}}u = h\left( x \right)$$

with $2 < p < \frac{{2N}}{{N - 2}}and2\gamma > 2N - (N - 2)p$.

We shall establish the existence of the smoothness barrier for these solutions, we shall deduce the sharp estimates for these solutions, for the gradients of solutions and for the second order derivatives of solutions. These a priori estimates are based on new integral identities.

We construct examples from which it follows that these a priori estimates are unimprovable ones.