A priori estimates for a class of degenerate elliptic equations

In this paper we investigate the regularity of solutions for the following degenerate partial differential equation {-Delta pu + u = f in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, when f is an element of L-q(Omega), p > 2 and q >= 2. If u is a weak solution in W-1,W-p (Omega), we obtain estimates for u in the Nikolskii space N-1+2/r,N-r (Omega), where r = q(p - 2) + 2, in terms of the L-q norm of f. In particular, due to embedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that parallel to u parallel to(r)(W1+2/r-epsilon,r(Omega)) <= C (parallel to f parallel to(q)(Lq(Omega)) + parallel to f parallel to(r)(Lq(Omega)) + parallel to f parallel to(2r/p)(Lq(Omega)) for every epsilon > 0 sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in W-1,W-r (Omega).