ON STRONG SOLVABILITY OF THE DIRICHLET PROBLEM FOR A CLASS OF SEMILINEAR ELLIPTIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

We study a strong solvability of the Dirichlet problem Sigma(n)(i,j=1) a(ij)(x)u(xixj) + g(x, u) = f(x), x is an element of Omega, u vertical bar(partial derivative Omega) = 0 for a class of semilinear elliptic equations with discontinuous coefficients satisfying the Cordes condition. For this problem we get the existence results in (W) over dot(2)(2) (Omega) Sobolev space whenever the norm parallel to f parallel to(L2(Omega)) is sufficiently small. We also have proved the strong solvability of the Dirichlet problem in space (W) over dot(p)(2) (Omega) with 1 < p < infinity for the equation considered above with continuous leading coefficients and a small parallel to f parallel to(Lp(Omega)) norm.