On the existence of strong solutions to some semilinear elliptic problems

Abstract

We study the following semilinear elliptic problem: [GRAPHICS] where B is a ball in R-N, N greater than or equal to 3, a(ij) = a(ij) (x, r) epsilon C-0,C-1 ((B) over bar x R), a(ij), partial derivativea(ij)/partial derivativex(i), partial derivativea(ij)/partial derivativer, b(i), c epsilon L-infinity (B x R), with i, j = 1, 2,(...),N and c less than or equal to 0, and f epsilon L-p(B). For each p, p greater than or equal to N, there exists a strong solution u epsilon W-2,W-p(B) boolean AND W-0(1,p)(B) provided the oscillations of a(ij) with respect to r are sufficiently small. Moreover, for N/2 < p < N, if f(Lp) is small enough, then the existence result remains hold.