Existence of strong solutions to some quasilinear elliptic problems on bounded smooth domains

Abstract

We consider the following quasilinear elliptic problems in a bounded smooth domain Z of R-N, N greater than or equal to 3: Lu = Sigma(i,j=1)(N) a(ij)(x,u) delta(2)u/deltax(i)deltax(j) + Sigma(i=1)(N)b(i)(x,u)deltau/deltax(i) + c(x, u)u = f (x) in Z, u = 0 on deltaZ, where f (x) is an element of L-p(Z) and all the coefficients a(ij), b(i), c are Carathedory functions. Suppose that a(ij) is an element of C-0,C-1((Z) over bar x R), a(ij), deltaa(ij)/deltax(i), deltaa(ij)/deltar, b(i), c is an element of L-infinity(Z x R), c less than or equal to 0 for i, j = 1,...N and the oscillations of a(ij) = a(ij)(x,r) with respect to r are sufficiently small. A global estimate for a solution u is an element ofW(2,p)(Z) boolean AND W-0(1,p)(Z) is established and the existence of a strong solution u is an element of W-2,W-p(Z) boolean AND W-0(1,p)(Z) is proved for p > N. Furthermore, we replace f(x) by f(x,r,xi) which is defined on Z x R x R-N and is a Carathedory function. Assume that [f(x, r, xi)] less than or equal to C-0 + h([r])[xi]theta, 0 less than or equal to theta < 2, where C-0 is a nonnegative constant, h([r]) is a locally bounded function, and -c greater than or equal to alpha(0) > 0 for some constant alpha(0). We prove the existence of solution u is an element of W-2,W-p(Z) boolean AND W-0(1,p)(Z) for the equation Lu = f (x, u, delu).