Potential theory and harmonic analysis methods for quasilinear and Hessian equations

Abstract

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems:-[delta]pu = uq + [mu], Fk[-u] = uq + [mu], u [greater than or equal to] 0, on Rn, or on a bounded domain [omega] [subset of or implied by] Rn. Here [delta]p is the p-Laplacian defined by [delta]pu = div ([delta]u [delta]u p-2), and Fk[u] is the k-Hessian defined as the sum of k x k principal minors of the Hessian matrix D2u (k = 1, 2, . . . , n); [mu] is a nonnegative measurable function (or measure) on [omega]. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data [mu] [such that element of] Ls([omega]), s [greater than] 1. Such results are deduced from our existence criteria with the sharp exponents s = (n(q-p+1)) / pq for the first equation, and s = (n(q-k) q-k) / 2kq for the second one. Furthermore, a complete characterization of removable singularities for each corresponding homogeneous equation is given as a consequence of our solvability results.