On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary

Abstract

We prove the existence of an appropriate function (very weak solution) u in the Lorentz space L(N') (,infinity)(Omega), N' = (N)(N - 1) satisfying Lu - Vu + g (x, u, del u) = mu in Omega an open bounded set of R(N), and u = 0 on partial derivative Omega in the sense that (u, L phi)(0) - (Vu, phi)(0) + (g(., u, del u),phi)(0) = mu(phi), for all phi is an element of C(c)(2)(Omega). The potential V <= lambda < lambda(1) is assumed to be in the weighted Lorentz space L(N,1)(Omega, delta), where delta(x) = dist (x, partial derivative Omega), mu is an element of M(1)(Omega, delta), the set of weighted Radon measures containing L(1)(Omega, delta), L is an elliptic linear self adjoint second order operator, and lambda(1) is the first eigenvalue of L with zero Dirichlet boundary conditions. If mu is an element of L(1)(Omega, delta) we only assume that for the potential V is in L(loc)(1) (Omega), V <= lambda < lambda(1). If mu is an element of M(1)(Omega, delta(alpha)), alpha is an element of [0, 1[, then we prove that the very weak solution vertical bar del u vertical bar is in the Lorentz space L(N/N-1+alpha,infinity)(Omega). We apply those results to the existence of the so called large solutions with a right hand side data in L(1)(Omega, delta). Finally, we prove some rearrangement comparison results.