RT Journal Article
T1 On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary
A1 Díaz Díaz, Jesús Ildefonso
A1 Rakotoson, Jean Michel Theresien
AB 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.
PB American Institute of Mathematical Sciences
SN 1553-5231
YR 2010
FD 2010-07
LK https://hdl.handle.net/20.500.14352/42148
UL https://hdl.handle.net/20.500.14352/42148
LA spa
NO Unión Europea. FP7
NO DGISPI (Spain)
NO UCM
DS Docta Complutense
RD 18 abr 2025