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

Díaz Díaz, Jesús Ildefonso and Rakotoson, Jean Michel Theresien (2010) On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete and Continuous Dynamical Systems. Series A., 27 (3). pp. 1037-1058. ISSN 1553-5231

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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.

Item Type:Article
Uncontrolled Keywords:Very weak solutions; semilinear elliptic equations distance to the boundary; weighted spaces measure unbounded potentials
Subjects:Sciences > Mathematics > Differential geometry
Sciences > Mathematics > Topology
ID Code:15046
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