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

We prove the existence of an appropriate function (very weak solution) $u$ in the Lorentz space $L^{N',\infty}(\Omega), \ N'=\frac N{N-1}$ satisfying $Lu-Vu+g(x,u,\nabla u)=\mu$ in $\Omega$ an open bounded set of $\R^N$, and $u=0$ on $\partial\Omega$ in the sense that

$(u,L\varphi)_0-(Vu,\varphi)_0+(g(\cdot,u,\nabla u),\varphi)_0=\mu(\varphi),\quad\forall\varphi\in C^2_c(\Omega).$

The potential $V \le \lambda < \lambda_1$ is assumed to be in the weighted Lorentz space $L^{N,1}(\Omega,\delta)$, where $\delta(x)= dist(x,\partial\Omega),\ \mu\in M^1(\Omega,\delta)$, 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 $\mu\in L^1(\Omega,\delta)$ we only assume that for the potential $V$ is in L¹_loc$(\Omega)$, $V \le \lambda<\lambda_1$. If $\mu\in M^1(\Omega,\delta^\alpha),\ \alpha\in$[$0,1[$[, then we prove that the very weak solution $|\nabla u|$ is in the Lorentz space $L^{\frac N{N-1+\alpha},\infty}(\Omega)$. 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.