We consider non-linear elliptic equations having a measure in the right-hand side, of the type $diva(x,Du)=\mu ,$ and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderón-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.

We consider a semilinear elliptic eigenvalues problem on a ball of ${\Re }^n$ and show that all the eigenfunctions and eigenvalues, can be obtained from the Lane-Emden function.

We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation $$-\Delta u+{\mathrm{e}}^u({\mathrm{e}}^u-1)=\mu \text{in}\Omega$$ with the Dirichlet boundary condition. Approximating $\mu$ by a sequence ${\left({\mu }_n\right)}_{n\in \mathbb{N}}$ of $L^1$ functions or finite signed measures such that this equation has a solution $u_n$ for each $n\in \mathbb{N}$, we are interested in establishing the convergence of the sequence ${\left(u_n\right)}_{n\in \mathbb{N}}$ to a function $u^{\#}$ and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by $u^{\#}$.