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 will consider the following problem$$-\Delta u-\lambda \frac{u}{{\left|x\right|}^2}={\left|\nabla u\right|}^p+cf,u\>0\text{in}\Omega ,$$where $\Omega \subset {ℝ}^N$ is a domain such that $0\in \Omega$, $N\ge 3$, $c\>0$ and $\lambda \>0$. The main objective of this note is to study the precise threshold $p_{+}=p_{+}\left(\lambda \right)$ for which there is novery weak supersolutionif $p\ge p_{+}\left(\lambda \right)$. The optimality of $p_{+}\left(\lambda \right)$ is also proved by showing the solvability of the Dirichlet problem when $1\le p\<p_{+}\left(\lambda \right)$, for $c\>0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe.

In this paper, we are concerned with the asymptotically linear elliptic problem -Δu + λ0u = f(u), u ∈ H01(Ω) in an exterior domain Ω = RnO (N ≥ 3) with O a smooth bounded and star-shaped open set, and limt→+∞ f(t)/t = l, 0 < l < +∞. Using a precise deformation lemma and algebraic topology argument, we prove under our assumptions that the problem possesses at least one positive solution.

Necessary and sufficient conditions have been found to force all solutions of the equation $${\left(r\left(t\right)y^{\text{'}}\left(t\right)\right)}^{(n-1)}+a\left(t\right)h\left(y\left(g\left(t\right)\right)\right)=f\left(t\right),$$ to behave in peculiar ways. These results are then extended to the elliptic equation $${\left|x\right|}^{p-1}\Delta y\left(\right|x\left|\right)+a\left(\right|x\left|\right)h\left(y\right(g\left(\right|x\left|\right)\left)\right)=f\left(\right|x\left|\right)$$ where $\Delta$ is the Laplace operator and $p\ge 3$ is an integer.

The paper is devoted to the description of some connections between the mean curvature in a distributional sense and the mean curvature in a variational sense for several classes of non-smooth sets. We prove the existence of the mean curvature measure of $\partial E$ by using a technique introduced in [4] and based on the concept of variational mean curvature. More precisely we prove that, under suitable assumptions, the mean curvature measure of $\partial E$ is the weak limit (in the sense of distributions) of the mean...

In this note we prove the existence of extremal solutions of the quasilinear Neumann problem $-\left(\right|x^{{}^{\text{'}}}\left(t\right){{|}^{p-2}{x}^{{}^{\text{'}}}\left(t\right))}^{{}^{\text{'}}}=f(t,x\left(t\right),x^{{}^{\text{'}}}\left(t\right))$, a.e. on $T$, $x^{{}^{\text{'}}}\left(0\right)=x^{{}^{\text{'}}}\left(b\right)=0$, $2\le p<\infty$ in the order interval $[\psi ,\varphi ]$, where $\psi$ and $\varphi$ are respectively a lower and an upper solution of the Neumann problem.