We give existence results and a priori estimates for a semilinear elliptic problem of the form \begin{equation*} \left\{ \begin{array}{l} -\Delta w=w^{Q}+\mu ,\qquad \text{in \thinspace }\Omega , \\ w=\lambda ,\qquad \qquad \qquad \quad \text{on }\partial \Omega , \end{array} \right. \end{equation*} where $Q>0,$ and $\mu $ and $\lambda $ are nonnegative Radon measures in $ \Omega $ and $\partial \Omega ,$ with $\int_{\Omega }\rho \,d\mu <+\infty ,$ where $\rho $ is the distance to $\partial \Omega .$ We extend the results to the case of systems \begin{equation*} \left\{ \begin{array}{l} -\Delta u=v^{p}+\mu ,\qquad -\Delta v=u^{q}+\eta ,\qquad \text{in }\Omega , \\ u=\lambda ,\qquad v=\kappa ,\qquad \text{on }\partial \Omega , \end{array} \right. \end{equation*} with $p,q>0,$ and the same assumptions on $\eta $ and $\kappa .$

We investigate the existence of sign-changing radial solutions for a class of singular equations: $$ -\Delta u(x)+ b(|x|)u(x) = |u(x)|^{\theta-1}u(x)+h(|x|)\hspace{1cm}x\in\mathbb R^N $$ where $b(|x|)$ may change sign and behaves like $|x|^{-\alpha}$ at infinity for some $\alpha\in(0,2)$, and $\theta>1$.

A general expression is derived for the isothermal minimum free energy of a compressible viscoelastic fluid with linear dependence on the history of strain and nonlinear dependence on the density. This involves the solution of a Wiener--Hopf integral equation, for which an existence and uniqueness theorem is proved. The factorization of a particular material tensor, which is fundamental to the methodology, can be carried out explicitly in this case.

In this note, we consider the following problem: \begin{eqnarray*} \left \{ \begin{array}{l} {\Delta} u = 2 u_x \wedge u_y \quad \mbox{in} \; \Omega, \quad u \in {H_0^1(\Omega ; {\bf R}^3)}, \\ u |_{\partial \Omega} = 0, \end{array} \right. \end{eqnarray*} where $\Omega \subset {\bf R}^2$ is a smooth bounded domain. We show that if the domain $\Omega$ is conformal equivalent to a $(K+1)$-ply connected domain satisfying some conditions, then the problem has at least $K$ distinct non-trivial solutions.