Let $u$ be a weak solution of a quasilinear elliptic equation of the growth $p$ with a measure right hand term $\mu$. We estimate $u\left(z\right)$ at an interior point $z$ of the domain $\Omega$, or an irregular boundary point $z\in \partial \Omega$, in terms of a norm of $u$, a nonlinear potential of $\mu$ and the Wiener integral of ${𝐑}^n\setminus \Omega$. This quantifies the result on necessity of the Wiener criterion.

The electric potential u in a solute of electrolyte satisfies the equation Δu(x) = f(u(x)), x ∈ Ω ⊂ ℝ³, ${u|}_{\partial \Omega }=0$. One studies the existence of a solution of the problem and its properties.

Consider a class of elliptic equation of the form $$-\Delta u-\frac{\lambda }{{\left|x\right|}^2}u=u^{2^{*}-1}+\mu u^{-q}\text{in}\Omega \setminus \left\{0\right\}$$ with homogeneous Dirichlet boundary conditions, where $0\in \Omega \subset {ℝ}^N$($N\ge 3$), $0<q<1$, $0<\lambda <{(N-2)}^2/4$ and $2^{*}=2N/(N-2)$. We use variational methods to prove that for suitable $\mu$, the problem has at least two positive weak solutions.

We shall prove an existence result for a class of quasilinear elliptic equations with natural growth. The model problem is $$\left\{\begin{array}{cc}-\text{div}\left(\left(1+{\left|u\right|}^r\right)\nabla u\right)+{\left|u\right|}^{m-2}u{\left|\nabla u\right|}^2=f\hfill & \text{in}\mathrm{\Omega }\hfill \\ u=0\hfill & \text{su}\partial \mathrm{\Omega }.\hfill \end{array}\right.$$

The existence of a positive radial solution for a sublinear elliptic boundary value problem in an exterior domain is proved, by the use of a cone compression fixed point theorem. The existence of a nonradial, positive solution for the corresponding nonradial problem is obtained by the sub- and supersolution method, under an additional monotonicity assumption.

For semilinear elliptic equations of critical exponential growth we establish the existence of positive solutions to the Dirichlet problem on suitable non-contractible domains.

We study the existence and nonexistence of positive solutions of nonlinear elliptic systems in an annulus with Dirichlet boundary conditions. In particular, $L^{\infty }$ a priori bounds are obtained. We also study a general multiple linear eigenvalue problem on a bounded domain.

The method of reliable solutions alias the worst scenario method is applied to the problem of von Kármán equations with uncertain initial deflection. Assuming two-mode initial and total deflections and using Galerkin approximations, the analysis leads to a system of two nonlinear algebraic equations with one or two uncertain parameters-amplitudes of initial deflections. Numerical examples involve (i) minimization of lower buckling loads and (ii) maximization of the maximal mean reduced stress.