Given a connected, bounded open set $\Omega_1 \subset \mathbb{R}^n$, we use a maximum principle, and compactness arguments to study the properties of the function $P(u,x)$ in (1.5) below associated to a weak solution of the exterior $p$-capacitary problem, \[ \hbox{\rm div\,}(|Du|^{p-2}Du) = 0\ \text{in} \ \Omega=\mathbb{R}^n \setminus \overline{\Omega_1}, \quad \quad 1<p<n, \] $u=1$ on $\partial{\Omega_1}$, $u(x)\to 0$ as $|x|\to \infty$. As a consequence of our results we prove spherical symmetry for the solution $u$ and for the condenser $\Omega_1$ when the overdetermined boundary condition $|Du|=c>0$ on $\partial\Omega_1$ is imposed. This provides a new proof of a recent result of Reichel \cite{31}.

Radial solutions of the Gelfand equation on an $N$--dimensional ball are studied for $3\le N\le 9$. It is shown that global classical solutions are uniformly bounded while unbounded global $L^1$--solutions are constructed for some parameter range.

We consider the problem \begin{alignat}{2} u_t-\Delta u^m+bu^\beta & =0, &\quad& (x,t)\ \epsilon\ Q_T =\Omega\times(0,T]\\ u(x,t) & =0, &\quad& (x,t)\ \epsilon\ S_T=\partial\Omega\times(0,T]\\ u(x,0) & =u_0(x), &\quad& x\ \epsilon\ \Omega, \end{alignat} where $m\geq 1,$ $ b \in R^1,$ $ \beta>0, $ $ T>0;$ $ \Omega$ is a bounded, connected domain in $R^N$ with compact boundary, which is assumed to be piecewise smooth and to satisfy the exterior sphere condition. Let $u_0 \in C(\bar\Omega),$ $ u_0>0$ for $x \in \Omega$ and $u_0=0$ for $x\in \partial\Omega$. Assume also that $u_0$ is smooth near $\partial\Omega$. We show that the small time behaviour of the solution near the boundary $\partial\Omega$ depends on the relative strength of the diffusion and reaction terms near the boundary, that is to say on the function $$ \gamma(\bar x)=\lim\limits_{x\to\bar x} \bigl(\Delta u_0^m\big/bu^\beta_0\bigr),\quad\bar x \in \partial\Omega,\quad x\in \Omega. $$ We are essentially interested in the initial development of the interface between the dead core and the positive $u(x,t)$ field. In all possible cases, the small time behaviour of the interface is found, together with the local solution.

The existence of the solution to an elliptic system arising in electrochemical modelling is proven here. The elliptic system of interest here is composed of two diffusion equations; one of them is posed with a Dirichlet condition which couples it to the other equation on an interface, and a Signorini condition on one of the boundaries. The other one is posed with Neumann conditions, which are also coupled at an interface. The existence of the solution is proven by using Schauder's fixed point theorem, which requires some previous local regularity properties of the solution to the ``Signorini problem".

For the problem $$ \begin{align} \Delta u + \lambda f(u) &= 0 \quad\text{ in } \ \Omega, \;\ \lambda \in {\mathbf {R}} ^{+}, \\ u & = 0 \quad\text{ on } \ \partial \Omega, \end{align} $$ if $\Omega$ and $f$ satisfy certain hypotheses, a parameterized curve of positive solutions $\alpha\mapsto (u(\alpha),\lambda(\alpha))$ has been shown to exist, where $\alpha=max_{\Omega}u$. If $\Omega\subset{\mathbf {R}}^{n}$ is translated by $1/\epsilon$ and then rotated about a coordinate axis to obtain a new domain $\Omega_{\epsilon}\subset{\mathbf {R}}^{n+1}$, it can be shown that a surface of positive rotationally invariant solutions $(\alpha,\epsilon)\mapsto (\hat{u}(\alpha,\epsilon),\hat{\lambda}(\alpha,\epsilon))$ exists for the resulting problem $$ \begin{align} \Delta_{\epsilon}\hat{u} + \hat{\lambda} f(\hat{u}) & = 0 \quad\text{ in } \ \Omega_{\epsilon}, \ \hat{\lambda} \in {\mathbf {R}} ^{+} \\ \hat{u} & = 0 \quad\text{ on } \ \partial \Omega_{\epsilon}, \end{align} $$ where $\Delta_{\epsilon}$ is the Laplacian in the new variables and $(\hat{u}(\alpha,0),\hat{\lambda}(\alpha,0))=(u(\alpha),\lambda(\alpha))$. From this, we can give various examples of problems on domains with a large hole for which the structure of solutions can be well described.

We prove general optimality principles for semicontinuous viscosity solutions of Hamilton-Jacobi equations. We also characterize the minimal nonnegative supersolution and the maximal subsolution null on a closed given set for a class of equations without uniqueness, including the degenerate eikonal equation and the Bellman equation of the linear quadratic control problem.