In this paper we consider a semilinear elliptic system of FitzHugh-Nagumo type on a bounded domain with the same diffusion constant $\lambda^{-1}$ under the Dirichlet boundary condition $-\Delta u=\lambda(f(u)-v)$, $-\Delta v=\lambda(\delta u-\gamma v)$ in $\Omega$. In some parameter range on $(\delta, \gamma)$ this system has at least two nontrivial solutions when $\lambda$ is sufficiently large, and these solutions are obtained by variational methods. We study the asymptotic profiles of these solutions for large $\lambda$ in some parameter range on $(\delta, \gamma)$, especially for small $\delta$ and large $\gamma$.

We study the decay estimates of the energy for the wave equation in an exterior domain with a localized dissipation. The dissipative term consists of the following two parts: The first part may be nonlinear and localized in a suitable bounded area, while the second part is linear in the outside of a big ball. So we may call such a dissipation ``half-linear" dissipation. We note that no geometrical condition is imposed on the boundary. As an application of the decay estimates we prove some global existence theorems for the wave equation with a nonlinear source term.

We consider steady, inviscid axisymmetric vortex flows with swirl in a bounded channel, possibly with obstacles. Such flows can be obtained by solving the nonlinear equation \begin{equation} -\frac{\partial ^2\psi }{\partial z^2}-r\frac \partial {\partial r}(\frac 1r\frac{\partial \psi }{\partial r})=r^2f(\psi )+h(\psi ), \tag*{(0.1)} \end{equation} where $f$ and $h$ are given functions of the stream function $\psi$, with $\psi$ prescribed on the boundary of the flow domain. We use an iterative procedure to prove the existence of solutions to this Dirichlet problem under certain conditions on $f$ and $h$. Solutions are not unique, and relations between different families of solutions are obtained. These families include not only vortex rings with swirl, but also flows with tubular regions of swirling vorticity as occur in models of vortex breakdown.

In this paper, we consider the nonlinear elliptic equation of the following type: $$ -\frac{1}{2} \Delta u(x) + \nabla w(x)\cdot \nabla u(x) + [\lambda + H(x,u(x))]u(x) = f(x),\qquad x \in {\mathbb R}^d, $$ where $\lambda$ is a given constant and $f$, $H$, and $w$ are given functions, respectively. The derivative $\nabla w$ of the function $w$ is unbounded on ${\mathbb R}^d$. Our purpose is to show the existence of a solution to this equation without sign conditions on $H$. Therefore, we can treat even the case that $H$ is unbounded below on ${\mathbb R}^d$. This is due to the existence of the term $\nabla w\cdot \nabla u$.

We study the nonlinear Schr\"{o}dinger equation \[ -\Delta u+\lambda a(x)u=\mu u+u^{2^{\ast }-1},\hbox{ \ }u\in \mathbb{R}^{N}, \] with critical exponent $2^{\ast }=2N/(N-2),$ $N\geq 4,$ where $a\geq 0$ has a potential well and is invariant under an orthogonal involution of $\mathbb{R} ^{N}$. Using variational methods we establish existence and multiplicity of solutions which change sign exactly once. These solutions localize near the potential well for $\mu $ small and $\lambda $ large.

The critical set generated by the Mountain-Pass Theorem involving functionals which are the sum of a locally Lipschitz continuous function and a convex, proper, lower semicontinuous function is studied. The classical Pucci-Serrin structure theory is reformulated in this framework.

In this paper, we considerably improve results of Ni, Takagi, and Wei on the shape of the mountain-pass solution and the location of its maximum for certain boundary-value problems for certain weakly nonlinear elliptic equations with small diffusion, on domains in $\mathbb R^2$ or $\mathbb R^3$. We remove geometric conditions on the domain and nondegeneracy and uniqueness assumptions on the solutions.