We prove the existence of a solution for a class of activator–inhibitor system of type $- \Delta u +u = f(u) -v$ , $-\Delta v+ v=u$ in $\mathbb{R}^{N}$ . The function f is a general nonlinearity which can grow polynomially in dimension $N\geq 3$ or exponentiallly if $N=2$ . We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.

This paper is concerned with the existence of solutions for a class of elliptic equations on the unit ball with zero Dirichlet boundary condition. The nonlinearity is supercritical in the sense of Trudinger–Moser. Using a suitable approximating scheme we obtain the existence of at least one positive solution.

We study a class of parabolic equations which can be viewed as a generalized mean curvature flow acting on cylindrically symmetric surfaces with a Dirichlet condition on the boundary. We prove the existence of a unique solution by means of an approximation scheme. We also develop the theory of asymptotic stability for solutions of general parabolic problems.

In this paper we prove existence and qualitative properties of solutions for a nonlinear elliptic system arising from the coupling of the nonlinear Schrödinger equation with the Poisson equation. We use a contraction map approach together with estimates of the Bessel potential used to rewrite the system in an integral form.

We study the problem −∆pu = f(x, u) + t in Ω with Neumann boundary condition |∇u|p−2(∂u/∂v) = 0 on ∂Ω. There exists a t0 ∈ ℝ such that for t > t0 there is no solution. If t ≤ t0, there is at least a minimal solution, and for t < t0 there are at least two distinct solutions. We use the sub–supersolution method, a priori estimates and degree theory.

We show the existence and non-existence of positive solutions to a system of singular elliptic equations with the Dirichlet boundary condition.

We find positive rapidly decaying solutions for the equation

$$ -\text{div}(K(x)\nabla u)=K(x)u^{2^*-1}+\lambda K(x)|x|^{\alpha-2}u $$

in $\mathbb{R}^N$, where $N\geq3$, the nonlinearity is given by the critical Sobolev exponent $2^*=2N/(N-2)$, the weight is $K(x)=\exp(\tfrac14|x|^\alpha)$, $\alpha\geq2$ and $\lambda$ is a parameter.

We find positive solutions to a nonlinear equation of Klein–Gordon type. Our analysis is carried out by truncating the related functional and estimating mountain pass solutions by Moser’s iterative scheme.

There exists a set $\cal U$ in the plane, such that elements of $\cal U$ correspond to minimal stable solutions of a two-parameter elliptic system. For points on the boundary $\Upsilon$ of $\cal U$, there exist weak solutions to the elliptic system. The paper studies the properties of the curve $\Upsilon$.