In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary‎. ‎We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose ...

We consider three types of semilinear second order PDEs on a cylindrical domain Ω × (0,∞), where Ω is a bounded domain in N , N 2. Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of Ω × (0,∞) is reserved for time t, the third type is an elliptic equation with a singled out unbounded variable t. We discuss the asymptotic behavior, as t ...

A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two. The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern–Simons Higgs model and have been studied recently by various authors.

The finite element approximation of optimal control problems for semilinear elliptic partial differential equation is considered, where the control belongs to a finite-dimensional set and state constraints are given in finitely many points of the domain. Under the standard linear independency condition on the active gradients and a strong second-order sufficient optimality condition, optimal er...

We study properties of the semilinear elliptic equation ∆u = 1 u on domains in R, with an eye toward nonnegative singular solutions as limits of positive smooth solutions. We prove the nonexistence of such solutions in low dimensions when we also require them to be stable for the corresponding variational problem. The problem of finding singular solutions is related to the general study of sing...

We use a shadowing-type lemma in order to analyze the singular, semilinear elliptic equation describing static self-dual abelian Higgs vortices. Such an approach allows us to construct new solutions having an infinite number of arbitrarily prescribed vortex points. Furthermore, we obtain the precise asymptotic profile of the solutions in the form of an approximate superposition rule, up to an e...

We consider the following semilinear elliptic equation with singular nonlinearity: u ? 1 u + h(x) = 0 in where > 1; h(x) 2 C 1 (() and is an open subset in R n ; n 2. Let u 2 C 0 (() be a nonnegative nite energy stationary solution and = fx 2 ju(x) = 0g be the rupture set of u. We show that the Hausdorr dimension of is less than or equal to (n?2)+(n+2) +1 .