Existence and uniqueness of large positive solutions are obtained for some semilinear elliptic Dirichlet problems in bounded smooth domains Ω with a large parameter λ. It is shown that the large positive solution has flat core. The distance of its flat core to the boundary ∂Ω is exactly measured as λ→∞.

Sufficient conditions ensuring weak lower semi-continuity of the optimal value function are presented. To this end, refined inner semi-continuity properties of set-valued maps are introduced which meet the needs of the weak topology in Banach spaces. The results are applied to prove the existence of solutions in various worst-case robust optimal control problems governed by semilinear elliptic ...

Abstract. The aim of this paper is to give an existence theorem for a semilinear equation of evolution in the case when the generator of semigroup of operators depends on time parameter. The paper is a generalization of [2]. Basing on the notion of a measure of noncompactness in Banach space, we prove the existence of mild solutions of the equation considered. Additionally, the applicability of...

This article concerns the existence of positive solutions of semilinear elliptic system −∆u = λa(x)f(v), in Ω, −∆v = λb(x)g(u), in Ω, u = 0 = v, on ∂Ω, where Ω ⊆ RN (N ≥ 1) is a bounded domain with a smooth boundary ∂Ω and λ is a positive parameter. a, b : Ω → R are sign-changing functions. f, g : [0,∞) → R are continuous with f(0) > 0, g(0) > 0. By applying LeraySchauder fixed point theorem, w...

We develop monotone iterative technique for a system of semilinear elliptic boundary value problems when the forcing function is the sum of Caratheodory functions which are nondecreasing and nonincreasing, respectively. The splitting of the forcing function leads to four different types of coupled weak upper and lower solutions. In this paper, relative to two of these coupled upper and lower so...

We review some recent results on construction of entire solutions to the classical semilinear elliptic equation ∆u+u−u = 0 in R . In various cases, large dilations of an embedded, complete minimal surface approximate the transition set of a solution that connects the equilibria ±1. In particular, our construction answers negatively a celebrated conjecture by E. De Giorgi in dimensions N ≥ 9.

In this paper, we deal with a class of semilinear elliptic equation in a bounded domain Ω ⊂ R , N ≥ 3, with C boundary. Using a new fixed point result of the Krasnoselskii’s type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.

We give here an extension of the recent result of Kwong (which in turn extended earlier results of Cooman and McLeod and Serrin) on the uniqueness of the positive radial solution of a semilinear elliptic equation. When reduced to the special case considered by Kwong, our proof is shorter.

A semilinear elliptic boundary-value problem on bounded multiconnected domains is studied. The authors prove that under suitable conditions, the problem may have no solutions in certain cases and many have one or two nonnegative solutions in some other cases. The radial solutions were also studied in annular domains.

We consider the uniqueness of the positive solution to a semilinear elliptic equation with Dirichlet boundary condition and the nonlinearity satisfying f(0) < 0 and having asymptotic sublinear growth rate. A similar idea is also applied to the nonexistence of a positive solution to a superlinear problem.