A multiscale method for semilinear elliptic equations

The Nehari Manifold for a Class of Indefinite Weight Semilinear Elliptic Equations

Singular Solutions for some Semilinear Elliptic Equations

We are concerned with the behavior of u near x = O. There are two distinct cases: 1) When p >= N / ( N -2) and (N ~ 3) it has been shown by BR~ZIS & V~RON [9] that u must be smooth at 0 (See also BARAS & PIERRE [1] for a different proof). In other words, isolated singularities are removable. 2) When 1-< p < N / ( N 2) there are solutions of (1) with a singularity at x ---0. Moreover all singula...

Topological Derivatives for Semilinear Elliptic Equations

1.1. Topological derivatives in shape optimization. Topological derivatives are introduced for linear problems in (Sokolowski and Zochowski, 1999) and for variational inequalities in (Sokolowski and Zochowski, 2005). The mathematical theory of asymptotic analysis is applied in (Nazarov and Sokolowski, 2003; 2006) for the derivation of topological derivatives in shape optimization of elliptic bo...

Some Limiting Situations for Semilinear Elliptic Equations

— The objective of this mini-course is to take a look at a standard semilinear partial differential equation −∆u = λf(u) on which we show the use of some basic tools in the study of elliptic equation. We will mention the maximum principle, barrier method, blow-up analysis, regularity and boot-strap argument, stability, localization and quantification of singularities, Pohozaev identities, movin...

Bubble towers for supercritical semilinear elliptic equations

Abstract : We construct positive solutions of the semilinear elliptic problem ∆u + λu + up = 0 with Dirichet boundary conditions, in a bounded smooth domain Ω ⊂ RN (N ≥ 4), when the exponent p is supercritical and close enough to N+2 N−2 and the parameter λ ∈ R is small enough. As p → N+2 N−2 , the solutions have multiple blow up at finitely many points which are the critical points of a functi...