The Ljapunov–Schmidt Reduction for Some Critical Problems

Abstract

We study a class of problems which are perturbation of some critical problems, namely

• the Brezis-Nirenberg problem

  $$-\Delta{u}=|u|^{2^{*}-2}u\;+\;\epsilon{u}\;\mathrm{in}\;\Omega,\quad u=0\;\mathrm{on}\;\partial\Omega$$

• the almost-critical problem

  $$-\Delta{u}=|u|^{2^{*}-2-\epsilon}u\;\mathrm{in}\;\Omega,\quad u=0\;\mathrm{on}\;\partial\Omega$$

• the Coron problem

  $$-\Delta{u}=|u|^{2^{*}-2}u\;\mathrm{in}\;\Omega\;\backslash B(x_{0},\epsilon),\quad u=0\;\mathrm{on}\;\partial(\Omega\backslash B(x_{0},\epsilon)),$$

where Ω is an open bounded domain in $\mathbb{R}^n,\;n\geq\;3,2^{*}=\;\frac{2n}{n-2},\epsilon$ is a small real parameter and $x_{0}\in\Omega$.In particular, we prove existence and multiplicity of positive and sign changing solutions which blow-up or blow-down at one or more points of the domain as the parameter є goes to zero. The main tool is the Ljapunov–Schmidt reduction method.

Mathematics Subject Classification (2010). 35B40, 35J20, 35J55.