Finite-dimensional reduction of a supercritical exponent equation

Abstract

We present a finite-dimensional reduction for a supercritical exponent PDE. We reduce the existence of a solution of the problem

$$-\mathrm{\Delta }u=K|u{|}^{4∕\left(n-2\right)+\epsilon }u\text{ in }\mathrm{\Omega }\text{ (with }\epsilon >0\text{)},u=0\text{ on }\partial \mathrm{\Omega },$$

to finding a critical point of a function defined in some set $\mathcal{V}\subset {ℝ}^N\times {ℝ}^N\times {\mathrm{\Omega }}^N$.