Perturbation theory and a topological approach to non-symmetric elliptic equations

Abstract

In this paper, we deal with the equations of the type {$-\Delta u =

u

^{p-1}u - f$ in $\Omega$, $u = 0

\partial \Omega$ $\Omega bounded domain in R^N$}

where $p \in (1, {N+2}/{N-2})$ with $N \geq 3$ and $f \in L^2(\Omega)$. Judging by the case of ordinary differential equations, it is predicted that the above equations have infinitely many solutions. However, in general the proof of this conjecture remains open. There were two types of approach to solve this problem. The first one is using the index theory and minimax method, and the other one is using the concepts of the algebraic topology, in particular, the Lefschetz-Hopf fixed point theorem. Both of these approaches partially solved the problem. The purpose of this paper is that to introduce and review these two approaches together.