Morse theory for indefinite nonlinear elliptic problems

Abstract

Using the heat flow as a deformation, a Morse theory for the solutions of the nonlinear elliptic equation:

in a bounded domain $\Omega \subset {\mathbf{R}}^N$ with the Dirichlet boundary condition is established, where $a_{\pm }⩾0$, $\sup \mathrm{p}(a_{-})\cap \sup \mathrm{p}(a_{+})=\varnothing$, $\sup \mathrm{p}(a_{+})\ne \varnothing$, $1<q<2^{\ast }-1$ and $p>1$. Various existence and multiplicity results of solutions are presented.