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: −\mathrm{\Delta }u−\lambda u = a_{ + }(x)|u|^{q−1}u−a_{−}(x)|u|^{p−1}u + h(x,u) in a bounded domain \Omega \subset \mathbf{R}^{N} with the Dirichlet boundary condition is established, where a_{ \pm }⩾0 , \mathrm{\sup p}(a_{−}) \cap \mathrm{\sup p}(a_{ + }) = \emptyset , \mathrm{\sup p}(a_{ + }) \neq \emptyset , 1 < q < 2^{ \ast }−1 and p > 1 . Various existence and multiplicity results of solutions are presented.