Applications of Morse Theory to Semilinear Elliptic Boundary Value Problems.

摘要

In this dissertation, we study the existence of solutions to the problem &cubl3;-Du=gx,u in W;u=0 on6W, where O is a smooth bounded domain in RNN≥2 and g:Wx R→R is a differentiable function with g(x,0) = 0 for all x ∈ O. By using minimax methods and Morse theory, we prove the existence of at least three nontrivial solutions for the case in which an asymmetric condition on the nonlinearity g is assumed. The first two nontrivial solutions are obtained by employing a cutoff technique used by Chang et al. For the existence of the third nontrivial solution, first we compute the critical group at infinity of the associated functional by using a technique used by Liu and Shaoping. The final result is obtained by using a standard argument involving the Morse relation. In the second part, we will assume that g( x,s) has a resonant behavior for large negative values of s and that the Landesmann-Lazer condition is satisfied. We also assume that g(x,s)