Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications

Assume that I is the functional defined on the Hilbert space concerning the problem: in and u on , where f is sublinear at and superlinear at 0, that is, , and is the first eigenvalue of in . Under very general conditions, I has at least two local minimizers u ₁ and u ₂ and one mountain pass point u ₃ , and . Assuming that u ₁ , u ₂ and u ₃ are the only three nontrivial critical points of I, we prove that the level set I _b is contractible for all . Using this conclusion, we extend one of Hofer's result concerning existence of four nontrivial solutions of the above problem to the case where I is not and the trivial critical point 0 may be degenerate. Since I is not , the local topological degree and the critical groups of u ₃ can not be clearly computed. The lack of topological information about 0 and u ₃ makes it impossible to use topological degree theory or Morse theory in obtaining the fourth nontrivial solution. To overcome these difficulties, we explore a new technique in this paper.