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

Abstract

Assume that I is the functional defined on the Hilbert space $ H_0^1(\Omega) $ concerning the problem: $ -\Delta u=f(u) $ in $ \Omega $ and u on $ \partial\Omega $ , where f is sublinear at $ \infty $ and superlinear at 0, that is, $ \limsup_{|t|\to\infty}f(t)/t \lambda_1 $ , and $ \lambda_1 $ is the first eigenvalue of $ -\Delta $ in $ H_0^1(\Omega) $ . Under very general conditions, I has at least two local minimizers u 1 and u 2 and one mountain pass point u 3 , and $ \max\{I(u_1),\ I(u_2)\} $<$ I(u_3)<0 $ . 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 $ b\geq I(u_3) $ . 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 $ {\cal C}^2 $ and the trivial critical point 0 may be degenerate. Since I is not $ {\cal C}^2 $ , 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.