Nonlinear eigenvalue problems for even functionals

Abstract

Let H be a Hilbert space and let g∈C ¹(H,ℝ) be an even Fréchet differentiable functional with completely continuous derivative. We introduce maximin values σ _k (t) which are critical values of g restricted to the sphere

and show that the functions σ _k (t) have right and left derivatives and that are eigenvalues of g', i.e. there exist such that

Applications of the result are given to semilinear elliptic equations.

Keywords:

• Nonlinear eigenvalue problem
• Krasnoselskii genus
• Elliptic semilinear equations

Keywords:

• 2000 Mathematics Subject Classifications: Primary 49J35
• 49J40
• Secondary 35J20
• 35J25