A min-max theorem for multiple integrals of the calculus of variations and applications
Summary: In this paper we deal with the existence of critical points for functionals defined on the Sobolev space \(W^{1,2}_0(\Omega)\) by \(f(v)= \int_\Omega J(x, v, Dv) dx\), \(v\in W^{1,2}_0(\Omega)\), where \(\Omega\) is a bounded, open subset of \(\mathbb{R}^N\). Since the differentiability can fail even for very simple examples of functionals defined through multiple integrals of calculus of variations, we give a suitable version of the Ambrosetti-Rabinowitz Mountain Pass theorem, which enables us to the study of critical points for functionals which are not differentiable in all directions. Then we present some applications of this theorem to the study of the existence and multiplicity of nonnegative critical points for multiple integrals of the calculus of variations.
Ambrosetti-Rabinowitz mountain pass theorem
scientific article; zbMATH DE number 817260