Nonlinear Neumann problems with constraints (Q2868894)

Let \(\Omega\) be a bounded domain with smooth boundary. For any \(1<p<+\infty\), let \(W_n^{1,p}(\Omega)\) denote the ``Neumann'' Sobolev space, that is, the completion of the space NEWLINE\[NEWLINEC^1_n(\overline\Omega):=\left\{ u\in C^1(\overline\Omega);\;\frac{\partial u}{\partial n}=0\;\text{on }\;\partial\Omega\right\}NEWLINE\]NEWLINE with respect to the norm of the Sobolev space \(W^{1,p}(\Omega)\). Let \(M\) be a \(C^1\)-manifold in \(W_n^{1,p}(\Omega)\) of codimension 1. Let \(\psi: W_n^{1,p}(\Omega)\rightarrow {\mathbb R}\) a functional of class \(C^1\).NEWLINENEWLINEThe main result of this paper establishes that any local \(C^1_n(\overline\Omega)\)-minimizer of \(\psi_{|M}\) is also a local \(W_n^{1,p}(\Omega)\)-minimizer. This abstract result is used in the final part of this paper to show that a nonlinear parametric problem driven by the \(p\)-Laplace operator and restricted on a sphere, has at least three distinct smooth solutions. The proofs combine several refined tools in critical point theory and the calculus of variations.NEWLINENEWLINEThe present paper is very well written and it opens perspectives to further results in the qualitative analysis of solutions of quasilinear elliptic equations with Neumann boundary condition.