Level set maxima and quasilinear elliptic equations (Q809267)

The paper studies an isoperimetric problem (1) \(\sigma (t)=\sup_{A(u)=t}B(u)\), where \(A(u)\) is a ``positive definite'' and \(B(u)\) is a ``completely continuous'' functional on a Banach space. It is found that \(\sigma\) (t) has left and right hand derivatives, \(0<\sigma '_ - (t)\leq \sigma '_+(t)\) and for every t there exist \(u_{\pm}\) such that \(A(u_{\pm})=t\), \(B(u_{\pm})=\sigma (t)\) and \((2)B'(u_{\pm})=\sigma '_{\pm}(t)A'(u_{\pm})\). The range of the eigenvalues \(\sigma '_{\pm}(t)\) is studied with application to quasilinear differential equations of form (2) in \(W^{1,p}\)-spaces providing solvability results.