Pucci, Patrizia; Serrin, James A general variational identity. (English) Zbl 0625.35027 Indiana Univ. Math. J. 35, 681-703 (1986). The authors consider extremals of the general variational problem, \(\delta \int_{\Omega}F(x,u,Du)du=0\) which are \(C^ 2\) and satisfy the Euler-Lagrange equation, \(div(F_ p(x,u,Du))=F_ u(x,u,Du),\quad x\in \Omega.\) Under certain assumptions on F, the Euler-Lagrange equation has no nontrivial solution \(C^ 2(\Omega)\cap C^ 1(\Omega)\) which vanishes on \(\partial \Omega\). Similar results are established for higher-order extremals and for vector-valued extremals, which gives some new results for certain elliptic systems. Reviewer: S.M.Lenhart Cited in 7 ReviewsCited in 222 Documents MSC: 35J50 Variational methods for elliptic systems 49J20 Existence theories for optimal control problems involving partial differential equations 35A15 Variational methods applied to PDEs 35J60 Nonlinear elliptic equations Keywords:extremals; Euler-Lagrange equation; higher-order extremals; vector-valued extremals PDF BibTeX XML Cite \textit{P. Pucci} and \textit{J. Serrin}, Indiana Univ. Math. J. 35, 681--703 (1986; Zbl 0625.35027) Full Text: DOI OpenURL