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A general variational identity. (English) Zbl 0625.35027
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

##### 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
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