On existence of variational multipliers \( M=M(x,u(x),D_ku(x)) \) and extremal variational principles for partial equations (Q2719445)

The paper addresses the following problem: Find for the given equation NEWLINE\[NEWLINE N[u]=f,\tag{1} NEWLINE\]NEWLINE where \(N\) is an operator acting in some Hilbert space \(H\) \((N: D(N)\subset H\to R(N)\subset H)\), \(\overline{D(N)} = H\) with a scalar product \((.,.)\), the variational multiplier \(B\) and the functional \(\Phi\) so that NEWLINE\[NEWLINE \delta\Phi[u] = (B^*(N[u]-f),\delta u),\quad u\in D(N) NEWLINE\]NEWLINE holds and \(B^*\) is a reversible operator. It is proved that the class of Euler-Lagrange functions is not sufficient for obtaining the semibounded solutions of converse problems of the variational calculus for some classes of partial equations.