On the Validity of the Euler–Lagrange Equation

Under some regularity assumptions on the boundary datum u0 (assumptions automatically satisfied in the classical case when the growth of the integrand is bounded by a+b ξ p), we prove the validity of the Euler–Lagrange equation for the functional∫Ω [f( u(x) )+g(x, u(x))] dxunder general growth assumptions on f, for instance for f( ξ )=P( ξ ) eK ξ with P a nontrivial polynomial and K 0, or for f ( ξ )= ξ ξ .