Is it possible to prove a Pohozaev identity and the related non-existence result for non-trivial critical points of the functional
$$\int_\Omega \left(A(x,u,\nabla u) -\frac{\lambda}{2} |u|^{2} - \frac{1}{2^*}|u|^{2^*} \right) dx,$$ where $2^*$ is the critical Sobolev exponent for the embedding of $H^1_0(\Omega)$ into Lebesgue spaces?
Note that the functional above is the Euler-Lagrange functional of a nonlinear elliptic problem.
Note that the classical Pohozaev result covers the case $A(x,u,\nabla u) = \frac{1}{2}|\nabla u|^2$.