A generalized Palais-Smale type compactness condition is applied to prove the existence of critical points of functionals (summation convection) \[ E(u)=frac{1}{2}\int_{\Omega}a^{\alpha \beta}(x,u)\partial_{\alpha}u^ i\partial_{\beta}u^ idx\quad on\quad H_ 0^{1,2}(\Omega,{\mathbb{R}}^ N) \] subject to a nonlinear constraint \(G(u)=1\). The results obtained extend well-known results on semilinear elliptic eigenvalue problems to variational problems of the type \[ -\partial_{\alpha}(a^{\alpha \beta}(x,u)\partial_{\beta}u^ i)=f^ i(x,u,\nabla u);\quad u|_{\partial \Omega}=0,\quad 1\leq i\leq N. \]