On the coercitive solvability of the non-linear Laplace–Beltrami equation in Hilbert space

The problem of separability of differential operators is considered for the first time in the works of V. N. Everitt and M. Hirz. Further development of this theory belongs to K. H. Boymatov, M. Otelbaev and their students. The main part of the published papers on this theory relates to linear operators. The nonlinear case was considered mainly when studied operator was a weak perturbation of the linear one. The case when the operator under study is not a weak perturbation of the linear operator is considered only in some works. The results obtained in this paper also relate to this little-studied case. The paper studies the coercive properties of the nonlinear Laplace-Beltrami operator in the space $L_2(R^n)$
$$ L[u]=-\frac{1}{\sqrt{det\, g(x)}}\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\left[\sqrt{det\, g(x)}g^{-1}(x)\frac{\partial u}{\partial x_j}\right]+V(x,u)u(x) $$
and proves its separability in this space by coercivity estimates. The operator under study is not a weak perturbation of the linear operator, i.e. it is strongly nonlinear. Based on the obtained coercive estimates and separability, the solvability of the nonlinear Laplace-Beltrami equation in the space $L_2(R^n)$ is studied.