Un majorant du nombre des valeurs propres négatives correspondantes à l'opérateur de Schrödinger généralisé. (Q443803)

In the paper under review, the author generalizes the upper bound on the number of negative eigenvalues for the Schrödinger operator \(-\Delta -V\) with nonnegative potential \(V\), obtained in [\textit{R. Kerman} and \textit{E. Sawyer}, Ann. Inst. Fourier 36, No. 4, 207--228 (1986; Zbl 0591.47037)], to operators of the form \((-\Delta)^m-V\), where \(-\Delta\) is the usual Laplacian on \(\mathbb{R}^d\), \(1<m<d/2\), and \(V\) is a multiplication operator by a nonnegative function.NEWLINENEWLINEThe nonnegative potential \(V\) is assumed to be locally in \(L^1(\mathbb{R}^d)\) and satisfies the following condition: there exist \(c(d)>0\) and a finite number \(N_0\) of dyadic cubes \(Q\) such that NEWLINE\[NEWLINE\int_{Q^2}\frac{V(x)V(y)}{|x-y|^{d-2m}}\, dxdy\;\geq \;c(d)\, \int_QV(x)\, dx\, .NEWLINE\]NEWLINE The author proves that there exists a constant \(C(d)\), depending only on \(d\), such that the self-adjoint realization of \((-\Delta)^m-V\) has at most \(C(d)N_0\) negative eigenvalues. The proof makes use of the Riesz potential and of the ellipticity of the operator. It is inspired by the one in [Kerman and Sawyer, loc. cit.]. Applications in nonlinear PDEs are also provided.