Universal bounds for eigenvalues of the polydrifting Laplacian operator in compact domains in the $$\mathbb {R}^{n}$$ R n and $$\mathbb {S}^{n}$$ S n

Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons

“…Remark 1.5 For the eigenvalue problem (1.2) with m > 2, Pereira et al [28] gave some universal inequalities on Euclidean spaces and unit spheres. However, here we give the universal inequality on a special cylinder shrinking soliton.…”

“…At the end of this section, we introduce two lemmas given by Pereira et al [28]. For more details, see for instance [23].…”

“…(32) in [28], we can get universal inequalities for lower order eigenvalues of poly-drifting Laplacian in unite spheres S n as follows:…”

“…By Lemma 2.1, (16) in[28] and a similar computation as in (3.11), we can get a stronger universal inequality on the Gaussian shrinking soliton than that of Pereira-Adriano-Pina (cf [28,. Theorem 3.2]) as follows:…”

“…In fact, when m = 1, Xia and Xu [34] investigated the eigenvalues of the Dirichlet problem of the drifting Laplacian on compact manifolds and got some universal inequalities; when m = 2, Du et al [16] obtained some universal inequalities of Yang type for eigenvalues of the bi-drifting Laplacian problem either on a compact Riemannian manifold with boundary (possibly empty) immersed in a Euclidean space, a unit sphere or a projective space, or on bounded domains of complete manifolds supporting some special function. Recently, when m is an arbitrary integer no less than 2, Pereira et al [28] have given some universal inequalities on bounded domains in a Euclidean space or a unit sphere. Based on the existing results mentioned above, in this paper, we will give some universal inequalities for eigenvalues of problem (1.2) when m is an arbitrary integer no less than 2.…”