In this paper we consider the eigenvalue problem − pu = λ(m)|u|p−2u, u ∈ W 1,p 0 ( ) where p > 1, p is the p-Laplacian operator, λ > 0, is a bounded domain in R(N ≥ 1) and m is a given positive function in L( ) (r depending on p and N ). We prove that the second positive eigenvalue admits exactly two nodal domains. AMS subject classification: 35J20, 35J70, 35P05, 35P30.

The paper deals with the existence and uniqueness of a non-trivial solution to non-homogeneous p(x)- Laplacian equations, managed by non polynomial growth operator in framework variable exponent Sobolev spaces on Riemannian manifolds. mountain pass Theorem is used.

Recently, Steinerberger (Potential Analysis, 2020) proved a uniform inequality for the Laplacian serving as counterpoint to standard sublevel set which is known fail Laplacian. In this paper, we observe that many inequalities of type follow from lower bound on L1 norm, and give an analogous result any linear differential operator, can non-linear operators. We consider bounds L p quasi-norms < 1...

To any compact Riemannian manifold (M, g) (with or without boundary), we can associate a second-order partial differential operator, the Laplace operator ∆, defined by ∆(f) = −div(grad(f)) for f ∈ L(M, g). We will also sometimes write ∆g for ∆ if we want to emphasize which metric the Laplace operator is associated with. The set of eigenvalues of ∆ (the spectrum of ∆, or of M), which we will wri...

We study the following third-order p-Laplacian m-point boundary value problems on time scales φp uΔ∇ ∇ a t f t, u t 0, t ∈ 0, T T , u 0 ∑m−2 i 1 biu ξi , u Δ T 0, φp uΔ∇ 0 ∑m−2 i 1 ciφp u Δ∇ ξi , where φp s is p-Laplacian operator, that is, φp s |s|p−2s, p > 1, φ−1 p φq, 1/p 1/q 1, 0 < ξ1 < · · · < ξm−2 < ρ T . We obtain the existence of positive solutions by using fixed-point theorem in cones....

1 Laplacian Methods: An Overview 2 1.1 De…nition: The Laplacian operator of a Graph . . . . . . . . . . 2 1.2 Properties of the Laplacian and its Spectrum . . . . . . . . . . . 4 1.2.1 Spectrum of L and e L: Graph eigenvalues and eigenvectors: 4 1.2.2 Other interesting / useful properties of the normalized Laplacian (Chung): . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Laplacians of Weight...