We study the spectrum of the scalar Laplacian on the five-dimensional toric Sasaki-Einstein manifolds Y . The eigenvalue equation reduces to Heun’s equation, which is a Fuchsian equation with four regular singularities. We show that the ground states, which are given by constant solutions of Heun’s equation, are identified with BPS states corresponding to the chiral primary operators in the dua...

In this paper we study the nonlocal ∞−Laplacian type diffusion equation obtained as the limit as p →∞ to the nonlocal analogous to the p−Laplacian evolution, ut(t, x) = ∫ RN J(x− y)|u(t, y)− u(t, x)|p−2(u(t, y)− u(t, x)) dy. We prove existence and uniqueness of a limit solution that verifies an equation governed by the subdifferetial of a convex energy functional associated to the indicator fun...

a concept related to the spectrum of a graph is that of energy. the energy e(g) of a graph g is equal to the sum of the absolute values of the eigenvalues of the adjacency matrix of g . the laplacian energy of a graph g is equal to the sum of distances of the laplacian eigenvalues of g and the average degree d(g) of g. in this paper we introduce the concept of laplacian energy of fuzzy graphs. ...

We prove the existence of a positive solution for nonlocal problems involving the fractional Laplacian and a critical growth power nonlinearity when the equation is set in a suitable contractible domain.

In this paper we prove the local L s regularity (where s depends on the summability of the data) for local " unbounded " weak solutions of a class of nonlinear parabolic equations including the p-Laplacian equation.

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.

In this paper, we establish Lyapunov-type inequalities for a single higher-order differential equation, a cycled system and a coupled system with one-dimensional p -Laplacian. Our result generalize some given results.

In this paper, we derive an anisotropic Picone identity for the anisotropic Laplacian, which contains some known Picone identities. As applications, a Sturmian comparison principle to the anisotropic elliptic equation and an anisotropic Hardy type inequality are shown.

Using Morse theory, we establish the existence of solutions to the equation −∆pu = f(x, u) with Dirichlet boundary conditions. We assume that ∫ s 0 f(x, t) dt lies between the first two eigenvalues of the p-Laplacian.

In this paper, we consider a kind of p-Laplacian neutral Rayleigh equation with singularity of attractive type, [Formula: see text] By applications of an extension of Mawhin's continuation theorem, sufficient conditions for the existence of periodic solution are established.