We review our joint result with E. Lenzmann about the uniqueness of ground state solutions of non-linear equations involving the fractional Laplacian and provide an alternate uniqueness proof for an equation related to the intermediate long-wave equation.

The inhomogeneous Laplace equation with internal Dirichlet boundary conditions has recently appeared in many applications arising from image segmentations, image colorization, image filtering and so on. Efficient solutions of (anisotropy) Laplacian equations have been studied intensively in numerical analysis world. In this project paper, I apply the known algorithms, especially an algebraic mu...

By means of Mawhin’s continuation theorem, we study a kind of third-order p-Laplacian functional differential equation with distributed delay in the form: ( φp(x ′(t)) )′′ = g ( t, ∫ 0 −τ x(t+ s) dα(s) ) + e(t), some criteria to guarantee the existence of periodic solutions are obtained. Keywords—p–Laplacian, Distributed delay, Periodic solution, Mawhin’s continuation theorem

In this article, we study eigenvalue problems with the p-Laplacian operator: −(|y′|p−2y′)′ = (p− 1)(λρ(x)− q(x))|y|p−2y on (0, πp), where p > 1 and πp ≡ 2π/(p sin(π/p)). We show that if ρ ≡ 1 and q is singlewell with transition point a = πp/2, then the second Neumann eigenvalue is greater than or equal to the first Dirichlet eigenvalue; the equality holds if and only if q is constant. The same ...

We propose a newmethod to analyze and represent data recorded on a domain of general shape in R by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problemon such a domain via theHelmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the ...

In this paper, we prove that the semigroup [Formula: see text] generated by the Cauchy problem of the evolution p-Laplacian equation [Formula: see text] ([Formula: see text]) is continuous form a weighted [Formula: see text] space to the continuous space [Formula: see text]. Then we use this property to reveal the fact that the evolution p-Laplacian equation generates a chaotic dynamical system...