We study here a class of quasilinear elliptic systems involving the p-Laplacian operator. Under some suitable assumptions on the nonlinearities, we show the existence result by using a fixed point theorem.

We show that Laplacian and symmetric diagonally dominant (SDD) matrices can be well approximated by linear-sized sparse Cholesky factorizations. Specifically, n × n matrices of these types have constant-factor approximations of the form LL , where L is a lowertriangular matrix with O(n) non-zero entries. This factorization allows us to solve linear systems in such matrices in O(n) work and O(lo...

Content The first paper by Jean Mawhin on critical point theory [] was published in  and was devoted to periodic solutions of a forced pendulum equation. One of the most recent papers in  [] concerns periodic solutions of difference systems with φ-Laplacian. It is impossible to describe all the contributions. We have selected  articles,  books and some fundamental topics: the force...

The main theme of the workshop is fast algorithms, particularly those that relate to fast solvers for linear systems involving the Laplacian of a graph. In my lectures, I will discuss three key technical ingredients that underlie those solvers. In this first lecture, I will discuss “low-stretch trees”. Given a graph, the goal is to find a spanning subtree such that, on average, distances in the...

We show that either superlinearity or sublinearity assumptions can guarantee the existence of positive radial solutions for quasilinear systems involving the p-Laplacian. 2005 Elsevier Ltd. All rights reserved.

Let G be a mixed graph and L(G) be the Laplacian matrix of G. In this paper, the coefficients of the Laplacian characteristic polynomial of G are studied. The first derivative of the characteristic polynomial of L(G) is explicitly expressed by means of Laplacian characteristic polynomials of its edge deleted subgraphs. As a consequence, it is shown that the Laplacian characteristic polynomial o...

We study certain generalized Leray-alpha magnetohydrodynamics systems. We show that the solution pairs of velocity and magnetic fields to this system in two-dimension preserve their initial regularity in two cases: dissipation logarithmically weaker than a full Laplacian and zero diffusion, zero dissipation and diffusion logarithmically weaker than a full Laplacian. These results extend previou...

This paper develops an algebraic multigrid preconditioner for the graph Laplacian. The proposed approach uses aggressive coarsening based on the aggregation framework in the setup phase and a polynomial smoother with sufficiently large degree within a (nonlinear) Algebraic Multilevel Iteration as a preconditioner to the flexible Conjugate Gradient iteration in the solve phase. We show that by c...

In this paper, we are concerned with stable solutions to the fractional elliptic equation $$\begin{aligned} (-\Delta )^s u=e^u \text{ in } {\mathbb {R}}^{N}, \end{aligned}$$ where $$(-\Delta )^s$$ is Laplacian $$0<s<1$$ . By developing a new technique non-compactly supported test functions, establish nonexistence of provided that $$N<10s$$ This result optimal when $$s\uparrow 1$$ On other hand,...