This article initiates the study of nonlinear elliptic partial difference equations (PdE) on graphs. We seek solutions u : V → R to the semilinear elliptic difference equation −Lu + f(u) = 0 on a graph G = (V,E), where L is the (negative) Laplacian on the graph G. We extend techniques used to prove existence theorems and derive numerical algorithms for the partial differential equation (PDE) ∆u...

In [14], a new local minimax method that characterizes a saddle point as a solution to a local minimax problem is established. Based on the local characterization, a numerical minimax algorithm is designed for finding multiple saddle points. Numerical computations of many examples in semilinear elliptic PDE have been successfully carried out to solve for multiple solutions. One of the important...

in this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $dsubset mathbb{r}^{n}$ with smooth boundary‎. ‎we give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of caffarelli and friedman regarding the representation of functions whose ...

We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve certain nonlinear in cases where solution a corresponding linear equation is not known. By using adaptation of this we show that results [24], [23] remain valid general

We seek solutions u ∈ R to the semilinear elliptic partial difference equation −Lu+ fs(u) = 0, where L is the matrix corresponding to the Laplacian operator on a graph G and fs is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: a) Nonlinear Elliptic Partial Difference Equations on Graphs (J. Experimental Mathematics, 2006),...

In this paper, we will investigate the existence of multiple solutions for the general inhomogeneous elliptic problem   − u+ u = f (x, u) + μh (x) , x ∈ R , u ∈ H (RN) , (1.1)μ where h ∈ H−1 (RN), N ≥ 2, |f (x, u)| ≤ C1up−1 + C2u with C1 > 0, C2 ∈ [0, 1) being some constants and 2 < p < +∞. ∗Research supported in part by the Natural Science Foundation of China and NSEC †Research supported ...