## 6 Citations

### A class of semilinear elliptic equations on lattice graphs

- Mathematics
- 2022

In this paper, we study the semilinear elliptic equation of the form −∆u+ a(x)|u|u− b(x)|u|u = 0 on lattice graphs Z , where N ≥ 2 and 2 ≤ p < q < +∞. By the Brézis-Lieb lemma and concentration…

### Extremal functions for the second-order Sobolev inequality on groups of polynomial growth

- Mathematics
- 2022

. In this paper, we prove the second-order Sobolev inequalities on Cayley graphs of groups of polynomial growth. We use the discrete Concentration- Compactness principle to prove the existence of…

### The existence of positive ground state solutions for the Choquard type equation on groups of polynomial growth

- Mathematics
- 2022

. In this paper, let G be a Cayley graph of a discrete group of polynomial growth with homogeneous dimension N ≥ 3 . We study the Choquard type equation on G : where α ∈ (0 ,N ) , p > N + α N − 2 and…

### The existence and convergence of solutions for the nonlinear Choquard equations on groups of polynomial growth

- Mathematics
- 2022

. In this paper, we study the nonlinear Choquard equation ∆ 2 u − ∆ u + (1 + λa ( x )) u = ( R α ∗ | u | p ) | u | p − 2 u on a Cayley graph of a discrete group of polynomial growth with the…

### Discrete Schwarz rearrangement in lattice graphs

- Mathematics, Computer Science
- 2022

A discrete version of the generalized Riesz inequality on Z d is proved, which will derive the extended Hardy-Littlewood and P´olya-Szeg¨o inequalities and establish cases of equality in the latter.

### Symmetrization inequalities on one-dimensional integer lattice

- Mathematics
- 2022

. In this paper, we develop a theory of symmetrization on the one dimensional integer lattice. More precisely, we associate a radially decreasing function u ∗ with a function u deﬁned on the integers…

## References

SHOWING 1-10 OF 56 REFERENCES

### Sobolev spaces on graphs

- Mathematics
- 2005

The present paper is devoted to discrete analogues of Sobolev spaces of smooth functions. The discrete analogues that we consider are spaces of functions on vertex sets of graphs. Such spaces have…

### An Extended Discrete Hardy-Littlewood-Sobolev Inequality

- Mathematics
- 2013

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: \mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case:…

### The best constant of discrete Sobolev inequality

- Mathematics
- 2009

A discrete version of Sobolev inequalities in Hilbert spaces ℓ2 and ℓ2N, which are equipped with an inner product defined by using 2Mth positive difference operators, is presented. Their best…

### The best constant in a weighted Hardy-Littlewood-Sobolev inequality

- Mathematics
- 2007

We prove the uniqueness for the solutions of the singular nonlinear PDE system: Formula math. In the special case when a = β and p = q, we classify all the solutions and thus obtain the best constant…

### A Relation Between Pointwise Convergence of Functions and Convergence of Functionals

- Mathematics
- 1983

We show that if f n is a sequence of uniformly L p-bounded functions on a measure space, and if f n → f pointwise a.e., then lim for all 0 < p < ∞. This result is also generalized in Theorem 2 to…

### OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM

- Mathematics
- 2014

### Qualitative properties of solutions for an integral equation

- Mathematics
- 2003

Let $n$ be a positive integer and let $ 0 < \alpha < n.$
In this paper, we study more general integral equation
$ u(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} K(y) u(y)^p dy.
We establish…

### Spectrum of the 1‐Laplacian and Cheeger's Constant on Graphs

- Mathematics, Computer ScienceJ. Graph Theory
- 2016

A nonlinear spectral graph theory is developed, in which the Laplace operator is replaced by the 1 − Laplacian Δ1, and Cheeger's constant equals to the first nonzero Δ1 eigenvalue for connected graphs.