The existence of extremal functions for discrete Sobolev inequalities on lattice graphs

@article{Hua2021TheEO,
  title={The existence of extremal functions for discrete Sobolev inequalities on lattice graphs},
  author={Bobo Hua and Ruowei Li},
  journal={Journal of Differential Equations},
  year={2021}
}
  • B. Hua, Ruowei Li
  • Published 30 June 2021
  • Mathematics
  • Journal of Differential Equations

A class of semilinear elliptic equations on lattice graphs

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

. 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

. 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

. 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

TLDR
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

. 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

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

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

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

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

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

Qualitative properties of solutions for an integral equation

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

  • K. Chang
  • Mathematics, Computer Science
    J. Graph Theory
  • 2016
TLDR
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.
...