This is a unique book that provides a comprehensive understanding of nonlinear equations
involving the fractional Laplacian as well as other nonlocal operators. Beginning from the …

We prove the uniqueness of the positive solutions of the following elliptic system:(1) -
Δ(u(x))=u(x)^αv(x)^β,(2) -Δ(v(x))=u(x)^βv(x)^α. Here x∈R^n, n\geq3, and 1≦α<β≦n+2n-2 …

We classify all positive solutions for the following integral system: Here fi (u), 1≤ i≤ m, are
real-valued functions of homogeneous degree and are monotone nondecreasing with …

In this paper, we prove that all the positive solutions for the PDE system (-\Delta)^{k} u_ {i}=
f_ {i}(u_ {1},..., u_ {m}), x\in R^{n}, i= 1, 2,..., m are super polyharmonic, ie (-\Delta)^{j} u_ {i}> …

We study the effective time evolution of a large quantum system consisting of a mixture of
different species of identical bosons in interaction. If the system is initially prepared so as to …

In this paper, we give sufficient conditions ensuring that any positive classical solution (u,v)
of an elliptic system in the whole space R^n has the symmetry property u=v. As an …

We establish the uniqueness of ground states of some coupled nonlinear Schrödinger
systems in the whole space. We firstly use Schwartz symmetrization to obtain the existence …

We study multi-component elliptic Schrödinger systems arising in nonlinear optics and Bose-
Einstein condensation phenomena. We prove new Liouville-type nonexistence theorems, as …

Let $ R^ n_+ $ be the $ n $-dimensional upper half Euclidean space, and let $\alpha $ be
any real number satisfying $0<\alpha< n. $ In this paper, we consider the integral …

Some focusing coupled Schrödinger equations are investigated. First, existence of ground
state is obtained. Second, global existence and finite-time blow-up of solutions are …