In this paper, we consider the existence of positive solutions of second-order periodic boundary value problem
u"+((1)/(2)+ε)^{2}u=λg(t)f(u), t∈[0, 2π], u(0)=u(2π), u'(0)=u'(2π), where 0 < ε < (1)/(2), g:[0, 2π]→R is continuous, f:[0, ∞)→R is continuous and λ > 0 is a parameter.
Differential-difference equations of the form =F_{n}(t, u_{n-1}, u_{n}, u_{n+1}, 2???-1, , +1) are classified according to their intrinsic Lie point symmetries, equivalence group and some low-dimensional Lie algebras including the Abelian symmetry algebras, nilpotent nonAbelian symmetry algebras, solvable symmetry algebras with nonAbelian nilradicals, solvable symmetry algebras with Abelian nilradicals and nonsolvable symmetry algebras. Here F_{n} is a nonlinear function of its arguments and the dot over u denotes differentiation with respect to t.
In this article, we study the weak dissipative Kirchhoff equation
u_{tt}-M(||∂u||_{2}^{2})△u+b(x)u_{t}+f(u)=0,
under nonlinear damping on the boundary
(∂u)/(∂ν)+α(t)g(u_{t})=0. We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping. Our result extends and improves some results in the literature such as the work by Zhang and Miao (2010) in which only exponential energy decay is considered and the work by Zhang and Huang (2014) where the energy decay has been not considered.
We discuss the properties of solutions for the following elliptic partial differential equations system in R^{n},
(-△)^{(α/(2))}u=u^{p1}v^{p2},
(-△)^{(α/(2))}v=u^{q1}v^{q2},
where 0 < α < n, p_{i} and q_{i} (i=1, 2) satisfy some suitable assumptions. Due to equivalence between the PDEs system and a given integral system, we prove the radial symmetry and regularity of positive solutions to the PDEs system via the method of moving plane in integral forms and Regularity Lifting Lemma. For the special case, when p1+p2=q1+q2=(n+α)/(n-α), we classify the solutions of the PDEs system.