The singularly perturbed problem for the semilinear elliptic equations is considered. Under appropriate conditions, by using the comparison theorem, the existence and asymptotic behavior of solution for the boundary value problems are studied.

In this paper, the existence of homoclinic orbits, for a perturbed cubic-quintic nonlinear Schrödinger equation with even periodic boundary conditions, under the generalized parameters conditions is established. More specifically, we combine geometric singular perturbation theory with Melnikov analysis and integrable theory to prove the persistence of homoclinic orbits.

The aim of this paper is to study Morrey regularity of the solution to the Dirichlet problem for second order elliptic equation of the form Lu = -(a_{ij}u_x_i + b_j_u)_x_j + du = e + (f_j)x_j in a bounded open subset of Rn(n ≥ 3), where b_j , d, e, f_j are assumed to be in some Morrey spaces.

In this paper the P_2 approximation to the one-group planar neutron transport theory is discussed. The stability of the solutions for P_2 equations with general boundary conditions, including the Marshak boundary condition, is proved. Moreover, the stability of the up-wind difference scheme for the P_2 equation is demonstrated.

The Boussinesq approximation, where the viscosity depends polynomially on the shear rate,finds more and more frequent use in geological practice. In this paper, we consider the periodic initial value problem and initial value problem for this modified Boussinesq approximation with the viscous part of the stress tensor τ^v = τ(e)- 2μΔe, where the nonlinear function τ(e) satisfies τ_{ij}(e)e_{ij} ≥ C|e|^p or τ_{ij}(e)e_{ij} ≥ C(|e|²+|e|^p). The existence, uniqueness and regularity of the weak solution is proved for p > \frac{2n}{n + 2}.

The author proves the uniqueness of the regularizable radial minimizers of a Ginzburg-Landau type functional in the case n - 1 < p < n,and the location of the zeros of the regularizable radial minimizers of this functional is discussed.

In this paper, we use the Wigner measure approach to study the semiclassical limit of nonlinear Schrödinger equation in small time. We prove that: the limits of the quantum density: ρ^∈ =: |ψ^∈|² and the quantum momentum: J^∈ =: ∈Im(\overline{ψ^∈}∇ψ^∈) satisfy the compressible Euler equations before the formation of singularities in the limit system.