A note on solutions for asymptotically linear elliptic systems

In this paper, we are concerned with the elliptic system of
{ -△u＋V（x）u=g（x,v）, x∈R^N,
-△v＋V（x）v=f（x,u）, x∈R^N,
where V（x） is a continuous potential well, f, g are continuous and asymptotically linear as t→∞. The existence of a positive solution and ground state solution are established via variational methods.     

A note on regular points for solutions of elliptic systems

A vector valued function u(x), solution of a quasilinear elliptic system cannot be too close to a straight line without being regular.     

Summability for solutions to some quasilinear elliptic systems

We prove local regularity in Lebesgue spaces for weak solutions \(u\) of quasilinear elliptic systems whose off-diagonal coefficients are small when \(|u|\) is large: the faster off-diagonal coefficients decay, the higher integrability of \(u\) becomes.     

Singular points of solutions of some elliptic systems on the plane

In the present paper, the structure of solutions of some important classes of singular elliptic systems on the plane are investigated. In particular, it is proved that the solutions of such systems have in principle nonanalytic behavior in the neighborhood of fixed singular points. These results make it possible to state correctly the boundary value problems and give their complete analysis. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

Positive solutions for some nonlocal and nonvariational elliptic systems

Using fixed point techniques, we study the existence and multiplicity of positive radial solutions for two classes of nonlocal elliptic systems defined on bounded annular domains or exterior domains. To this end, we reduce our problem to second-order functional ordinary elliptic systems. Our approach also allows us to study systems involving various orders, which serve as models for the suspension bridge equations.     

Existence of positive bounded solutions for some nonlinear elliptic systems

We prove some existence results of positive bounded continuous solutions to the semilinear elliptic system Δu=λp(x)g(v), Δv=μq(x)f(u) in domains D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f,g are nonnegative continuous monotone on (0,∞) and the potentials p, q are nonnegative and satisfy some hypotheses related to the Kato class K(D).     

A note on solutions of some differential-difference equations

This research is a continuation of the recent paper by X. Qi and L. Yang [15]. In this paper, we continue our study concerning existence of solutions of a Fermat type differentialdifference equation, and improve the results obtained by K. Liu et al. in [8, 10].     

Existence of positive radial solutions for some nonvariational superlinear elliptic systems

The system under consideration is

     

Layer and unlayer solutions for some singularly perturbed semilinear elliptic systems

In this paper the existence of a transition layer and unlayer solution for the stationary system ɛ ²Δu = f(u, υ), Δυ = g(u, υ) with x ∈ Ω ⊂ R ^N (N ≥ 2) is studied by using the degree-theoretical argument.     

The stability of the elliptic equilibrium of planar quasi-periodic Hamiltonian systems

In this paper, we study the planar Hamiltonian system  = J (A(θ)x + ▽f(x, θ)), θ = ω, x ∈ R2 , θ∈ Td , where f is real analytic in x and θ, A(θ) is a 2 × 2 real analytic symmetric matrix, J = (1-1 ) and ω is a Diophantine vector. Under the assumption that the unperturbed system  = JA(θ)x, θ = ω is reducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system.     

A note on periodic solutions of nonautonomous second-order systems

A multiplicity theorem is obtained for periodic solutions of nonautonomous second-order systems with partially periodic potentials by the minimax methods.

     

A note on some similarity solutions of the diffusion equation

Summary A first integral of the diffusion equation exists for similarity solutions when the diffusivity obeys a power or exponential law. Structure of the solutions in both cases and connection to an optimization result are discussed for an arbitrary diffusivity.

---

Résumé Une intégrale première de l'équation de diffusion existe pour les solutions similaires quand le coefficient de diffusion obéit une loi de puissance arbitraire ou une loi exponentielle. La structure des solutions est discutée dans les deux cas ainsi que leur relation avec les résultats d'optimisation pour une diffusion arbitraire.

     

A note on Besov regularity of layer potentials and solutions of elliptic PDE's

Let be a second order, (variable coefficient) elliptic differential operator and let , , 0$">, satisfy in the Lipschitz domain . We show that can exhibit more regularity on Besov scales for which smoothness is measured in with . Similar results are valid for functions representable in terms of layer potentials.

     

A note on regularity of solutions for degenerate singular elliptic boundary value problems

We prove the \(C^{1,\beta }\)-boundary regularity and a comparison principle for weak solutions of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u-\lambda \psi _{p}(u)=f(x)&{}\quad \text {in }\Omega , \\ u=0&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb {R}^{N},N>1\ \)with smooth boundary \(\partial \Omega ,\ \ \Delta _{p}u=\mathrm{div}(|\nabla u|^{p-2}\nabla u),\psi _{p}(u)=|u|^{p-2}u,p>1,\ \)and f is allowed to be unbounded.