Singular Elliptic Systems Research Articles

Existence of solution for a singular elliptic system with convection terms

In this paper we use the dual approach introduced by Colin and Jeanjean (2004) and Liu et al. (2003) combined with a Rabinowitz’s result, Galerkin’s method and an approximation argument to show the existence of solution for the following quasilinear Schrödinger elliptic system with both singular and convection terms −Δz−Δ(z2)z=μ1wθ1z−γ1+zα1+|∇w|η1inΩ,−Δw−Δ(w2)w=μ2zθ2w−γ2+wα2+|∇z|η2inΩ,z,w>0 inΩ,z=w=0 on ∂Ω,where Ω is a bounded domain of RN(N≥3) with smooth boundary, μi,θi,γi,αi,ηi>0,i=1,2 are real parameters.

The fibering map approach for a singular elliptic system involving the $p(x)$-Laplacian and nonlinear boundary conditions

The purpose of this work is to study the existence and multiplicity of positive solutions for a class of singular elliptic systems involving the p(x)- Laplace operator and nonlinear boundary conditions.

Existence and multiplicity of solutions for singular fractional elliptic system via the Nehari manifold approach

Existence and multiplicity of solutions for singular fractional elliptic system via the Nehari manifold approach

Homogenization of singular elliptic systems with nonlinear conditions on the interfaces

We prove existence and homogenization results for elliptic problems involving a singular lower order term and defined on a domain with interfaces having a nonlinear response. The considered system of equations can model heat or electrical conduction in composite media.

On Positive Solution for a Class of Singular Elliptic System involving Critical Coupling Terms

On Positive Solution for a Class of Singular Elliptic System involving Critical Coupling Terms

A singular elliptic system involving the p(x)-Laplacian and generalized Lebesgue–Sobolev spaces

The purpose of this work is to study a class of singular elliptic system involving the [Formula: see text]-Laplace operator of the form [Formula: see text] where [Formula: see text] [Formula: see text] is a bounded domain with [Formula: see text] boundary, [Formula: see text] are two parameters, [Formula: see text] are non-negative weight functions with compact support in [Formula: see text] and [Formula: see text] are assumed to satisfy the assumptions (A0)–(A2) in Sec. 1. We employ the Nehari manifold approach combined with some variational techniques in order to show the existence and the multiplicity of positive solutions.

Nonlocal singular elliptic system arising from the amoeba–bacteria population dynamics

In this paper, we prove the existence of coexistence states for a nonlocal singular elliptic system that arises from the interaction between amoeba and bacteria populations. Our study is based on fixed point arguments using a version of the Bolzano’s theorem, for which we will first analyze a local system by bifurcation theory. Moreover, we study the behavior of the coexistence region obtained and we interpret our results according to the growth rate of both species.

Multiple positive solutions for a class of quasilinear singular elliptic systems

In this paper we establish the existence of two positive solutions for a class of quasilinear singular elliptic systems. The main tools are sub and supersolution method and Leray–Schauder Topological degree.

Singular Generalized Analytic Functions

In this paper, we consider solution spaces for some class of singular elliptic systems on Riemann surfaces and boundary-value problems for solution spaces of such systems. We also discuss some relations for the kernels of the Carleman–Vekua equation. In particular, representations of these kernels in the form of generalized power functions are completely analogous to the classical Cauchy kernel expansion. The obtained results are applied to some problems of the theory of generalized analytic functions.

Existence of positive weak solutions for a nonlocal singular elliptic system

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with $C^{1, 1}$ boundary, and let $s\in\left(0, 1\right) $ be such that $s 0$ in $\Omega, $ $v>0$ in $\Omega, $ where $a$ and $b$ are nonnegative bounded measurable functions such that $\inf_{\Omega}a>0$ and $\inf_{\Omega}b>0.$ For the found weak solution $\left(u, v\right), $ the behavior of $u$ and $v$ near $\partial\Omega$ is also investigated.

A critical surface for the solutions to singular elliptic systems

In this paper, we mainly study the radially-symmetric and strictly-decreasing solutions to a system of critical elliptic equations in R N , which involves strongly-coupled critical nonlinearities and different Hardy-type terms. By the ODEs analysis methods, the asymptotic behaviors at the origin and infinity of solutions are proved. By the way, the asymptotic properties of radial and decreasing minimizers to the related best Sobolev constant are established. We find the existence of a critical surface, above which and below which the asymptotic properties of solutions are deferent.

Existence of Positive Solutions for a Class of Quasilinear Singular Elliptic Systems Involving Caffarelli-Kohn-Nirenberg Exponent with Sign-Changing Weight Functions

The paper deals with the existence of weak positive solutions for a new class of quasilinear singular elliptic systems involving critical Caffarelli–Kohn–Nirenberg exponent with sign-changing weight functions using the method of sub-super solutions. Our results are natural extensions from the previous ones in [3].

Existence and a priori estimates of solutions for quasilinear singular elliptic systems with variable exponents

This article sets forth results on the existence, a priori estimates and boundedness of positive solutions of a singular quasilinear systems of elliptic equations involving variable exponents. The approach is based on Schauder’s fixed point Theorem. A Moser iteration procedure is also obtained for singular quasilinear systems involving variable exponents establishing a priori estimates and boundedness of solutions.

Multiple solutions of critical singular degenerate elliptic system with concave-convex nonlinearities

This paper is devoted to study a class of critical singular degenerate elliptic system with concave-convex nonlinearities and sign-changing weight functions. The existence and multiplicity of nontrivial nonnegative solutions are obtained by the variational.

On G-invariant solutions of a singular biharmonic elliptic system involving multiple critical exponents in R^{N}

In this work, a biharmonic elliptic system is investigated in mathbb{R}^{N}, which involves singular potentials and multiple critical exponents. By the Rellich inequality and the symmetric criticality principle, the existence and multiplicity of G-invariant solutions to the system are established. To our best knowledge, our results are new even in the scalar cases.

On the existence results for a class of singular elliptic system involving indefinite weight functions and asymptotically linear growth forcing term

In this work, we study the existence of positive solutions to the singular system$$\left\{\begin{array}{ll}-\Delta_{p}u = \lambda a(x)f(v)-u^{-\alpha} & \textrm{ in }\Omega,\\-\Delta_{p}v = \lambda b(x)g(u)-v^{-\alpha} & \textrm{ in }\Omega,\\u = v= 0 & \textrm{ on }\partial \Omega,\end{array}\right.$$where $\lambda $ is positive parameter, $\Delta_{p}u=\textrm{div}(|\nabla u|^{p-2} \nabla u)$, $p>1$, $ \Omega \subset R^{n} $ some for $ n >1 $, is a bounded domain with smooth boundary $\partial \Omega $ , $ 0<\alpha< 1 $, and $f,g: [0,\infty] \to\R$ are continuous, nondecreasing functions which are asymptotically $ p $-linear at $\infty$. We prove the existence of a positive solution for a certain range of $\lambda$ using the method of sub-supersolutions.

Existence and Nonexistence of Ground State Solutions to Singular Elliptic Systems

In this paper, a system of semi-linear elliptic equations is investigated, which involves multiple critical nonlinearities and multiple singular points. By variational methods, the existence and nonexistence of minimizers to the Rayleigh quotient and ground state solutions to the system are established.

Positive solutions for singular elliptic systems with convection term

We prove the existence and regularity of solutions for a quasi-linear elliptic system with convection terms that can be singular in the solution and its gradient. Comparison properties and a priori estimates are also obtained. Our approach relies on invariance, regularity, strong maximum principle, and fixed-point arguments.

Positive solutions for a class of singular elliptic system

Positive solutions for a class of singular elliptic system

On a singular elliptic system with quadratic growth in the gradient

In this paper, we deal with the following singular elliptic system:{−Δu+α|∇u|2u=pp+qa(x)|v|q|u|p−2u+f,x∈RN,−Δv+β|∇v|2v=qp+qa(x)|u|p|v|q−2v+g,x∈RN, where N≥3, α,β>N+24, p,q>1 and p+q≤N+2N−2. We show through the sub- and supersolutions method, the existence of a nonnegative solution for an approximated system. The limit of the approximated solution is a positive solution. In the case, α=β=0, p=q and f=g, we prove the uniqueness of a solution. Among others, we prove some existence and uniqueness results for some auxiliary problems by using the comparison principle, a minimization method and with the help of Nehari manifold. The proofs rely on the concentration-compactness principle.