Nonlinear Elliptic Systems Research Articles

Bellman Systems with Mean Field Dependent Dynamics

The authors deal with nonlinear elliptic and parabolic systems that are the Bellman like systems associated to stochastic differential games with mean field dependent dynamics. The key novelty of the paper is that they allow heavily mean field dependent dynamics. This in particular leads to a system of PDE’s with critical growth, for which it is rare to have an existence and/or regularity result. In the paper, they introduce a structural assumptions that cover many cases in stochastic differential games with mean field dependent dynamics for which they are able to establish the existence of a weak solution. In addition, the authors present here a completely new method for obtaining the maximum/minimum principles for systems with critical growths, which is a starting point for further existence and also qualitative analysis.

Nonuniformly elliptic energy integrals with [formula omitted]-growth

We study the local boundedness of minimizers of a nonuniformly energy integral of the form ∫Ωf(x,Dv)dx under p,q-growth conditions of the type λ(x)|ξ|p≤f(x,ξ)≤μ(x)1+|ξ|qfor some exponents q≥p>1 and with nonnegative functions λ,μ satisfying some summability conditions. We use here the original notation introduced in 1971 by Trudinger [26], where λ(x) and μ(x) had the role of the minimum and the maximum eigenvalues of an n×n symmetric matrix aijx and f(x,ξ)=∑i,j=1naijxξiξjwas the energy integrand associated to a linear nonuniformly elliptic equation in divergence form. In this paper we consider a class of energy integrals, associated to nonlinear nonuniformly elliptic equations and systems, with integrands f(x,ξ) satisfying the general growth conditions above.

Gradient [formula omitted] theory for a class of non-diagonal nonlinear elliptic systems

We consider a class of nonlinear non-diagonal elliptic systems with p-growth and establish the Lq-integrability for all q∈[p,p+2] of any weak solution provided the corresponding right hand side belongs to the corresponding Lebesgue space and the involved elliptic operator asymptotically satisfies the p-uniform ellipticity, the so-called splitting condition and it is continuous with respect to the spatial variable. For operators satisfying the uniform p-ellipticity condition the higher integrability is known for q∈[p,dp∕(d−2)] and for operators having the so-called Uhlenbeck structure, the theory is valid for all q∈[p,∞). The key novelty of the paper is twofold. First, the statement uses only the information coming from the asymptotic operator and second, and more importantly, by using the splitting condition, we are able to extend the range of possible q’s significantly whenever p<d−2.

Twice Periodic Solutions of a Nonlinear Elliptic Second-Order Systems

The problem of the existence of twice periodic solutions of a nonlinear elliptic system of second-order equations on a plane whose main periods are given is studied. Conditions for the existence of solutions with prescribed poles, as well as with prescribed zeros and poles, are obtained.

Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian

In this paper, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: [Formula: see text] where [Formula: see text] denotes the fractional Laplacian and [Formula: see text] is a smooth bounded domain in [Formula: see text]. It shown that under some assumptions on [Formula: see text] and [Formula: see text], the problem has at least one positive solution [Formula: see text]. Our proof is based on the classical scaling method of Gidas and Spruck and topological degree theory.

On the existence of positive solutions to semilinear elliptic systems involving gradient term

ABSTRACTIn this work, we analyse the existence of solutions to the nonlinear elliptic system: where is a bounded domain of and , with . f, g are nonnegative measurable functions with additional hypotheses and . As a consequence we show that the fourth-order problem has a solution for all , under suitable conditions on and .

Positive solutions to a non-homogeneous elliptic system of fourth order

In this work, the existence of positive solutions to a non-linear elliptic system of fourth order with positive parameters is studied. We establish an existence of an interval of parameters such the system has at least two positive solutions. The behaviour of energy corresponding to these positive solutions, with respect to the real parameters is proved.

Positive solutions to a non-homogeneous elliptic system of fourth order

In this work, the existence of positive solutions to a non-linear elliptic system of fourth order with positive parameters is studied. We establish an existence of an interval of parameters such the system has at least two positive solutions. The behaviour of energy corresponding to these positive solutions, with respect to the real parameters is proved.

An application of a theorem of G. Zwirner to a class of non-linear elliptic systems in divergence form

A theorem on the solutions of the problem \(U'(w)={\gamma }F(U(w),w), U(w_1)=u_1, U(w_2)=u_2\) is applied for finding the functional solutions of the system of partial differential equations $$\begin{aligned} {\nabla }\cdot (a(u,w){\nabla }u)= & {} 0,\quad u=u_1{ \hbox {on}\quad {{\Gamma }}_1},\quad u=u_2{ \hbox {on}\quad {{\Gamma }}_2},\quad \frac{{\partial }u}{{\partial }n}=0{ \hbox {on}\quad {{\Gamma }}_3} {\nabla }\cdot (b(u,w){\nabla }w)= & {} 0,\quad w=w_1{ \hbox {on}\quad {{\Gamma }}_1},\quad w=w_2{ \hbox {on}\quad {{\Gamma }}_2},\quad \frac{{\partial }w}{{\partial }n}=0{ \hbox {on}\quad {{\Gamma }}_3}. \end{aligned}$$ The problem of existence and uniqueness of solutions is considered.

A Liouville type theorem for non-linear elliptic systems involving advection terms

In this paper, we study the following elliptic system where a is a smooth divergence-free vector field satisfying some decay condition and . We establish a Liouville type result for stable positive solutions of the system in the case and . In particular, when , our result is a natural extension of Cowan [Nonlinear Anal. 2014;104:1–11] to the equation with advection.

Boundedness of the solutions to non-linear systems with Morrey data

ABSTRACTWe consider non-linear elliptic systems satisfying componentwise coercivity condition. The non-linear terms have controlled growths with respect to the solution and its gradient, while the behaviour in x is governed by functions in Morrey spaces. We firstly prove essential boundedness of the weak solution and then we obtain Morrey regularity of its gradient.

Bubbling mixed type solutions of the SU(3) models on a torus

We consider a nonlinear elliptic system arising in the study of the SU(3) Chern-Simons model on a two-dimensional flat torus Ω. Solutions of this SU(3) Chern Simons system could be classified as topological, mixed-type, and non-topological solutions. In this paper, we succeed to construct bubbling mixed type solutions. This is the first result for such example in the literature. The analysis for the existence of such solution provides some important insights for us to develop the asymptotic analysis of classifying all mixed-type solution.

Liouville type theorem for nonlinear elliptic system involving Grushin operator

We study the degenerate elliptic system of the form{−ΔGu=vp−ΔGv=uq on RN=RN1×RN2, where ΔG:=Δx+|x|2αΔy is the Grushin operator, α≥0 and p≥q>1. We establish some Liouville type results for stable solutions of the system. In particular, we prove the comparison principle – a crucial step to establish such results. As consequences, we obtain a Liouville type theorem for the scalar equation and provide a counterpart of the previous result in C. Cowan (2013) [7].

Erratum to: A boundary value problem for a nonlinear elliptic system relevant in general relativity

Erratum to: A boundary value problem for a nonlinear elliptic system relevant in general relativity

Existence and regularity results for weak solutions to (p,q)-elliptic systems in divergence form

Abstract We prove existence and regularity results for weak solutions of non-linear elliptic systems with non-variational structure satisfying ( p , q ) {(p,q)} -growth conditions. In particular, we are able to prove higher differentiability results under a dimension-free gap between p and q.

On the Existence of Integrable Solutions to Nonlinear Elliptic Systems and Variational Problems with Linear Growth

We investigate the properties of certain elliptic systems leading, a~priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic Uhlenbeck structure, then the solution can in fact be understood as a standard weak solution, with one proviso: analogously as in the case of minimal surface equations, the attainment of the boundary value is penalized by a measure supported on (a subset of) the boundary, which, for the class of problems under consideration here, is the part of the boundary where a Neumann boundary condition is imposed.

Solitary waves for coupled nonlinear elliptic system with nonhomogeneous nonlinearities

In present paper we consider a class of coupled elliptic system with nonhomogeneous nonlinearities. This type of system is related to the Raman amplification in a plasma. We make rigorous study and find the threshold conditions to guarantee the existence, nonexistence and multiplicity of nontrivial solutions for both two and three coupled system by using Morse theory, direct analysis methods and Krasnosel’skii–Rabinowitz global bifurcation theorem. Moreover, we study the asymptotical behavior of positive solutions, and prove some interesting phenomena for these solutions. Comparing to our previous works Wang and Shi (standing waves for weakly coupled Schrodinger equations with quadratic nonlinearities. Preprint, 2015) on the homogeneous case, we encounter some new challenges in proving the existence and multiplicity of nontrivial solutions. We overcome these difficult by combining the Mountain–Pass theorem in convex set and the Nehari constraint methods.

Generalised solutions for fully nonlinear PDE systems and existence–uniqueness theorems

We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of Distributions to PDEs and is not based on either integration by parts or on the maximum principle. Instead, our starting point builds on the probabilistic representation of derivatives via limits of difference quotients in the Young measures over a toric compactification of the space of jets. After developing some basic theory, as a first application we consider the Dirichlet problem and we prove existence–uniqueness–partial regularity of solutions to fully nonlinear degenerate elliptic 2nd order systems and also existence of solutions to the ∞-Laplace system of vectorial Calculus of Variations in L∞.

Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent

This paper deals with the following system linearly coupled by nonlinear elliptic equations{−Δu+λ1u=|u|2⁎−2u+βv,x∈Ω,−Δv+λ2v=|v|2⁎−2v+βu,x∈Ω,u=v=0on∂Ω. Here Ω is a smooth bounded domain in RN(N≥3), λ1,λ2>−λ1(Ω) are constants, λ1(Ω) is the first eigenvalue of (−Δ,H01(Ω)), 2⁎=2NN−2 is the Sobolev critical exponent and β∈R is a coupling parameter. By variational method, we prove that this system has a positive ground state solution for some β>0. Via a perturbation argument, we show that this system also admits a positive higher energy solution when |β| is small. Moreover, the asymptotic behaviors of the positive ground state and higher energy solutions as β→0 are analyzed.

Nonexistence and multiplicity of solutions for nonlinear elliptic systems in [formula omitted

In the whole space RN, we will study the following nonlinear elliptic system in two cases: −△u+V1(x)u=f(x,u,v),x∈RN,−△v+V2(x)v=g(x,u,v),x∈RN,u(x)→0,v(x)→0,|x|→∞.Case 1 : The periodic case (i.e., V1, V2, f and g are periodic in xi for i=1,…,N), we obtain the nonexistence of nontrivial solutions for the system. The existence result has been obtained in Chen and Ma (2013). Case 2 : The non-periodic case (i.e., V1, V2, f and g are non-periodic), we mainly focus on the case where the nonlinearities f and g are superlinear at infinity, and we obtain infinitely many nontrivial solutions of the system by variational methods. To the best of our knowledge, there is no work focusing on the system in Cases 1–2. The main novelties of this paper can be summarized as follows: (1) The system is defined in the whole space RN; (2) The potentials Vi(x) (i=1,2) can be sign-changing; (3) The nonexistence of nontrivial solutions in the periodic case is obtained; (4) Infinitely many nontrivial solutions in the non-periodic case are obtained.