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In this paper, we prove the existence of solutions for an elliptic system. More precisely, we combine the potential theory with the sub-super solution method and use the properties of the well-known Kato class to justify our existence results. The novelty of our study is that we consider either the bounded or the exterior domain; Also, the nonlinearities may be singular near the boundary. Some examples are presented to validate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study some systems of elliptic PDEs involving the 1 - Laplacian operator. In the first one, we deal with the subcritical regime, while in the second, we study a system with nonlinearities with critical growth. The approach is based on an approximation argument, in which the solutions are obtained as the limit of related problems with the p - Laplacian operator. In order to overcome the lack of compactness in the critical case, a version of the Concentration of Compactness Principle of Lions is proved. [ABSTRACT FROM AUTHOR]

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following system: - w Δ λ u - ∇ λ w. ∇ λ u = ρ v p - w Δ λ v - ∇ λ w. ∇ λ v = ρ u q in R N , where w , ρ are nonnegative continuous functions satisfying some growth conditions at infinity and p , q > 1 . Here, Δ λ is the sub-elliptic operator introduced in [A.E. Kogoj and E. Lanconelli. Nonlinear Anal. 2012;75(12): 4637–4649] and is of the form Δ λ = ∑ i = 1 N ∂ x i (λ i 2 ∂ x i). Our purpose is to establish a Liouville-type theorem for the class of positive stable solutions of the system. On one hand, our result generalizes the result in Duong and Nguyen (Electron J Differ Equ Paper No. 108, 11 pp, 2017) from the equation to the system, and on the other hand, it extends that of Hu (NoDEA Nonlinear Differ Equ Appl 25(1):7, 2018) to the sub-elliptic case. [ABSTRACT FROM AUTHOR]

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

The paper deals with the three-dimensional Dirichlet boundary value problem (BVP) of the couple-stress elasticity theory for anisotropic inhomogeneous solids and develops the generalized potential method based on the localized parametrix method. Using Green's integral representation formula and the properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to the system of localized boundary-domain integral equations (LBDIE). The equivalence between the Dirichlet BVP and the corresponding LBDIE system is studied. We state that the obtained localized boundary-domain integral operator belongs to the Boutet de Monvel algebra and, using the Wiener-Hopf factorization method, we investigate the corresponding Fredholm properties and prove the invertibility of the localized operator in appropriate Sobolev function spaces. [ABSTRACT FROM AUTHOR]

In a recent joint paper with Hwang and Mitrea we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domain above a Lipschitz graph, satisfying Lp-boundary data for p near 2. The main novel aspect of our result is that it applies to operators with coefficients of limited regularity and applies to operators satisfying a natural Carleson condition that has been first considered in the scalar case. In this paper we extend this result in several directions. We improve the range of solvability of the Lp Dirichlet problem to the interval 2 − ε < p < 2 (n − 1) (n − 3) + ε , for systems in dimension n = 2, 3 in the range 2 − ε < p < ∞. We do this by considering solvability of the Regularity problem (with boundary data having one derivative in Lp) in the range 2 − ε < p < 2 + ε. Secondly, we look at perturbation type-results where we can deduce solvability of the Lp Dirichlet problem for one operator from known Lp Dirichlet solvability of a "close" operator (in the sense of Carleson measure). This leads to improvement of the main result of our previous paper; we establish solvability of the Lp Dirichlet problem in the interval 2 − ε < p < 2 (n − 1) (n − 2) + ε under a much weaker (oscillation-type) Carleson condition. A particular example of the system where all these results apply is the Lamé operator for isotropic inhomogeneous materials with Poisson ratio ν < 0.396. In this specific case further improvements of the solvability range are possible. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the symmetry and monotonicity of positive solutions for linear coupling elliptic systems in a ball in $ \mathbf{R}^N $ R N . Using the Alexandrov–Serrin method of moving planes combined with the strong maximum principle, we prove that the solutions of elliptic systems with linear couplings in a ball are symmetric w.r.t. 0 and radially decreasing. For our problems, the tangential gradient of solutions and the coupling conditions play important roles in using the moving plane method. Our results on the symmetry of solutions are further research based on the existence of solutions in [1]. [ABSTRACT FROM AUTHOR]

This paper studies a stochastic control problem with regime switching in a fairly general abstract setting. Such problems may arise from production planning management. We perform a full mathematical analysis of this stochastic control problem via the HJB equation and verification. The connection of the optimal controls and subgame perfect controls is discussed, and it is shown that the optimal controls solve the generalized HJB equation as well. In a special case we provide a closed form solution. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose some results which we hope could contribute to understand better Lavrentiev's phenomenon for energy integrals as in (1.1) under some p; qgrowth conditions as in (1.2); in fact, we expect that Lavrentiev's phenomenon does not occur if the quotient q=p is not too large in dependence of n, for instance, as in the cases -- either scalar or vectorial ones -- that we consider in this manuscript. [ABSTRACT FROM AUTHOR]

In this paper, we consider the homogenization problem for generalized elliptic systems$ \mathcal{L}_{ \varepsilon} = -\operatorname{div}[A(x/ \varepsilon)\nabla+V(x/ \varepsilon)]+B(x/ \varepsilon)\nabla+c(x/ \varepsilon)+\lambda I $with dimension two. Precisely, we will establish the $ W^{1, p} $ estimates, Hölder estimates, Lipschitz estimates and $ L^p $ convergence results for $ \mathcal{L}_{ \varepsilon} $ with dimension two. The operator $ \mathcal{L}_{ \varepsilon} $ has been studied by Qiang Xu with dimension $ d\geq 3 $ in [22,23] and the case $ d = 2 $ is remained unsolved. As a byproduct, we will construct the Green functions for $ \mathcal{L}_{ \varepsilon} $ with $ d = 2 $ and their convergence rates. [ABSTRACT FROM AUTHOR]

In this paper we prove existence (and summability properties) of solutions for the following elliptic system$ \left\{ \begin{array}{cl} -{\rm{div}}(A(x)\,{\nabla} u) + u^{{\lambda}} = -{\rm{div}}(u^{{\lambda}} \, M(x)\,{\nabla}\psi) + f(x)\,, & {\rm{in }}\; \Omega , \\ -{\rm{div}}(M(x)\,{\nabla}\psi) = u^{\rho}\,, & {\rm{in }}\; \Omega , \\ u = 0 = \psi & \;{\rm{on}}\; \partial\Omega , \end{array} \right. $under some assumptions on $ {\lambda} > 0 $, $ \rho > 0 $ and $ f(x) $ in $ L^{{m}}(\Omega) $, $ m \geq 1 $. 'Return of the Patriarca' (see [3]) [ABSTRACT FROM AUTHOR]

In this paper, we establish a weighted Trudinger–Moser type inequality and as the application of this result by using the Galerkin methods and a linking theorem, we prove the existence of weak solutions for the following class of elliptic systems: { (− Δ) 1 / 2 u + V (x) u = g (x , v) , v > 0 in R , (− Δ) 1 / 2 v + V (x) v = f (x , u) , u > 0 in R , when the potential V is neither bounded away from zero, nor bounded from above. The nonlinear terms g (x , s) and f (x , s) are superlinear at infinity and have subcritical or critical exponential growth. [ABSTRACT FROM AUTHOR]

The paper deals with finding of the necessary and sufficient conditions for the local quasilinearizability of elliptic Monge–Ampère systems. [ABSTRACT FROM AUTHOR]

In this paper we prove, using a duality method, existence of solutions for the nonlinear elliptic mean field games type system {− div(M(x)∇u) + u − div(u A(x)∇ψ) = ƒ(x) in Ω , − div(M(x)∇ψ) + ψ + A(x)∇ψ⋅∇ψ = up−1 in Ω, u = 0 = ψ on ∂Ω, under different assumptions on p > 1, and the function ƒ(x) in Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this paper, we establish the estimates for solutions of second order elliptic problems with drift terms in bounded Lipschitz domains by using a real variable method. For scalar equations, we prove that the estimates hold for for , and the range for is sharp. For elliptic systems, we prove that the estimates hold for under the assumption that the Lipschitz constant of the domain is small. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we prove a partial Hölder regularity result for weak solutions , , to non-autonomous elliptic systems with general growth of the type: The crucial point is that the operator satisfies very weak regularity properties and a general growth, while the inhomogeneity has a controllable growth. [ABSTRACT FROM AUTHOR]

In this paper we prove existence of solutions for an elliptic system related to the stationary thermistor problem with bounded sources, and thermal and electrical condictivities growing as powers of the temperature. [ABSTRACT FROM AUTHOR]

In this paper, we establish uniform W1,p estimates for composite material problems which can be described by a divergence form elliptic system on a nonsmooth domain composed of a finite number of subdomains. We want to derive global W1,p regularity under the assumption that the coefficients are almost (δ, R)-vanishing of codimension 1 (see Definition 1.2) in each of multiple subdomains and the boundaries of subdomains are Reifenberg flat, moreover the estimates do not depend on the distance between these subdomains. [ABSTRACT FROM AUTHOR]

In this paper, we prove existence results for distributional solutions of anisotropic nonlinear elliptic systems with variable growth and degenerate coercivity. The functional setting involves anisotropic Sobolev spaces with variable exponents as well as weak Lebesgue (Marcinkiewicz) spaces with variable exponents. [ABSTRACT FROM AUTHOR]

In this paper we deal with the following class of Hamiltonian elliptic systems {−Δu = g(v) in Ω, −Δv = ƒ(u) in Ω, u = v = 0 on ∂Ω, where Ω ⊂ R2 is a bounded domain and g is a nonlinearity with exponential growth condition. We derive the maximal growth conditions allowed for ƒ, proving that it can be of exponential type, double-exponential type, or completely arbitrary, depending on the conditions required for g. Under the hypothesis of arbitrary growth conditions or else when ƒ has a double exponential growth, we prove existence of nontrivial solutions for the system. [ABSTRACT FROM AUTHOR]

The paper is devoted to the homogenization of elliptic systems in divergence form. We obtain uniform interior as well as boundary Lipschitz estimates in a bounded C1, γ domain when the coefficients are Dini continuous, inhomogeneous terms are divergence of Dini continuous functions and the boundary functions have Dini continuous derivatives. The results extend Avellaneda and Lin's work [Comm. Pure Appl. Math., 40: 803–847 (1987)], where H¨older continuity is the main assumption on smoothness of the data. [ABSTRACT FROM AUTHOR]

In this paper we give a classification of positive radial solutions of the following system: ▵u =vm, ▵v =h(∣x∣)g(u)f(∣▿u∣), in the open ball BR, with m > 0 and f , g, h nonnegative nondecreasing continuous functions. In particular, we deal with both explosive and bounded solutions. Our results involve, as in (27), a generalization of the well-known Keller-Osserman condition, namely, ∫∝1 (∫s0 F(t) dt)-m/(2m+1) ds < ∝, where F(t) =∫t0 f(s) ds Moreover, in the second part of the paper, the p-Laplacian version, given by ▵pu =vm, ▵pv =f(∣▿u∣), is treated. When p £2, we prove a necessary condition for the existence of a solution with at least a blow up component at the boundary, precisely ∫∝1 (∫s0 F(t) dt)-m/(mp+p-1)s(p-2)(p-1)/(mp+p-1) ds < ∝. [ABSTRACT FROM AUTHOR]

This work deals with the system (−Δ) u = a( x) v, (−Δ) v = b( x) u with Dirichlet boundary condition in a domain $${\Omega\subset\mathbb{R}^n}$$ , where Ω is a ball if n ≥ 3 or a smooth perturbation of a ball when n = 2. We prove that, under appropriate conditions on the parameters ( a, b, p, q, m, n), any nonnegative solution ( u, v) of the system is bounded by a constant independent of ( u, v). Moreover, we prove that the conditions are sharp in the sense that, up to some border case, the relation on the parameters are also necessary. The case m = 1 was considered by Souplet (Nonlinear Partial Differ Equ Appl 20:464-479, ). Our paper generalize to m ≥ 1 the results of that paper. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of solutions to systems of strongly coupled integral equations with oscillatory coefficients. The system of equations is motivated by a peridynamic model of the deformation of heterogeneous media that additionally accounts for shortrange forces. We consider the vanishing nonlocality limit on the same length scale as the heterogeneity and show that the system's effective behavior is characterized by a coupled system of local equations that are elliptic in the sense of Legendre--Hadamard. This effective system is characterized by a fourth-order tensor that shares properties with Cauchy elasticity tensors that appear in the classical equilibrium equations for linearized elasticity. [ABSTRACT FROM AUTHOR]

In this paper, we prove the validity of Gibbons' conjecture for a coupled competing Gross–Pitaevskii system. We also provide sharp a priori bounds, regularity results and additional Liouville-type theorems. [ABSTRACT FROM AUTHOR]

In this paper, we study existence and properties of solutions for some nonlinear elliptic systems like { − div (A (x) ∇ u) + u = − div (u M (x) ∇ ψ) + f (x) , − div (M (x) ∇ ψ) = u θ. The above Dirichlet problem is related to evolution systems connected with the mathematical study of partial differential equations models for chemotaxis and to systems connected with the mathematical analysis of models of flows in porous media. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of weak solutions for a quasilinear elliptic system involving the fractional pseudo-differential (p(.), q(.))-Laplacian operators. The approach is based on mountain pass theorem in which we prove that the energy functional associated to our problem satisfies the Palais–Smale and two mountain pass geometric conditions. [ABSTRACT FROM AUTHOR]

An optimal control of a steady state thermistor problem is considered where the convective boundary coefficient is taken to be the control variable. A distinct feature of this paper is that the problem is considered in R d , where d > 2 , and the electrical conductivity is allowed to vanish above a threshold temperature value. Existence of the state system is proved. An objective functional is introduced, existence of the optimal control is proved, and the optimality system is derived. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic behavior of local solutions for the semilinear elliptic system −Δu = |u|p−1u with boundary isolated singularity, where 1 < p < n+2/n−2, n ≥ 2 and u is a C2 nonnegative vector-valued function defined on the half space. This work generalizes the correspondence results of Bidaut-Véron-Ponce-Véron on the scalar case, and Ghergu-Kim-Shahgholian on the internal singularity case. [ABSTRACT FROM AUTHOR]

In this paper, we present the abstract results for the existence and uniqueness of the solution of nonlinear elliptic systems, parabolic systems and integro-differential systems involving the generalized $(p,q)$-Laplacian operator. Our method makes use of the characteristics of the ranges of linear and nonlinear maximal monotone operators and the subdifferential of a proper, convex, and lower-semi-continuous functional, and we employ some new techniques in the construction of the operators and in proving the properties of the newly defined operators. The systems discussed in this paper and the method used extend and complement some of the previous work. [ABSTRACT FROM AUTHOR]

In this paper, we show that W1,p (1 ≤ p < ∞ ) weak solutions to divergence form elliptic systems are Lipschitz and piecewise C¹ provided that the leading coefficients and data are of piecewise Dini mean oscillation, the lower-order coefficients are bounded, and interfacial boundaries are C1,Dini. This extends a result of Li and Nirenberg [Comm. Pure Appl. Math., 56 (2003), pp. 892-925]. Moreover, under a stronger assumption on the piecewise L¹-mean oscillation of the leading coefficients, we derive a global weak-type (1,1) estimate with respect to A1 Muckenhoupt weights for the elliptic systems without lower-order terms. [ABSTRACT FROM AUTHOR]

In this paper, we analyse the semilinear fourth-order problem (− Δ)2 u = g (u) in exterior domains of ℝ N . Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δ u > 0 exist if and only if N ≥ 5 and $$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$ for some δ > 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝ N and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δ u > 0 in ℝ N , and they do not exist when the previous condition fails. [ABSTRACT FROM AUTHOR]

In this review article is considered the comparison principle for linear and quasi-linear weakly coupled systems of elliptic and of parabolic PDE. It is demonstrated that a cooperativeness is a kind of a watershed quality for the comparison principle. Roughly speaking comparison principle holds for cooperative systems, while it does not hold for every noncooperative one. Considering a cooperative system one can apply the theory of a positive operator in a positive cone and prove the validity of the comparison principle. One particularly important result for cooperative systems is the existence of positive first eigenvalue and positive first eigenvector. Investigation of the validity of the comparison principle for non-cooperative system is more complicated. In this paper is mentioned the idea of division of the non-cooperative system in a cooperative and competitive part. Then the spectral properties of the cooperative part are employed in order to derive conditions for validity of comparison principle for the non-cooperative system. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of coexistence states for an elliptic system modeling two subpopulations of the same species competing for resources. Our results generalize and complement the work of Bouguima et al. (Nonlinear Anal., Real World Appl. 9:1184-1201, 2008). In particular, an open problem proposed by Bouguima et al. is partially solved. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for an elliptic system. More precisely, we combine the potential theory with the sub-super solution method and use the properties of the well-known Kato class to justify our existence results. The novelty of our study is that we consider either the bounded or the exterior domain; Also, the nonlinearities may be singular near the boundary. Some examples are presented to validate our main results.

In this paper, we establish the existence and multiplicity of solutions for a class of superlinear elliptic systems without Ambrosetti and Rabinowitz growth condition. Our results are based on minimax methods in critical point theory. [ABSTRACT FROM AUTHOR]

The paper concerns regularity theory for linear elliptic systems with divergence form arising from transmission problems. Estimates in BMO, Dini and Hölder spaces are derived simultaneously and the gaps among of them are filled by using Campanato-John-Nirenberg spaces. Results obtained in the paper are parallel to the classical regularity theory for elliptic systems. [ABSTRACT FROM AUTHOR]

In this paper, we study a Neumann problem for elliptic systems with variable exponents. We obtain the existence of at least three nontrivial solutions by using an equivalent variational approach to a recent Ricceri's three critical points theorem (Ricceri in Nonlinear Anal TMA 70:3084-3089, ). [ABSTRACT FROM AUTHOR]