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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 finding of the necessary and sufficient conditions for the local quasilinearizability of elliptic Monge–Ampère systems. [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]

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 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 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]

This paper focuses on uniform boundary estimates in homogenization of a family of higher-order elliptic operators Lε, with rapidly oscillating periodic coefficients. We derive uniform boundary Cm-1,λ(0<λ<1) and Wm,p estimates in C1 domains, as well as uniform boundary Cm-1,1 estimate in C1,θ(0<θ<1) domains without the symmetry assumption on the operator. The proof, motivated by the works "Armstrong and Smart in Ann Sci Éc Norm Supér (4) 49(2):423-481 (2016) and Shen in Anal PDE 8(7):1565-1601 (2015)," is based on a suboptimal convergence rate in Hm-1(Ω). Compared to "Kenig et al. in Arch Ration Mech Anal 203(3):1009-1036 (2012) and Shen (2015)," the convergence rate obtained here does not require the symmetry assumption on the operator, nor additional assumptions on the regularity of u0 (the solution to the homogenized problem), and thus might be of some independent interests even for second-order elliptic systems. [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]

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]

The main aim of this paper is to discuss the problem concerning the analyticity of the solutions of analytic non-linear elliptic boundary value problems. It is proved that if the corresponding first variation is regular in Lopatinskiĭ sense, then the solution is analytic up to the boundary. The method of proof really covers the case that the corresponding first variation is regularly elliptic in the sense of Douglis-Nirenberg-Volevich, and hence completely generalize the previous result of C. B. Morrey. The author also discusses linear elliptic boundary value problems for systems of elliptic partial differential equations where the boundary operators are allowed to have singular integral operators as their coefficients. Combining the standard Fourier transform technique with analytic continuation argument, the author constructs the Poisson and Green’s kernel matrices related to the problems discussed and hence obtain some representation formulae to the solutions. Some a priori estimates of Schauder type and L p type are obtained. [ABSTRACT FROM AUTHOR]

In this paper,nonnegative solutions for the degenerate elliptic systems are considered. First,nonnegative solutions for scalar equation with spatial discontinuities are studied. Then the method developed for scalar equation is applied to study elliptic systems. At last,the existence criteria of nonnegative solutions of elliptic systems are given. [ABSTRACT FROM AUTHOR]

In this paper, the Dirichlet-type problem for the system of elliptic equations of second order with the degeneracy at a line crossing the domain is considered. The Dirichlet-type problem with additionally given asymptotics of the solution at this line is discussed. The uniqueness and the existence of the solution of this problem in the class of Hölder functions is proved. [ABSTRACT FROM AUTHOR]

Our aim in this paper was to establish by a variational method the existence of nontrivial solutions for strongly coupled Hamiltonian systems of the form 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 exponential subcritical or critical growth of the Trudinger-Moser type. Typical features of this class of problems are the lack of compactness because of the unboundedness of the domain and the critical growth. Moreover, the Lagrangian functional associated with this class of systems is strongly indefinite, that is, it has a saddle-point geometry where both positive and negative subspaces of the quadratic form are infinite-dimensional. To overcome these difficulties, we use an inequality of Trudinger-Moser type combined with Galerkin methods and a linking theorem. [ABSTRACT FROM AUTHOR]

We obtain multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals. There is no assumption to the effect that the nonlinear terms have to satisfy the Ambrosetti-Rabinowitz superquadratic condition. To establish the existence of solutions, a new version of the symmetric mountain pass theorem for strongly indefinite functionals is presented in this paper. This theorem is subsequently applied to deal with cases where all the Palais-Smale sequences of the energy functional may be unbounded. [ABSTRACT FROM AUTHOR]

This paper deals with an elliptic system of the form -Δu = θ1a(x)v + f (λ, x, v) in Ω, -Δv = λθ2a(x)u + g(λ, x, u) in Ω, u = 0 = v on ∂Ω, where λ ∈ R is a parameter and Ω RN (N ≥ 1) is a bounded domain with C2,ξ -boundary ∂Ω, ξ ∈ (0,1) (a bounded open interval if N = 1). Here a(x) ∈ L∞(Ω) with a(x) > 0 a.e. in Ω and θ1, θ2 > 0 are constants. The nonlinear perturbations f, g : R×Ω×R→R are Carathéodory functions that are sublinear at infinity. We provide sufficient conditions for determining the λ-direction to which a continuum of positive and negative solutions emanates from infinity at the first eigenvalue of the associated linear problem. Furthermore, as a consequence of main results, we also provide sufficient condition for the solvability of a class of asymptotically linear system near and at resonance satisfying Landesman-Lazer type conditions. [ABSTRACT FROM AUTHOR]

The control of a Cauchy system for an elliptic operator seems to be globally an open problem. In this paper, we analyze this problem using a regularization method which consists in viewing a singular problem as a limit of a family of well-posed problems. Following this analysis and assuming that the interior of considered convex is non-empty, we obtain a singular optimality system (S.O.S.) for the considered control problem. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of solution for two kinds of hemivariational inequalities: the first of them is of elliptic type, the second one of hamiltonian type. In those problems the energy functional is indefinite, so the classical variational principles can’t be used in a direct way. The results are an application of two theorems of existence of critical points for non-differentiable functionals recently obtained. [ABSTRACT FROM AUTHOR]

In this paper we consider the weakly coupled elliptic system with critical growthwherea,b,c,dareC1-functions defined in a bounded regular domain O of RN. Here we construct families of solutions which blow-up and concentrate at some points in O as the positive parameter e goes to zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with the formulation of a phenomenological model of finite elastoplasticity valid for small elastic strains for initially isotropic polycrystalline material. As a basic we assume the multiplicative split F = Fe Fp of the deformation gradient into elastic and plastic part. A key feature of the model is the introduction of an independent field of 'elastic' rotations Re which eliminate the remaining geometrical nonlinearities coming from finite elasticity in the presence of small elastic strains. In contrast to micro-polar theories an evolution equation for Re is presented which relates Re to Fe making use of a new device found by the author to perform the polar decomposition asymptotically. The model is shown to be invariant under both change of frame and rotation of the so called intermediate configuration. The corresponding equilibrium equations at frozen plastic and viscoelastic configuration constitute then a linear, elliptic system with nonconstant coefficients which makes this model amenable to a rigorous mathematical analysis. The introduced hysteresis effects within the elastic region are related to viscous elastic rotations of the grains of the polycrystal due to internal friction at the grain boundaries and constitute as such a rate dependent transient texture effect. The inclusion of work hardening will be addressed in future work. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R N be a bounded domain with N ≥ 3 . We address the existence and nonexistence of solutions for the following class of elliptic systems: - Δ u = a u + b v + μ | u | p - 2 u in Ω - Δ v = b u + a v + | v | 2 ∗ - 2 v in Ω with Dirichlet boundary condition, where a , b ∈ R , the exponent p > 1 , μ ∈ R is a parameter and 2 ∗ = 2 N / (N - 2) is the critical Sobolev exponent. By exploiting minimization arguments, minimax techniques and a Pohozaev identity, we obtain various results for the above system by analyzing the interplay between a , b , μ and the exponent p. [ABSTRACT FROM AUTHOR]

Let O ⊂ R be a bounded domain of class C . Let 0 < ε - 1. In L ( O;C) we consider a positive definite strongly elliptic second-order operator B with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent ( B − ζQ (·/ ε)) as ε → 0. Here the matrix-valued function Q is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L ( O;C)-operator norm and in the norm of operators acting from L ( O;C) to the Sobolev space H ( O;C) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q (x/ ε)∂v( x, t) = −( B v)( x, t). [ABSTRACT FROM AUTHOR]

Let Ω and Ω be two bounded open subsets of $${{\mathbb{R}}^{n}}$$ containing 0. Let G be a (nonlinear) map from $${\partial\Omega^{i}\times {\mathbb{R}}^{n}}$$ to $${{\mathbb{R}}^{n}}$$ . Let a be a map from ∂Ω to the set $${M_{n}({\mathbb{R}})}$$ of n × n matrices with real entries. Let g be a function from ∂Ω to $${{\mathbb{R}}^{n}}$$ . Let γ be a positive valued function defined on a right neighborhood of 0 in the real line. Let T be a map from $${]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})}$$ to $${M_{n}({\mathbb{R}})}$$ . Then we consider the problemwhere $${\nu_{\epsilon\Omega^{i}}}$$ and ν denote the outward unit normal to $${\epsilon\partial \Omega^{i}}$$ and ∂Ω, respectively, and where $${\epsilon >0 }$$ is a small parameter. Here (ω − 1) plays the role of ratio between the first and second Lamé constants and T(ω, ·) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor. Under the condition that $${\lim_{\epsilon\to 0^{+}}{\gamma(\epsilon)} \epsilon^{-1} (\log\epsilon)^{-\delta_{2,n}}=0}$$ and $${\lim_{\epsilon\to 0^{+}}{\epsilon^{n-1}}{\gamma(\epsilon)^{-1}}=0}$$ , we prove that under suitable assumptions the above problem has a family of solutions $${\{u(\epsilon,\cdot)\}_{\epsilon\in ]0,\epsilon'[}}$$ for $${\epsilon'}$$ sufficiently small and we analyze the behaviour of such a family as $${\epsilon}$$ approaches 0 by an approach which is alternative to those of asymptotic analysis. Here δ denotes the Kronecker symbol. [ABSTRACT FROM AUTHOR]

We provide sharp estimates in Lorentz spaces for the solution of the Dirichlet problem associated to the system where Ω is an open bounded subset of $${\mathbb R^n}$$ ( n ≥ 3) with sufficiently regular boundary, A( u) is an elliptic operator with VMO-coefficients and f is not in the natural dual space. Moreover, when the coefficients belong to $${{{C}^{0,\alpha}}(\alpha\in ]0,1[)}$$, we study the differentiability of the solution in Besov-Morrey spaces. [ABSTRACT FROM AUTHOR]

We establish the existence of weak solution to the following class of fractional elliptic systems (- Δ) s u + a (x) u = F u (x , u , v) , x ∈ R N , (- Δ) s v + b (x) v = F v (x , u , v) , x ∈ R N , where s ∈ (0 , 1) , the potentials a, b are bounded from below and may change sign. The nonlinear term F ∈ C 1 (R N × R 2 , R) can be asymptotically linear or superlinear at infinity. It interacts with the eigenvalues of the linearized problem. In the proofs we apply variational methods by considering both the resonant and nonresonant case. We notice that our results are new even in the local case s = 1 . [ABSTRACT FROM AUTHOR]

A two-sided game for the control of a stationary semilinear competitive system with autonomous sources is considered, where the controls are the kernels of the nonlocal interaction terms. The saddle point (the optimal solution of the game) is characterized as the unique solution of the associated optimality system, which is solved by an iterative scheme. [ABSTRACT FROM AUTHOR]

In a finite domain D of the complex plane bounded by a smooth contour Γ, we consider the Riemann–Hilbert boundary-value problem ReCU+ = f for the first-order elliptic system ∂ U ∂ y − A ∂ U ∂ x + a z U z + b z U z ¯ = F z with constant leading coefficients. Here + means the boundary value of the function U on Γ, the constant matrices A1,A2 ∈ ℂl×l and the (l × l)-matrix coefficients a and b belong to the Hölder class Cμ, 0 < μ < 1, and (l × l)-matrix function C belongs to the class Cμ(Γ). We prove that in the class U ∈ Cμ( D ¯ ) ∩ C1(D), this problem is a Fredholm problem and its index is given by the formula ϰ = − ∑ j = 1 m 1 π arg det G Γ j + 2 − m l. [ABSTRACT FROM AUTHOR]

Let Ω be a C1,s bounded domain (s > 1/2) in ℝd, and let A ε = − div A (x , x / ε) ∇ be a matrix elliptic operator on Ω with Dirichlet boundary condition. We suppose that ε is small and the function A is Lipschitz in the first variable and periodic in the second one, so the coefficients of A ε are locally periodic. For μ in the resolvent set, we are interested in finding the rates of approximations, as ε → 0, for (A ε − μ ρ ε) − 1 and ∇ (A ε − μ ρ ε) − 1 in the operator topology on L2. Here ρε(x)= ρ(x,x/ε) is a positive definite locally periodic function with ρ satisfying the same assumptions as A. Keeping track of the rate dependence on both ε and μ, we then proceed to similar questions for the solution to the initial boundary-value problem ρ ε ∂ t v ε = − A ε v ε . [ABSTRACT FROM AUTHOR]

It is shown that the Poisson kernel for the Lamé equation in a disk can be interpreted as a bi-univalent mapping of the projection of an elliptic cone onto the projection of the surface of revolution of a hyperbola. The corresponding mapping of these surfaces is bijective. Such an interpretation sheds light on the nature of the well-known special property of solutions of elliptic systems on a plane to map points to curves and vice versa. In particular, a singular point of the kernel under study can be considered as the projection of the cone element so that the mapping is regular in the sense that this element is bijectively mapped into a curve. [ABSTRACT FROM AUTHOR]

Method of sub- and super-solutions is applied in investigation of solvability in classical $$C^2(\Omega ) \bigcap C(\overline{\Omega })$$ sense of quasi-linear non-cooperative weakly coupled systems of elliptic second-order PDE. The result is illustrated on a model example arising in differential geometry. [ABSTRACT FROM AUTHOR]

Multiple solutions for a class of superquadratic elliptic systems near resonance with high eigenvalues are obtained by using the nabla theorem due to Marino and Saccon in (Topol. Methods Nonlinear Anal. 17:213-237, 2001) and the linking theorem. [ABSTRACT FROM AUTHOR]

For a strongly elliptic system of second-order equations of a special form, formulas for the Poisson integral and Green's function in a circle and an ellipse are obtained. The operator under consideration is represented by the sum of the Laplacian and a residual part with a small parameter, and the solution to the Dirichlet problem is found in the form of a series in powers of this parameter. The Poisson formula is obtained by the summation of this series. [ABSTRACT FROM AUTHOR]

In the general vector-valued case $$N \ge 1$$ , we prove the Lipschitz continuity of local minimizers to some integrals of the calculus of variations of the form $$ \int _{\Omega } g(x, |Du|)\,\mathrm{d}x$$ , with p, q-growth conditions only for $$|Du| \rightarrow +\infty $$ and without further structure conditions on the integrand $$g=g(x,|Du|)$$ . We apply the regularity results to weak solutions to nonlinear elliptic systems of the form $$\sum _{i=1}^{n}\frac{\partial }{\partial x_{i}}a_{i}^{\alpha }\left( x,Du\right) =0$$ , $$\alpha =1,2,\ldots ,N$$ . [ABSTRACT FROM AUTHOR]

We consider the Riemann-Hilbert (Hilbert) problem in classes similar to the Hardy class for general first-order elliptic systems on a plane. We establish basic properties ofHardy classes for solutions of that systems and solvability conditions for boundary-value problems. We construct the example demonstrating that for discontinuous coefficients the solvability features differ from the pictures of solvability of analogous problems for holomorphic and generalized analytic functions. In particular, the problem with positive index can be unsolvable. [ABSTRACT FROM AUTHOR]

We consider a region unbounded in at least one direction on which is defined an abstract elliptic system subject to non-homogeneous Neumann data on part $$\Gamma $$ of the surface that is otherwise free. We show that a quadratic form, analogous to the Dirichlet integral, becomes increasingly small on sub-regions increasingly remote from $$\Gamma $$ . The technique is illustrated by various specific linear and nonlinear examples, including the p-Laplacian, for some of which the method must be slightly modified. [ABSTRACT FROM AUTHOR]

Existence of weak solutions for a class of nonlinear elliptic systems is obtained under the certain Landesman-Lazer-type conditions by variational method. [ABSTRACT FROM AUTHOR]

We prove the nonexistence of positive solutions to the system where $$B$$ is the open unit ball in $$\mathbb {R}^{N}$$ , $$\ N>1,\ \lambda ,\mu $$ are positive constants bounded away from $$0$$ with $$\lambda \mu $$ large, $$f,g$$ are smooth functions with $$f(0),g(0)<0,\ f\circ (cg)$$ and $$g\circ (cf)$$ growing at least linearly at $$\infty $$ . [ABSTRACT FROM AUTHOR]

We study the competition of two species for a single resource in a chemostat. In the simplest space-homogeneous situation, it is known that only one species survives, namely the best competitor. In order to exhibit coexistence phenomena, where the two competitors are able to survive, we consider a space dependent situation: we assume that the two species and the resource follow a diffusion process in space, on top of the competition process. Besides, and in order to consider the most general case, we assume each population is associated with a distinct diffusion constant. This is a key difficulty in our analysis: the specific (and classical) case where all diffusion constants are equal, leads to a particular conservation law, which in turn allows to eliminate the resource in the equations, a fact that considerably simplifies the analysis and the qualitative phenomena. Using the global bifurcation theory, we prove that the underlying 2-species, stationary, diffusive, chemostat-like model, does possess coexistence solutions, where both species survive. On top of that, we identify the domain, in the space of the relevant bifurcation parameters, for which the system does have coexistence solutions. [ABSTRACT FROM AUTHOR]

A waveguide occupies a domain G in ℝ, n ⩾ 1, having several cylindrical outlets to infinity. The waveguide is described by a general elliptic boundary value problem that is self-adjoint with respect to the Green formula and contains a spectral parameter µ. As an approximation to a row of the scattering matrix S(µ) we suggest a minimizer of a quadratic functional J(·, µ). To construct such a functional, we solve an auxiliary boundary value problem in the bounded domain obtained by cutting off, at a distance R, the waveguide outlets to infinity. It is proved that, if a finite interval [µ, µ] of the continuous spectrum contains no thresholds, then, as R → ∞, the minimizer tends to the row of the scattering matrix at an exponential rate uniformly with respect to µ ∈ [µ, µ]. The interval may contain some waveguide eigenvalues whose eigenfunctions exponentially decay at infinity. [ABSTRACT FROM AUTHOR]

The aim of this work is to show that the ellipticity is a sufficient condition for the elliptic iterate property to hold for systems of linear differential operators in the anisotropic Roumieu classes. [ABSTRACT FROM AUTHOR]

In this article we prove solvability results for L boundary value problems of some elliptic systems Lu=0 on the upper half-space ${\mathbb{R}} ^{n+1}_{+}, n\ge1$, with transversally independent coefficients. We use the first order formalism introduced by Auscher-Axelsson-McIntosh and further developed with a better understanding of the classes of solutions in the subsequent work of Auscher-Axelsson. The interesting fact is that we prove only half of the Rellich boundary inequality without knowing the other half. [ABSTRACT FROM AUTHOR]

We prove the existence of the ground state for the spinor Bose-Einstein condensates in the one-dimensional case. [ABSTRACT FROM AUTHOR]

The authors establish a general monotonicity formula for the following elliptic system where Ω ⊂⊂ ℝ n is a regular domain, ( f i( x, u1, ..., u m)) = ∇ $$ \vec u $$ F( x, $$ \vec u $$), F( x, $$ \vec u $$) is a given smooth function of x ∈ ℝ n and $$ \vec u $$ = ( u1, ..., u m) ∈ ℝ m. The system comes from understanding the stationary case of Ginzburg-Landau model. A new monotonicity formula is also set up for the following parabolic system , where t1 < t2 are two constants, ( f i( x, $$ \vec u $$)), is given as above. The new monotonicity formulae are focused more attention on the monotonicity of nonlinear terms. The new point of the results is that an index β is introduced to measure the monotonicity of the nonlinear terms in the problems. The index β in the study of monotonicity formulae is useful in understanding the behavior of blow-up sequences of solutions. Another new feature is that the previous monotonicity formulae are extended to nonhomogeneous nonlinearities. As applications, the Ginzburg-Landau model and some different generalizations to the free boundary problems are studied. [ABSTRACT FROM AUTHOR]

We establish a priori bounds for positive solutions of semilinear elliptic systems of the form where Ω is a bounded and smooth domain in $${\mathbb{R}}^2$$ . We obtain results concerning such bounds when f and g depend exponentially on u and v. Based on these bounds, existence of positive solutions is proved. [ABSTRACT FROM AUTHOR]