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This paper aims to show that there exists a weak solution to the following quasilinear system driven by the M-Laplacian 0.1 (- Δ m 1) u = F u (x , u , v) i n Ω , (- Δ m 2) v = F v (x , u , v) i n Ω , u = v = 0 i n ∂ Ω , where Ω is a bounded open subset in R N and (- Δ m) is the M-Laplacian operator. Here we consider the non-reflexive case taking into account the Orlicz and Orlicz-Sobolev Space. The non-reflexive case occurs when the N-function M ¯ does not verify the Δ 2 -condition. We consider an approximated quasilinear elliptic problem driven by the M ϵ -Laplacian and using the Mountain Pass Theorem to obtain the existence of a nontrivial and nonnegative solution for the above system in reflexive case. By tending ϵ → 0 we get the solution in the non-reflexive case. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following two-component elliptic system with critical growth - Δ u + (V 1 (x) + λ) u = μ 1 u 3 + β u v 2 , x ∈ R 4 , - Δ v + (V 2 (x) + λ) v = μ 2 v 3 + β v u 2 , x ∈ R 4 , where V j (x) ∈ L 2 (R 4) are nonnegative potentials and the nonlinear coefficients β , μ j , j = 1 , 2 , are positive. Here we also assume λ > 0 . By variational methods combined with degree theory, we prove some results about the existence and multiplicity of positive solutions under the hypothesis β > max { μ 1 , μ 2 } . These results generalize the results for semilinear Schrödinger equation on half space by Cerami and Passaseo (SIAM J Math Anal 28:867–885, 1997) to the above elliptic system, while extending the existence result from Liu and Liu (Calc Var Partial Differ Equ 59:145, 2020). [ABSTRACT FROM AUTHOR]

This paper concerns the steady Euler-Poisson system with self-gravitation in a two dimensional annulus of finite radius, which can be used to describe the expansion of disk-like stars and ring galaxies. Via the stream function, the Euler-Poisson system in the subsonic region is reformulated into a second order nonlinear elliptic system. One of the crucial ingredients of the analysis is to obtain the well-posedness of the boundary value problem for the associated linearized elliptic system, which is achieved through a special structure of the elliptic system to gain an energy estimate. By combining the iteration method with a priori estimates of solutions to the elliptic system, we establish the existence, uniqueness and nonlinear structural stability of subsonic flows. [ABSTRACT FROM AUTHOR]

The ground states of two-component Bose--Einstein condensates with a Josephson junction can be described by minimizers of the problem e(a1, a2, β) := inf ∫R² (|u1|²+| u2|²)dx=1 Ea1,a2, β(u1,u2), where Ea1,a2,β (.) is the Gross--Pitaevskii energy functional Ea1,a2,β (u1,u2) .... By introducing some technical blow-up arguments, we classify how the Rabi frequency λ and potential Vi(x) (i = 1, 2) affect the existence and nonexistence of minimizers at one of the endpoints ai = a* or β = β*, where a* and β* can be given explicitly. This improves the results of [W. Bao and Y. Cai, East Asian J. Appl. Math., 1 (2011), pp. 49--81] and [Y. Guo et al., J. Funct. Anal., 276 (2019), pp. 183--230], where λ = 0, or both ai and β are nonendpoints. We also investigate the uniqueness of minimizers for certain local homogeneous potentials as β ↗ β*. Different from the case of λ = 0 in [Y. Guo et al., J. Funct. Anal., 276 (2019), pp. 183--230], λ ≠ 0 causes an uncontrolled error term in estimating the difference of Euler--Lagrange multipliers. To overcome this difficulty, we first establish the relationship between the L4-norm of minimizers and the partial derivatives of e(a1, a2, β) for almost all (a1, a2, β) and then show that the Euler--Lagrange multipliers are independent of the choice of minimizers. [ABSTRACT FROM AUTHOR]

The Cauchy problem for a second-order parabolic system with coefficients and a right-hand side that belong to a Zygmund anisotropic space is considered. A smoothness scale of the Cauchy problem solutions in anisotropic Zygmund spaces is obtained. A priori estimates for solutions to uniformly elliptic systems in isotropic Zygmund spaces are derived. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the boundedness of solutions of the Ginzburg–Landau system where and is the subelliptic operator In the stationary case, where the solutions are independent of the time variable, our result can be seen as an extension of some results in [A. Farina, B. Sciunzi, and N. Soave, Commun. Contemp. Math. 22 (5), Article no. 1950044 (2020)] from the Laplace operator to the subelliptic operator . [ABSTRACT FROM AUTHOR]

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This article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space of vector functions that are bounded and uniformly H¨older continuous in the entire complex plane. It was revealed that the problem of solving the system in the specified space may not be Noetherian. An example of a homogeneous system with an infinite number of linearly independent solutions was given. As is known, for many classes of elliptic systems, the Noetherianity of boundary value problems in a compact domain is equivalent to the presence of a priori estimates in the corresponding spaces. In this regard, it is important to study the issues related to the establishment of a priori estimates for the system under consideration in the above space. In the case of coefficients weakly oscillating at infinity, necessary and sufficient conditions for the validity of the a priori estimate were found. These conditions were written out in the language of the spectrum of limit matrices formed by the partial limits of the coefficient matrix at infinity. Specific examples were provided to illustrate how the limit matrices are constructed and what the above conditions look like.

This paper discusses the existence of multiple high‐energy solutions for a p$$ p $$‐Laplacian system involving critical Hardy‐Sobolev nonlinearity in ℝN$$ {\mathrm{\mathbb{R}}}^N $$. Considering that the "double" lack of compactness in the system is caused by the unboundedness of ℝN$$ {\mathrm{\mathbb{R}}}^N $$ and the presence of the critical Hardy–Sobolev exponent, we demonstrate the version to ℝN$$ {\mathrm{\mathbb{R}}}^N $$ of Struwe's classical global compactness result for double p$$ p $$‐Laplace operator. In virtue of the quantitative deformation lemma, a barycenter function, and the Brouwer degree theory, the existence of multiple high‐energy solutions is established. The results of this paper extend and complement the recent work. [ABSTRACT FROM AUTHOR]

In a bounded open domain $ \Omega \subset \mathbb {R}^{N} $ Ω ⊂ R N , where $ N\geq 2 $ N ≥ 2 , with Lipschitz boundary $ \partial \Omega $ ∂Ω , we consider the Dirichlet problem for the elliptic system given by \[ \left \{ \begin{array}{@{}ll} \displaystyle-\sum^{N}_{i=1}D_{i}\left(a_{i}(x,u(x),D_{i}u(x))\right)+F(x,u)=f(x), & x\quad \in \Omega,\\ u(x)=0, & x\quad \in \partial \Omega, \end{array} \right. \] { − ∑ i = 1 N D i (a i (x , u (x) , D i u (x))) + F (x , u) = f (x) , x ∈ Ω , u (x) = 0 , x ∈ ∂Ω , here, $ u:\Omega \rightarrow \mathbb {R}^{n} $ u : Ω → R n , $ n\geq 2 $ n ≥ 2 , represents a vector-valued function, $ D_{i}u=\frac {\partial u}{\partial x_{i}} $ D i u = ∂ u ∂ x i denotes the partial derivative of u with respect to $ x_i $ x i , and the vector fields $ a_i:\Omega \times \mathbb {R}^{n}\times \mathbb {R}^{n} \rightarrow \mathbb {R}^{n} $ a i : Ω × R n × R n → R n and $ F:\Omega \times \mathbb {R}^n\rightarrow \mathbb {R}^n $ F : Ω × R n → R n are Carathédory functions. In this paper, we focus on nonlinear degenerate anisotropic elliptic systems with variable growth and $ L^{m} $ L m data, where m is small. Specifically, we consider the case where the right-hand side term f belongs to $ L^{m}(\Omega ;\mathbb {R}^{n}) $ L m (Ω ; R n) with 1

In this paper, we consider the following elliptic system involving critical Sobolev and Hardy-Sobolev exponents - Δ u = 2 α 2 ∗ | u | α - 2 u | v | β + | u | 2 ∗ (s) - 2 u | x | s + a (x) | u | q - 2 u in Ω , - Δ v = 2 β 2 ∗ | u | α | v | β - 2 v + | v | 2 ∗ (s) - 2 v | x | s + b (x) | v | q - 2 v in Ω , u = v = 0 on ∂ Ω , where Ω is a bounded domains in R N satisfying some geometric condition, N ≥ 3 , 0 < s < 2 , 1 < q < 2 , 2 ∗ (s) : = 2 (N - s) (N - 2) and α , β > 1 such that α + β = 2 ∗ : = 2 ∗ (0) . Our main result asserts that, if N > max (4 , ⌊ 2 s ⌋ + 2) , then our problem has two disjoint and infinite sets of solutions. The present work may be seen as a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in Ambrosetti et al. (J Funct Anal 122:519–543, 1994) for an elliptic systems. [ABSTRACT FROM AUTHOR]

We consider the gradient flow of the Ambrosio–Tortorelli functional, proving existence, uniqueness and L2t(H2x) ∩ L∞t(H1x) ∩ H1t(L2x) regularity of the solution in dimension 2. Such functional is an approximation in the sense of Γ-convergence of the Mumford–Shah functional often used in problems of image segmentation and fracture mechanics. The strategy of the proof essentially follows the one of []Feng and Prohl, M2AN Math. Model. Numer. Anal.38 (2004) 291–320] but the crucial estimate is attained employing a different technique, and in the end it allows to prove better estimates than the ones obtained in [Feng and Prohl, M2AN Math. Model. Numer. Anal.38 (2004) 291–320]. In particular we prove that if U ⊂ ℝ2 is a bounded C2 domain, the initial data (u0, z0) ∈ [H1(U)]2 with 0 ≤ z0 ≤ 1, then for every T > 0 there exists a unique gradient flow (u(t), z(t)) of the Ambrosio–Tortorelli functional such that (u, z) ∈ [L2(0, T; H2(U)) ∩ L∞(0,T;H1(U)) ∩H1(0,T;L2(U)]2 while previously such regularity was known only for short times. [ABSTRACT FROM AUTHOR]

The paper deals with the equilibrium solutions of two-equation systems in which each component equation has a variational structure. Solutions are obtained that can be in one of the following generalized Nash situations: (a) one component of the solution represents a mountain pass type critical point and the other is a minimizer; (b) both components of the solution are mountain pass type; (c) both components are minimizers, that is, the solution is a proper Nash equilibrium. The simultaneous treatment of critical points of the mountain pass type and of the minimum ones is achieved by using a unifying notion of linking. The theory is applied to a system of four elliptic equations in which the subsystems formed by the first two and the last two equations, respectively, are of gradient type. An example shows that the conditions found are non-contradictory. The theory could be applied to other classes of systems. [ABSTRACT FROM AUTHOR]

Consider a source detection problem for a diffusion system at its stationary status, which is stated as the inverse source problem for an elliptic equation from the measurement of the solution specified only in part of the domain. For this linear ill-posed problem, we propose to reconstruct the interior source applying neural network algorithm, which projects the problem into a finite-dimensional space by approximating both the unknown source and the corresponding solution in terms of two neural networks. By minimizing a novel loss function consisting of PDE-fit and data-fit terms but without the boundary condition fit, the modified deep Galerkin method (MDGM) is applied to solve this problem numerically. Based on the stability result for the analytic extension of the solution, we strictly estimate the generalization error caused by the MDGM algorithm employing the property of conditional stability and the regularity of the solution. Numerical experiments show that we can obtain satisfactory reconstructions even in higher-dimensional cases, and validate the effectiveness of the proposed algorithm for different model configurations. Moreover, our algorithm is stable with respect to noisy inversion input data for the noise in various structures. [ABSTRACT FROM AUTHOR]

In this article, we study the following Hamiltonian elliptic system with coefficients:$ \begin{align} \notag \left\{\begin{aligned} -\varepsilon^2\Delta u +c(x)u = b(x) |{v}|^{q-1}v, &\, \, \text{ and } -\varepsilon^2\Delta v +c(x)v = a(x) |{u}|^{p-1}u \quad\text{in } \Omega, \\ u>0, \ v>0 \text{ in } \Omega, &\, \, \text{ and }\quad\frac{ \partial u}{ \partial\nu} = 0 = \frac{ \partial v}{ \partial\nu}\quad \text{on } \partial\Omega, \end{aligned} \right. \end{align} $where $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n, n\ge 3 $. The coefficients $ a(x), b(x), $ and $ c(x) $ are smooth, positive, and bounded. We study the existence of point-concentrating solutions and the influence of the coefficients on their concentration profiles. As an application of this result, we establish the existence of higher-dimensional concentrating solutions for a class of elliptic systems. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to formulate sufficient conditions guaranteeing the existence of decaying solutions of the system of elliptic equations in a certain exterior domain. The asymptotic behavior of these solutions and their gradients will be characterized. We focus on the construction of uncountable sets of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. Our method can be applied for nonlinearities which are negative at the origin (semipositone problem). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence of a weak solution to a non-local elliptic system, driven by the fractional m (⋅) -Laplacian operator in fractional Orlicz–Sobolev space that may be non-reflexive. The non-reflexive case occurs when the Orlicz function M ¯ does not verify the Δ 2 -condition. [ABSTRACT FROM AUTHOR]

In this article, we study the elliptic system: −Δu+μ1u=∣x∣αu3+λv,x∈Ω−Δv+μ2v=∣x∣αv3+λu,x∈Ωu,v>0,x∈Ω,u=v=0,x∈∂Ω,\left\{\begin{array}{ll}-\Delta u+{\mu }_{1}u=| x\hspace{-0.25em}{| }^{\alpha }{u}^{3}+\lambda v,& x\in \Omega \\ -\Delta v+{\mu }_{2}v=| x\hspace{-0.25em}{| }^{\alpha }{v}^{3}+\lambda u,& x\in \Omega \\ u,v\gt 0,x\in \Omega ,u=v=0,x\in \partial \Omega ,\end{array}\right. where Ω⊂R3\Omega \subset {{\mathbb{R}}}^{3} is the unit ball. By the variational method, we prove that if α\alpha is sufficiently small, the ground state solutions of the system are radial symmetric, and if α>0\alpha \gt 0 is sufficiently large, the ground state solutions are nonradial; however, the solutions are Schwarz symmetry.

In this article, we study a Hardy-Sobolev critical elliptic system involving coupled perturbation terms: (0.1)−Δu+V1(x)u=η1η1+η2∣u∣η1−2u∣v∣η2∣x′∣+αα+βQ(x)∣u∣α−2u∣v∣β,−Δv+V2(x)v=η2η1+η2∣v∣η2−2v∣u∣η1∣x′∣+βα+βQ(x)∣v∣β−2v∣u∣α,\left\{\begin{array}{c}-\Delta u+{V}_{1}\left(x)u=\frac{{\eta }_{1}}{{\eta }_{1}+{\eta }_{2}}\frac{{| u| }^{{\eta }_{1}-2}u{| v| }^{{\eta }_{2}}}{| x^{\prime} | }+\frac{\alpha }{\alpha +\beta }Q\left(x)| u{| }^{\alpha -2}u| v{| }^{\beta },\\ -\Delta v+{V}_{2}\left(x)v=\frac{{\eta }_{2}}{{\eta }_{1}+{\eta }_{2}}\frac{{| v| }^{{\eta }_{2}-2}v{| u| }^{{\eta }_{1}}}{| x^{\prime} | }+\frac{\beta }{\alpha +\beta }Q\left(x){| v| }^{\beta -2}v{| u| }^{\alpha },\end{array}\right. where n≥3n\ge 3, 2≤m1{\eta }_{1},{\eta }_{2}\gt 1, and η1+η2=2(n−1)n−2{\eta }_{1}+{\eta }_{2}=\frac{2\left(n-1)}{n-2}, α,β>1\alpha ,\beta \gt 1 and α+β

We are concerned with the existence of solution for elliptic system - Δ u = λ u + δ v + f (u +) + k 1 (x) in Ω , - Δ v = δ u + γ v + g (v +) + k 2 (x) in Ω , u = v = 0 on ∂ Ω , where Ω ⊂ R N , N ≥ 3 , is a smooth bounded domain, s + : = max { s , 0 } , λ , δ and γ are real parameters such that max { λ , γ } > 0 and δ > 0 . By topological degree arguments, we show the existence of nontrivial solutions for above system under certain superlinear growth conditions on f, g and an one-sided Landesman−Lazer condition on k 1 , k 2 . Also, a priori bound for the solutions are obtained by adapting the method of Brezis−Turner. [ABSTRACT FROM AUTHOR]

This paper shows the existence of positive solutions for a class of p-Laplacian systems with critical homogeneous nonlinearity. We prove that the equation has at least one positive solution by applying variational and topological methods. Moreover, we establish a version of the global compactness result in unbounded domain. [ABSTRACT FROM AUTHOR]

We study the nonexistence of positive solutions to the degenerate system of elliptic inequalities { − G α u ≥ v p − G α v ≥ u q in R N 1 × R N 2 , where p , q ∈ R and G α is the Grushin operator. Recently, it has been established in Le et al. (Liouville-type theorems for sub-elliptic systems involving Δ λ -Laplacian. Complex Var Ellip Equ. DOI:) that the system has no positive classical solution in the case where the exponents satisfy one of the following conditions p ≤ 0 or q ≤ 0 , p, q>0 and p q ≤ 1 , p, q>0, pq>1 and max { 2 (p + 1) p q − 1 , 2 (q + 1) p q − 1 } > N α − 2. Here, N α = N 1 + (1 + α) N 2 is the homogeneous dimension associated to the Grushin operator G α . The nonexistence of positive solutions is left open in the borderline case p , q > 0 , p q > 1 , max { 2 (p + 1) p q − 1 , 2 (q + 1) p q − 1 } = N α − 2. In this paper, we shall prove that the system has no positive classical solution in this case. In particular, when α = 0 , this result agrees with the well-known one for the system involving the Laplace operator. Our approach is based on the development of the idea of Serrin and Zou in (Non-existence of positive solutions of Lane–Emden systems. Differ Integral Equ. 1996;9(4):635–653) and the spherical mean formula for the Grushin operator. [ABSTRACT FROM AUTHOR]

We examine well-posedness questions in the Sobolev spaces of inverse problems of recovering the source functions in stationary convection-diffusion models represented in a form of a finite segment of a series in some basis. The overdetermination conditions are the values of a solution at some collection of points lying inside the domain and on its boundary. The conditions obtained ensure existence and uniqueness of solutions to these problems in the Sobolev classes and the Fredholm property of the problem. [ABSTRACT FROM AUTHOR]

For a Jordan domain with sufficiently smooth boundaries, the solution to the Dirichlet problem for second-order skew-symmetric strongly elliptic system with constant coefficients and regular enough boundary data is constructed in the form of a power series of a small parameter describing the perturbation of the given system from the Laplace one. The coefficients of this series are the functions that are determined sequentially as solutions to special Dirichlet problems for the usual Laplace and Poisson equations. The obtained series converges uniformly in the closure of the domain under consideration. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence and multiplicity of component-wise nontrivial critical points (i.e., whose components are both nonzero) for an abstract C 1 -functional J defined in X × X with X being a reflexive Banach space. Applying the Nehari manifold method and the minimization method along with the geometric properties of the constructed functions, we investigate the existence and multiplicity of component-wise nontrivial critical points of J. As applications, we prove the existence and multiplicity of component-wise nontrivial solutions for the second-order and fourth-order elliptic systems. Our theoretical results improve and extend the relevant results in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the elliptic system of competitive type with nonhomogeneous terms $ \Delta u = u^pv^q+h_1(x) $, $ \Delta v = u^rv^s+h_2(x) $ in $ \Omega $ with two types of boundary conditions: (Ⅰ) $ u = v = +\infty $ and (SF) $ u = +\infty $, $ v = f $ on $ \partial\Omega $, where $ f > 0 $, $ (p-1)(s-1)-qr > 0 $, and $ \Omega \subset \mathbb{R^N} $ is a smooth bounded domain. The nonhomogeneous terms $ h_1(x) $ and $ h_2(x) $ may be unbounded near the boundary and may change sign in $ \Omega $. First, for a single semilinear elliptic equation with a singular weight and nonhomogeneous term, boundary asymptotic behaviour of large positive solutions is established. Using this asymptotic behaviour, we show existence of large positive solutions for this elliptic system with the boundary condition (SF), existence of maximal solution, boundary asymptotic behaviour and uniqueness of large positive solutions for this elliptic system with (Ⅰ).

В работе рассматриваются двумерные сингулярные интегральные операторы по ограниченной области с коэффициентами при интегралах, содержащими в нескольких точках существенный разрыв и операторы с ядрами, имеющими в нескольких точках фиксированные особенности типа однородных функций порядка -2. Такие операторы широко применяются при изучении различных краевых задач для эллиптических систем уравнений первого и второго порядка с сингулярными коэффициентами на плоскости (см. напр. [1]-[4]). Одно из таких приложений приведено в конце настоящей работы. Сначала излагаются результаты исследования разрешимости (нетеровости) двумерного сингулярного интегрального уравнения с коэффициентом при интеграле, содержащим в одной точке существенный разрыв.

We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of nonnegative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.

In this article, using Nehari manifold method we study the multiplicity of solutions of the nonlocal elliptic system involving variable exponents and concave-convex nonlinearities, $$\displaylines{ (-\Delta)_{p(\cdot)}^{s} u=\lambda a(x)| u|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x) +\beta(x)}c(x)| u|^{\alpha(x)-2}u| v| ^{\beta(x)}, \quad x\in \Omega; \cr (-\Delta)_{p(\cdot)}^{s} v=\mu b(x)| v|^{q(x)-2}v+\frac{\alpha(x)}{\alpha(x) +\beta(x)}c(x)| v|^{\alpha(x)-2}v| u| ^{\beta(x)},\quad x\in \Omega; \cr u=v=0,\quad x\in \Omega^c:=\mathbb R^N\setminus\Omega, }$$ where $\Omega\subset\mathbb R^N$, $N\geq2$ is a smooth bounded domain, $\lambda,\mu>0$ are parameters, and $s\in(0,1)$. We show that there exists $\Lambda>0$ such that for all $\lambda+\mu

In this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth:

Abstract We study the existence and multiplicity of positive radial solutions for a coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the boundary conditions are nonlocal. We use a new cone to establish the existence of solutions by means of fixed point index theory.

Abstract We examine the weighted Grushin system involving advection terms given by { Δ G u − a ⋅ ∇ G u = ( 1 + ∥ z ∥ 2 ( α + 1 ) ) γ 2 ( α + 1 ) v − p in R n , Δ G v − a ⋅ ∇ G v = ( 1 + ∥ z ∥ 2 ( α + 1 ) ) γ 2 ( α + 1 ) u − q in R n , $$ \textstyle\begin{cases} \Delta _{G} u - a \cdot \nabla _{G} u =(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)})^{ \frac{\gamma }{2(\alpha +1)}} v^{-p} &\text{in $\mathbb {R}^{n}$}, \\ \Delta _{G} v - a \cdot \nabla _{G} v =(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)})^{ \frac{\gamma }{2(\alpha +1)}} u^{-q} &\text{in $\mathbb {R}^{n}$}, \end{cases} $$ where Δ G u = Δ x u + | x | 2 α Δ y u $\Delta _{G} u= \Delta _{x} u+ |x|^{2\alpha } \Delta _{y} u$ , z = ( x , y ) ∈ R n : = R n 1 × R n 2 $\mathbf{z}=(x,y) \in \mathbb {R}^{n}:= \mathbb {R}^{n_{1}} \times \mathbb {R}^{n_{2}}$ is the Grushin operator, α ≥ 0 $\alpha \geq 0$ , p ≥ q > 1 $p \geq q >1$ , ∥ z ∥ 2 ( α + 1 ) = | x | 2 ( α + 1 ) + | y | 2 $\|\mathbf{z}\|^{2(\alpha +1)}= |x|^{2(\alpha +1)} + |y|^{2} $ , γ ≥ 0 $\gamma \geq 0$ and a is a smooth divergence-free vector that we will specify later. Inspired by recent progress in the study of the Lane–Emden system, we establish some Liouville-type results for bounded stable positive solutions of the system. In particular, we prove the comparison principle to establish our result. As consequences, we obtain a Liouville-type theorem for the weighted Grushin equation involving advection terms Δ G u − a ⋅ ∇ G u = ( 1 + ∥ z ∥ 2 ( α + 1 ) ) γ 2 ( α + 1 ) u − p in R n . $$ \Delta _{G} u - a \cdot \nabla _{G} u =\bigl(1+ \Vert \mathbf{z} \Vert ^{2(\alpha +1)}\bigr)^{ \frac{\gamma }{2(\alpha +1)}} u^{-p} \quad \mbox{in } \mathbb {R}^{n}. $$ The main tools in the proof of the main result are the comparison principle, nonlinear integral estimates via the stability assumption and the bootstrap argument. Our results generalize and improve the previous work in (Duong et al. in Complex Var. Elliptic Equ. 64(12):2117–2129, 2019).

This study deals with the construction of conservation laws of the elliptic systems of differential equations in three independent variables. The variational approach is used, that is, the Noether point symmetries and their corresponding conserved vectors are obtained.

In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system − ε 2 Δ u + W (x) u = Q u (u , v) in R N , − ε 2 Δ v + V (x) v = Q v (u , v) in R N , u , v ∈ H 1 (R N) , u (x) , v (x) > 0 for each x ∈ R N , where ε > 0 , W and V are positive potentials of C 2 class and Q is a p-homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V. Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme. [ABSTRACT FROM AUTHOR]

We consider fully coupled cooperative systems on with coefficients that decay exponentially at infinity. Expanding some results obtained previously on bounded domain, we prove that the existence of a strictly positive supersolution ensures the first eigenvalue to exist and to be nonzero. This result is applied to show that the topological solutions for a Chern-Simons model, described by a semilinear system on with exponential nonlinearity, are nondegenerate. [ABSTRACT FROM AUTHOR]

In this paper, an optimal boundary control problem for a distributed elliptic system on Lipschitz domains with boundary homogeneous Dirichlet conditions and independently with Neumann conditions is analysed. The necessary and sufficient optimality conditions for such problems with the quadratic cost functionals are obtained. A Jacobi spectral Galerkin method is introduced to develop a direct solution technique for the numerical solution of elliptic problems subject to Dirichlet and Neumann conditions in one and two dimensions. The numerical examples are included to demonstrate the validity and applicability of the techniques and comparison is made with the existing results. The method is easy to implement and yields very accurate results.

Consider the system

In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully nonlinear operators, such as Pucci's extremal operators, being new even for scalar equations. [ABSTRACT FROM AUTHOR]

In this work, we study the following problem (𝒫) − Δ u = λ g (x) u + 1 2 ∗ f (x) H u (u , v) , in   ℝ N , − Δ v = μ h (x) v + 1 2 ∗ f (x) H v (u , v) , in   ℝ N , where N ≥ 4 , 2 ∗ : = 2 N / (N − 2) , H u and H v are the partial derivatives of the homogeneous function H ∈ C 1 (ℝ + 2 , ℝ) , where ℝ + 2 = [ 0 , ∞) × [ 0 , ∞) , g + ≢ 0 , h + ≢ 0 , f + ≢ 0 , g , h , f ∈ L ∞ (ℝ N) , g + , h + ∈ L N / 2 (ℝ N) and λ , μ > 0. [ABSTRACT FROM AUTHOR]

For elliptic systems with discontinuous nonlinearities, we study the existence of strong solutions whose values are points of continuity with respect to the state variables for almost all values of the spatial variable. Such solutions are said to be semiregular. An upper-and-lower-solution principle is established for the existence of semiregular solutions to elliptic systems with discontinuous nonlinearities. This principle is used to prove theorems on the existence of semiregular solutions of elliptic systems with discontinuous nonlinearities, in particular, nontrivial solutions of problems with a parameter. Examples of classes of nonlinearities with separated variables satisfying the conditions of our theorems are given. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of multiple non-radial sign-changing solutions for − Δ u + u + α K (| x |) Φ (x) u = | u | p − 2 u , x ∈ R 3 , − Δ Φ = K (| x |) u 2 , x ∈ R 3 , where 2

In this paper, we study the second-order boundary asymptotic behaviour for large positive solutions to an elliptic system of competitive type. First, we derive a second-order estimate to a related single weighted equation with boundary blow-up data. Then, by relaxing the system and iterating the estimate of the single equation, we establish second-order estimates of the solutions. [ABSTRACT FROM AUTHOR]

In this article we consider a viscous regularization of a p-system with a Van der Waals pressure law, which presents both hyperbolic and elliptic zones. Even if the purely hyperbolic Van der Walls system is strongly ill-posed, we prove that the solutions of the regularized equation exist and experience a transition from ellipticity to hyperbolicity, i.e. solutions issued from initial data in the elliptic zone will enter the hyperbolic zone at some time T>0, and viceversa.