Polyharmonic Systems Research Articles

Optimal Liouville-type theorems for polyharmonic elliptic gradient systems and applications

In this paper, we investigate the following polyharmonic systems (d≥1) in RN,{(−Δ)du1=f1(u1,u2),(−Δ)du2=f2(u1,u2). By using the method of Rellich-Pohozaev type identities, Sobolev and interpolation inequalities on SN−1 and feedback and measure arguments, we prove Liouville-type theorems for nonnegative classical solutions under an optimal Sobolev growth condition on f1 and f2. Furthermore, we prove priori bounds, decay and singularity estimates of solutions. This system can be applied to noncooperative Schrödinger system with polyharmonic operator which arises from nonlinear optics and Bose-Einstein condensates.

An Existence Result for a Class of Coupled Polyharmonic Systems

This work deals with the existence of positive continuous solutions for a nonlinear coupled polyharmonic system. Our analysis is based on some potential theory tools, properties of functions in the Kato class Km, n and the Schauder fixed point theorem.

Method of scaling spheres for integral and polyharmonic systems

We establish a method of scaling spheres for the integral system { u ( x ) = ∫ R n | y | a v p ( y ) | x − y | n − α d y , x ∈ R n , v ( x ) = ∫ R n | y | b u q ( y ) | x − y | n − β d y , x ∈ R n , where 0 < α , β < n , a > − α , b > − β and p , q > 0 . By using this method, we obtain a Liouville theorem for nonnegative solutions when 0 < p ≤ n + α + 2 a n − β , 0 < q ≤ n + β + 2 b n − α and ( p , q ) ≠ ( n + α + 2 a n − β , n + β + 2 b n − α ) . As an application, we derive a Liouville theorem for nonnegative solutions of the polyharmonic Hénon-Hardy system { ( − Δ ) m u ( x ) = | x | a v p ( x ) in R n , ( − Δ ) l v ( x ) = | x | b u q ( x ) in R n , where m and l are integers in ( 0 , n 2 ) .

Liouville-type theorem for higher-order Hardy-Hénon system

&lt;p style='text-indent:20px;'&gt;In this paper, we study higher-order Hardy-Hénon elliptic systems with weights. We first prove a new theorem on regularities of the positive solutions at the origin, then study equivalence between the higher-order Hardy-Hénon elliptic system and a proper integral system, and we obtain a new and interesting Liouville-type theorem by methods of moving planes and moving spheres for integral system. We also use this Liouville-type theorem to prove the Hénon-Lane-Emden conjecture for polyharmonic system under some conditions.&lt;/p&gt;

Polyharmonic systems involving critical nonlinearities with sign-changing weight functions

This article concerns the existence of multiple solutions of the polyharmonic system involving critical nonlinearities with sign-changing weight functions $$\displaylines{ (-\Delta)^mu = \lambda f(x) |u|^{r-2}u+ \frac{\beta}{\beta+\gamma} h(x) |u|^{\beta-2}u |v|^{\gamma}\quad \text{in }\Omega,\cr (-\Delta)^mv = \mu g(x) |v|^{r-2}v+ \frac{\gamma}{\beta+\gamma} h(x) |u|^{\beta} |v|^{\gamma-2} v \quad \text{in }\Omega, \cr D^ku=D^kv=0\quad \text{for all }|k|\leq m-1\quad \text{on }\partial\Omega, }$$ where \((-\Delta)^m\) denotes the polyharmonic operators, \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial \Omega\), \(m\in \mathbb N\), \(N\geq {2m+1}\), \(1&lt;r&lt;2\) and \(\beta&gt;1\), \(\gamma&gt;1\) satisfying \(2&lt;\beta+\gamma\leq 2_{m}^{*}\) with \(2_{m}^{*}=\frac{2N}{N-2m}\) as a critical Sobolev exponent and \(\lambda\), \(\mu&gt;0\). The functions f, g and \(h:\overline{\Omega}\to \mathbb R\) are sign-changing weight functions satisfying f, \(g\in L^{\alpha}(\Omega)\) and \(h\in L^{\infty}(\Omega)\) respectively. Using the variational methods and Nehari manifold, we prove that the system admits at least two nontrivial solutions with respect to parameter \((\lambda, \mu)\in \mathbb R^2_{+} \setminus \{(0, 0)\}\).&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/119/abstr.html

Existence and asymptotic behaviour of positive solutions for semilinear polyharmonic coupled systems

Let m be a positive integer. We investigate the existence and the asymptotic behaviour of positive continuous solutions to the following coupled polyharmonic system in the unit ball B of , ,where , , r, such that and the functions a, b are nonnegative measurable on B satisfying some assumptions related to Karamata regular variation theory.

Nonexistence of positive solutions for polyharmonic systems in $\mathbb{R}^N_+$

In this paper, we study themonotonicity and nonexistence of positive solutions for polyharmonicsystems $\left\{\begin{array}{rlll}(-\Delta)^m u&=&f(u, v)\\(-\Delta)^m v&=&g(u, v)\end{array}\right.\;\hbox{in}\;\mathbb{R}^N_+.$ By using the Alexandrov-Serrin method of moving plane combined with integral inequalities and Sobolev's inequality in a narrow domain, we prove the monotonicity of positive solutions for semilinear polyharmonic systems in $\mathbb{R_+^N}.$ As a result, the nonexistence for positive weak solutions to the system are obtained.

A Liouville Type Theorem for Poly-harmonic System with Dirichlet Boundary Conditions in a Half Space

Abstract In this paper, we consider the following poly-harmonic system with Dirichlet boundary conditions in a half space ℝn +: where for i = 1, 2. First, we show that, under some mild growth conditions, (0.1) is equivalent to the IE system where is the Green’s function in ℝn + with the same Dirichlet boundary conditions. Then, inspired by the work [12] of Y. Fang and W. Chen on the Dirichlet problem for (−Δ)mu = up in ℝn +, we use method of moving planes in integral forms to prove the nonexistence of nontrivial nonnegative solutions for IE system (0.2), and as a consequence, we derive the nonexistence of nontrivial nonnegative classical solutions for problem (0.1).

Qualitative properties of solutions for an integral system related to the Hardy–Sobolev inequality

This article carries out a qualitative analysis on a system of integral equations of the Hardy–Sobolev type. Namely, results concerning Liouville type properties and the fast and slow decay rates of positive solutions for the system are established. For a bounded and decaying positive solution, it is shown that it either decays with the slow rates or the fast rates depending on its integrability. Particularly, a criterion for distinguishing integrable solutions from other bounded and decaying solutions in terms of their asymptotic behavior is provided. Moreover, related results on the optimal integrability, boundedness, radial symmetry and monotonicity of positive integrable solutions are also established. As a result of the equivalence between the integral system and a system of poly-harmonic equations under appropriate conditions, the results translate over to the corresponding poly-harmonic system. Hence, several classical results on semilinear elliptic systems are recovered and further generalized.

Shooting with degree theory: Analysis of some weighted poly-harmonic systems

In this paper, the author establishes the existence of positive entire solutions to a general class of semilinear poly-harmonic systems, which includes equations and systems of the weighted Hardy–Littlewood–Sobolev type. The novel method used implements the classical shooting method enhanced by topological degree theory. The key steps of the method are to first construct a target map which aims the shooting method and the non-degeneracy conditions guarantee the continuity of this map. With the continuity of the target map, a topological argument is used to show the existence of zeros of the target map. The existence of zeros of the map along with a non-existence theorem for the corresponding Navier boundary value problem imply the existence of positive solutions for the class of poly-harmonic systems.

Existence and multiplicity results for critical growth polyharmonic elliptic systems

In the present paper, we deal with the existence and multiplicity of nontrivial solutions for a class of polyharmonic elliptic systems with Sobolev critical exponent in a bounded domain. Some new existence and multiplicity results are obtained. Our proofs are based on the Nehari manifold and Ljusternik–Schnirelmann theory. Copyright © 2013 John Wiley & Sons, Ltd.

Blow up at infinity of solutions of polyharmonic Kirchhoff systems

This article concerns the blow up at infinity of global solutions of strongly damped polyharmonic Kirchhoff systems, involving lower order terms, a time dependent nonlinear dissipative function Q and a driving force f, under homogeneous Dirichlet boundary conditions. Some applications are presented in special subcases of f and Q.

On the existence of positive continuous solutions for some polyharmonic elliptic systems on the half

We study the existence of positive continuous solutions of the nonlinear polyharmonic system \((-\Delta)^m u + \lambda q g(v) = 0\); \((-\Delta)^m v + \mu p f(u) = 0\) in the half space \(\mathbb{R}^n_+:=\{x = (x_1,...,x_n) \in \mathbb{R}^n : x_n \gt 0\}\), where \(m \geq 1\) and \(n \gt 2m\). The nonlinear term is required to satisfy some conditions related to the Kato class \(K^{\infty}_{m,n}(\mathbb{R}^n_+)\). Our arguments are based on potential theory tools associated to \((-\Delta)^m\) and properties of functions belonging to \(K^{\infty}_{m,n}(\mathbb{R}^n_+)\).

Local asymptotic stability for polyharmonic Kirchhoff systems†

In this article we treat the question of local asymptotic stability for polyharmonic Kirchhoff systems, governed by time-dependent source forces and nonlinear damping terms. Even the simplest case (𝒫3) studied here includes several systems of great interest, as the physical models for vibrating beams of the Woinowsky–Krieger type. This article extends and generalizes some local stability results of Autuori et al. [Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl. 352 (2009), pp. 149–165] to higher order systems, and also of Nakao and Zhu [Decay rate of solutions of a wave equation with damping and external force, Nonlinear Anal. 46 (2001), pp. 335–345; An attractor for a nonlinear dissipative wave equation of Kirchho type, J. Math. Anal. Appl. 353 (2009), pp. 652–659].

Existence of Positive Radial Solutions for a Nonvariational Polyharmonic System

In this work we first prove a new Liouville-type result for a polyharmonic system in $$\mathbb {R}^{N}$$ , then prove the existence of positive radial solutions for a polyharmonic system without variational structure in bounded domains via topological method.

Asymptotic stability for nonlinear Kirchhoff systems

We study the asymptotic stability for solutions of the nonlinear damped Kirchhoff system, with homogeneous Dirichlet boundary conditions, under fairly natural assumptions on the external force f and the distributed damping Q . Then the results are extended to a more delicate problem involving also an internal dissipation of higher order, the so called strongly damped Kirchhoff system. Finally, the study is further extended to strongly damped Kirchhoff–polyharmonic systems, which model several interesting problems of the Woinowsky–Krieger type.

Nonexistence of Positive Solutions for Polyharmonic Systems in ℝN

Abstract This work is devoted to the nonexistence of positive solutions for polyharmonic systems (−Δ)mu = f(u, v), (−Δ)mv = g(u, v). Byusing the method of moving plane combined with integral inequalities and Hardy’s inequality, we prove some new Liouville type theorems for the above semilinear polyharmonic systems in ℝN.

On the nonexistence of positive solutions of polyharmonic systems in [formula omitted

In this work, we consider the polyharmonic system { ( − Δ ) m u = v p , ( − Δ ) m v = u q , in R N , with m ≥ 1 , N > 2 m , p , q > 0 , where ( − Δ ) m is the polyharmonic operator. We obtain some nonexistence results for positive solutions for this system.

A Liouville type theorem for polyharmonic elliptic systems

In this paper, we consider the polyharmonic system ( − Δ ) m U = V q , ( − Δ ) m V = U p in R N , for m > 1 , N > 2 m , with p ⩾ 1 , q ⩾ 1 , but not both equal to 1, where ( − Δ ) m is the polyharmonic operator. Set α = 2 m ( q + 1 ) / ( p q − 1 ) , β = 2 m ( p + 1 ) / ( p q − 1 ) , for α , β ∈ [ ( N − 2 ) / 2 , N − 2 m ) , we prove the nonexistence of positive solutions.

On the existence of radial positive entire solutions for polyharmonic systems

Two-dimensional nonlinear, polyharmonic systems of the type Δ ( | Δ n u j | ( p j − 2 ) Δ n u j ) = f j ( | x | , u 1 , u 2 , | ∇ u 1 | , | ∇ u 2 | ) , x ∈ R 2 , j = 1 , 2 , are considered, where p j > 1 is a constant and f j : [ 0 , ∞ ) × ( 0 , ∞ ) 2 × [ 0 , ∞ ) 2 → [ 0 , ∞ ) is a continuous function for j = 1 , 2 . Some sufficient conditions are obtained for the existence of infinitely many radial positive entire solutions of the system with the prescribed asymptotic behavior at infinity.