The biharmonicity of the product map Φ2=φ×ψ and the two generalized projections φ-and ψ-are analyzed. Some results are obtained, that is, Φ2 is a proper biharmonic map if and only if b is a non-constant sol...The biharmonicity of the product map Φ2=φ×ψ and the two generalized projections φ-and ψ-are analyzed. Some results are obtained, that is, Φ2 is a proper biharmonic map if and only if b is a non-constant solution of -1/f2 Jφ（dφ（grad（lnb）））＋n/2 grad｜dφ（grad（lnb））｜2=0 and f is a non-constant solution of -1/b2Jψ（dψ（grad（lnf）））＋m/2grad｜dψ（grad（lnf））｜2=0, and Φ2=φ×ψ is a proper biharmonic map if and only if φ-and ψ-are proper biharmonic maps.展开更多
This article considers the equation △2u = f（x,u）with boundary conditions either u｜aΩ = au/an｜aΩ = 0 or u｜aΩ = △u｜aΩ = 0, where f（x, t） is asymptotically linear with respect to t at infinity, and Ω is a ...This article considers the equation △2u = f（x,u）with boundary conditions either u｜aΩ = au/an｜aΩ = 0 or u｜aΩ = △u｜aΩ = 0, where f（x, t） is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in R^N, N 〉 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f（x, t）.展开更多
In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequal...In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequality.展开更多
The purpose of this article is to establish the regularity of the weak solutions for the nonlinear biharmonic equation {△^2u ＋ a（x）u = g（x, u）u∈ H^2（R^N）, where the condition u∈ H^2（R^N） plays the role o...The purpose of this article is to establish the regularity of the weak solutions for the nonlinear biharmonic equation {△^2u ＋ a（x）u = g（x, u）u∈ H^2（R^N）, where the condition u∈ H^2（R^N） plays the role of a boundary value condition, and as well expresses explicitly that the differential equation is to be satisfied in the weak sense.展开更多
This paper is dedicated to studying the existence and multiplicity of symmetric solutions for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in RN. By virtue of variational methods and t...This paper is dedicated to studying the existence and multiplicity of symmetric solutions for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in RN. By virtue of variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions.展开更多
In this paper a semilinear biharmonic problem involving nearly critical growth with Navier boundary condition is considered on an any bounded smooth domain. It is proved that positive solutions concentrate on a point ...In this paper a semilinear biharmonic problem involving nearly critical growth with Navier boundary condition is considered on an any bounded smooth domain. It is proved that positive solutions concentrate on a point in the domain, which is also a critical point of the Robin’s function corresponding to the Green’s function of biharmonic operator with the same boundary condition. Similar conclusion has been obtained in [6] under the condition that the domain is strictly convex.展开更多
Let the elastic body only be acted by gravity. By investigating the relations of bianalytic functions and biharmonic functions, the uniqueness and existence of the stress functions (Airy functions) are established in ...Let the elastic body only be acted by gravity. By investigating the relations of bianalytic functions and biharmonic functions, the uniqueness and existence of the stress functions (Airy functions) are established in planar simple connected region. Moreover, the integral representation formula of the stress functions in the unit disk of the plane is obtained.展开更多
This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouvill...This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.展开更多
In this paper, we study a class of p（x）-biharmonic equations with Navier boundary condition. Using the mountain pass theorem, fountain theorem, local linking theorem and symmetric mountain pass theorem, we establish...In this paper, we study a class of p（x）-biharmonic equations with Navier boundary condition. Using the mountain pass theorem, fountain theorem, local linking theorem and symmetric mountain pass theorem, we establish the existence of at least one solution and infinitely many solutions of this problem, respectively.展开更多
In this article, by using the method of invariant sets of descending flow, we obtain the existence of sign-changing solutions of p-biharmonic equations with Hardy potential in RN.
In this article, we focus on the eigenvalue problem of the following linear biharmonic equation in R^N：△^2u-αu＋λg（x）u=0 with u ∈H^2（R^N）,u≠0,N≥5Note that there are two parameters α and λ in it, which is ...In this article, we focus on the eigenvalue problem of the following linear biharmonic equation in R^N：△^2u-αu＋λg（x）u=0 with u ∈H^2（R^N）,u≠0,N≥5Note that there are two parameters α and λ in it, which is different from the usual eigenvalue problems. Here, we consider λ as an eigenvalue and seek Ior a sulble range of parameter α, which ensures that problem （＊） has a maximal eigenvalue. As the loss of strong maximum principle for our problem, we can only get the existence of non-trivial solutions, not positive solutions, in this article. As an application, by using these results, we studied also the existence of non-trivial solutions for an asymptotically linear biharmonic equation in R^N.展开更多
Schwarz methods are an important type of domain decomposition methods. Using the Fourier transform, we derive error propagation matrices and their spectral radii of the classical Schwarz alternating method and the add...Schwarz methods are an important type of domain decomposition methods. Using the Fourier transform, we derive error propagation matrices and their spectral radii of the classical Schwarz alternating method and the additive Schwarz method for the biharmonic equation in this paper. We prove the convergence of the Schwarz methods from a new point of view, and provide detailed information about the convergence speeds and their dependence on the overlapping size of subdomains. The obtained results are independent of any unknown constant and discretization method, showing that the Schwarz alternating method converges twice as quickly as the additive Schwarz method.展开更多
基金The National Natural Science Foundation of China(No.10971029)
文摘The biharmonicity of the product map Φ2=φ×ψ and the two generalized projections φ-and ψ-are analyzed. Some results are obtained, that is, Φ2 is a proper biharmonic map if and only if b is a non-constant solution of -1/f2 Jφ（dφ（grad（lnb）））＋n/2 grad｜dφ（grad（lnb））｜2=0 and f is a non-constant solution of -1/b2Jψ（dψ（grad（lnf）））＋m/2grad｜dψ（grad（lnf））｜2=0, and Φ2=φ×ψ is a proper biharmonic map if and only if φ-and ψ-are proper biharmonic maps.
基金This work was supported by NSFC(10571174,10631030)and CAS(KJCX3-SYW-S03)
文摘This article considers the equation △2u = f（x,u）with boundary conditions either u｜aΩ = au/an｜aΩ = 0 or u｜aΩ = △u｜aΩ = 0, where f（x, t） is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in R^N, N 〉 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f（x, t）.
基金Supported by NSFC(10471047)NSF Guangdong Province(05300159).
文摘In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequality.
基金Supported by the National Natural Science Foundation of China (10631030)PHD specialized grant of Ministry of Education of China (20060511001) and supported in part by the Xiao-Xiang Special Fund, Hunan
文摘The purpose of this article is to establish the regularity of the weak solutions for the nonlinear biharmonic equation {△^2u ＋ a（x）u = g（x, u）u∈ H^2（R^N）, where the condition u∈ H^2（R^N） plays the role of a boundary value condition, and as well expresses explicitly that the differential equation is to be satisfied in the weak sense.
基金Supported by the Natural Science Foundation of China(11471235,11601052)funded by Chongqing Research Program of Basic Research and Frontier Technology(cstc2017jcyj BX0037)
文摘This paper is dedicated to studying the existence and multiplicity of symmetric solutions for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in RN. By virtue of variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions.
基金The research work was supported by the National Natural Foundation of China (10371045)Guangdong Provincial Natural Science Foundation of China (000671).
文摘In this paper a semilinear biharmonic problem involving nearly critical growth with Navier boundary condition is considered on an any bounded smooth domain. It is proved that positive solutions concentrate on a point in the domain, which is also a critical point of the Robin’s function corresponding to the Green’s function of biharmonic operator with the same boundary condition. Similar conclusion has been obtained in [6] under the condition that the domain is strictly convex.
文摘Let the elastic body only be acted by gravity. By investigating the relations of bianalytic functions and biharmonic functions, the uniqueness and existence of the stress functions (Airy functions) are established in planar simple connected region. Moreover, the integral representation formula of the stress functions in the unit disk of the plane is obtained.
基金partially supportedby Ministerio de Ciencia e Innovacion-SPAINFEDER,project MTM2010-15314supported by the Ministry of Science and Education of the Republic of Kazakhstan through the Project No.0713 GF
文摘This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.
基金supported by the National Natural Science Foundation of China(11071198)Scientific Research Fund of SUSE(2011KY03)Scientific Reserch Fund of Sichuan Provincial Education Department(12ZB081)
文摘In this paper, we study a class of p（x）-biharmonic equations with Navier boundary condition. Using the mountain pass theorem, fountain theorem, local linking theorem and symmetric mountain pass theorem, we establish the existence of at least one solution and infinitely many solutions of this problem, respectively.
基金Supported by NSFC 11361077Young Academic and Technical Leaders Program(2015HB028)Yunnan Normal University,Lian Da Scholar Program
文摘In this article, by using the method of invariant sets of descending flow, we obtain the existence of sign-changing solutions of p-biharmonic equations with Hardy potential in RN.
基金supported by the National Science Foundation of China (11071245)
文摘In this article, we focus on the eigenvalue problem of the following linear biharmonic equation in R^N：△^2u-αu＋λg（x）u=0 with u ∈H^2（R^N）,u≠0,N≥5Note that there are two parameters α and λ in it, which is different from the usual eigenvalue problems. Here, we consider λ as an eigenvalue and seek Ior a sulble range of parameter α, which ensures that problem （＊） has a maximal eigenvalue. As the loss of strong maximum principle for our problem, we can only get the existence of non-trivial solutions, not positive solutions, in this article. As an application, by using these results, we studied also the existence of non-trivial solutions for an asymptotically linear biharmonic equation in R^N.
基金supported by the National Natural Science Foundation of China (No. 10671154)the Na-tional Basic Research Program (No. 2005CB321703)the Science and Technology Foundation of Guizhou Province of China (No. [2008]2123)
文摘Schwarz methods are an important type of domain decomposition methods. Using the Fourier transform, we derive error propagation matrices and their spectral radii of the classical Schwarz alternating method and the additive Schwarz method for the biharmonic equation in this paper. We prove the convergence of the Schwarz methods from a new point of view, and provide detailed information about the convergence speeds and their dependence on the overlapping size of subdomains. The obtained results are independent of any unknown constant and discretization method, showing that the Schwarz alternating method converges twice as quickly as the additive Schwarz method.
