In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive...In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.展开更多
In this paper,the 16-parameter nonconforming tetrahedral element which has an energy-orthogonal shape function space is presented for the discretization of fourth order elliptic partial differential operators in three...In this paper,the 16-parameter nonconforming tetrahedral element which has an energy-orthogonal shape function space is presented for the discretization of fourth order elliptic partial differential operators in three spatial dimensions.The newly constructed element is proved to be convergent for a model biharmonic equation.展开更多
This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽...This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.展开更多
In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities wi...In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.展开更多
In this paper, we consider the following nonlinear elliptic problem ： △2u =｜u｜n-4^-u＋u｜u｜q-1u,in Ω,△u=u=0 on ЭΩ,where Ω is a bounded and smooth domain in R^n,n∈{5,6,7},u is a parameter and q∈]4/（n-4）...In this paper, we consider the following nonlinear elliptic problem ： △2u =｜u｜n-4^-u＋u｜u｜q-1u,in Ω,△u=u=0 on ЭΩ,where Ω is a bounded and smooth domain in R^n,n∈{5,6,7},u is a parameter and q∈]4/（n-4）,（12-n）/（n-4）].We study the solutions which concentrate around two points of Ω. We prove that the concentration speeOs are the same order and the distances of the concentration points from each other and from the boundary are bounded. For Ω=（Ωa）a a smooth ringshaped open set, we establish the existence of positive solutions which concentrate at two points of Ω. Finally, we show that for u〉0, large enough, the problem has at least many positive solutions as the Ljusternik-Schnirelman category of Ω.展开更多
In this paper, the authors prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R^N, as well as Liouville type results for some solutions converging to the same value at in...In this paper, the authors prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R^N, as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. The authors also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.展开更多
In this paper, Saint-Venant's principle for anisotropic material in an end-loaded. semi-infinite elastic strip is established. Energy method is used to establish the lower bounds of the decay estimate of stress ef...In this paper, Saint-Venant's principle for anisotropic material in an end-loaded. semi-infinite elastic strip is established. Energy method is used to establish the lower bounds of the decay estimate of stress effect. An explicit estimate formula in terms of the elastic constants of the anisotropic materials is presented. Finally, a numerical example for an end-loaded, off-axis, graphite-epoxy strip is given to illustrate the results.展开更多
In this paper, we consider a p（x）-biharmonic problem with Navier boundary conditions. The existence of infinitely many solutions which tend to zero is investigated based on the symmetric Mountain Pass lemma. Our app...In this paper, we consider a p（x）-biharmonic problem with Navier boundary conditions. The existence of infinitely many solutions which tend to zero is investigated based on the symmetric Mountain Pass lemma. Our approach relies on the theory of variable exponent Sobolev space.展开更多
基金supported by Natural Science Foundation of China(11271372)Hunan Provincial Natural Science Foundation of China(12JJ2004)
文摘In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ^2u-（a＋bfR^3｜↓△u｜^2dx）△u＋V（x）u=f（x,u）,x∈R^3, u∈H^2（R3）,where a, b 〉 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti- Rabinowitz type condition.
文摘In this paper,the 16-parameter nonconforming tetrahedral element which has an energy-orthogonal shape function space is presented for the discretization of fourth order elliptic partial differential operators in three spatial dimensions.The newly constructed element is proved to be convergent for a model biharmonic equation.
基金supported by the National Natural Science Foundation of China(12271296,12271195).
文摘This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.
基金supported by National Natural Science Foundation of China(Grant No.11401310)supported by National Natural Science Foundation of China(Grant No.11531005).
文摘In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.
文摘In this paper, we consider the following nonlinear elliptic problem ： △2u =｜u｜n-4^-u＋u｜u｜q-1u,in Ω,△u=u=0 on ЭΩ,where Ω is a bounded and smooth domain in R^n,n∈{5,6,7},u is a parameter and q∈]4/（n-4）,（12-n）/（n-4）].We study the solutions which concentrate around two points of Ω. We prove that the concentration speeOs are the same order and the distances of the concentration points from each other and from the boundary are bounded. For Ω=（Ωa）a a smooth ringshaped open set, we establish the existence of positive solutions which concentrate at two points of Ω. Finally, we show that for u〉0, large enough, the problem has at least many positive solutions as the Ljusternik-Schnirelman category of Ω.
基金carried out in the framework of the Labex Archimède(ANR-11-LABX-0033)the A*MIDEX project(ANR-11-IDEX-0001-02)+6 种基金funded by the "Investissements d’Avenir" French Government program managed by the French National Research Agency(ANR)funding from the European Research Council under the European Union’s Seventh Framework Programme(FP/2007-2013)ERC Grant Agreement n.321186-ReaDiReaction-Diffusion Equations,Propagation and Modelling and from the ANR NONLOCAL project(ANR-14-CE25-0013)supported by INRIA-Team MEPHYSTOMIS F.4508.14(FNRS)PDR T.1110.14F(FNRS)ARC AUWB-2012-12/17-ULB1-IAPAS
文摘In this paper, the authors prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R^N, as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. The authors also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.
文摘In this paper, Saint-Venant's principle for anisotropic material in an end-loaded. semi-infinite elastic strip is established. Energy method is used to establish the lower bounds of the decay estimate of stress effect. An explicit estimate formula in terms of the elastic constants of the anisotropic materials is presented. Finally, a numerical example for an end-loaded, off-axis, graphite-epoxy strip is given to illustrate the results.
基金supported in part by the NNSF of China(Grant No.11101145)Research Initiation Project for Highlevel Talents(201031)of North China University of Water Resources and Electric Power
文摘In this paper, we consider a p（x）-biharmonic problem with Navier boundary conditions. The existence of infinitely many solutions which tend to zero is investigated based on the symmetric Mountain Pass lemma. Our approach relies on the theory of variable exponent Sobolev space.
