In this paper,we study the following coupled nonlinear logarithmic Hartree system{-Δu+λ_(1)u=μ_(1)(-1/2πln|x|*u^(2))u+β(-1/2πln|x|*v^(2))u,x∈R^(2),-Δv+λ_(2)v=μ_(2)(-1/2πln|x|*v^(2))v+β(-1/2πln|x|*u^(2))v,...In this paper,we study the following coupled nonlinear logarithmic Hartree system{-Δu+λ_(1)u=μ_(1)(-1/2πln|x|*u^(2))u+β(-1/2πln|x|*v^(2))u,x∈R^(2),-Δv+λ_(2)v=μ_(2)(-1/2πln|x|*v^(2))v+β(-1/2πln|x|*u^(2))v,x∈R^(2),where β,μ_(i),λ_(i)(i=1,2)are positive constants,* denotes the convolution in R^(2).By considering the constraint minimum problem on the Nehari manifold,we prove the existence of ground state solutions for β>0 large enough.Moreover,we also show that every positive solution is radially symmetric and decays exponentially.展开更多
We propose a fully discrete fast Fourier-Galerkin method for solving an integral equation of the first kind with a logarithmic kernel on a smooth open arc,which is a reformulation of the Dirichlet problem of the Lapla...We propose a fully discrete fast Fourier-Galerkin method for solving an integral equation of the first kind with a logarithmic kernel on a smooth open arc,which is a reformulation of the Dirichlet problem of the Laplace equation in the plane.The optimal convergence order and quasi-linear complexity order of the proposed method are established.A precondition is introduced.Combining this method with an efficient numerical integration algorithm for computing the single-layer potential defined on an open arc,we obtain the solution of the Dirichlet problem on a smooth open arc in the plane.Numerical examples are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method.展开更多
基金partially supported by the Natural Science Foundation of China(Grant No.12061012)the special foundation for Guangxi Ba Gui Scholars.
文摘In this paper,we study the following coupled nonlinear logarithmic Hartree system{-Δu+λ_(1)u=μ_(1)(-1/2πln|x|*u^(2))u+β(-1/2πln|x|*v^(2))u,x∈R^(2),-Δv+λ_(2)v=μ_(2)(-1/2πln|x|*v^(2))v+β(-1/2πln|x|*u^(2))v,x∈R^(2),where β,μ_(i),λ_(i)(i=1,2)are positive constants,* denotes the convolution in R^(2).By considering the constraint minimum problem on the Nehari manifold,we prove the existence of ground state solutions for β>0 large enough.Moreover,we also show that every positive solution is radially symmetric and decays exponentially.
基金supported by the President Fund of GUCAS and the US National Science Foundation(Grant No.CCR-0407476,DMS-0712827)National Natural Science Foundation of China(Grant No.10371122,10631080)
文摘We propose a fully discrete fast Fourier-Galerkin method for solving an integral equation of the first kind with a logarithmic kernel on a smooth open arc,which is a reformulation of the Dirichlet problem of the Laplace equation in the plane.The optimal convergence order and quasi-linear complexity order of the proposed method are established.A precondition is introduced.Combining this method with an efficient numerical integration algorithm for computing the single-layer potential defined on an open arc,we obtain the solution of the Dirichlet problem on a smooth open arc in the plane.Numerical examples are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method.
