We are interested in the existence and asymptotic behavior for the least energy solutions of the following fractional eigenvalue problem (P)(-△)^(s)u+V(x)u=μu+am(x)|u|^(4s/N)u,∫_(R^(N))|u|^(2)dx=1,u∈H^(s)(R^(N)),w...We are interested in the existence and asymptotic behavior for the least energy solutions of the following fractional eigenvalue problem (P)(-△)^(s)u+V(x)u=μu+am(x)|u|^(4s/N)u,∫_(R^(N))|u|^(2)dx=1,u∈H^(s)(R^(N)),where s∈(0,1),μ∈R,a>0,V(x)and m(x)are L^(∞)(R^(N))functions with N≥2.We prove that there is a threshold a^(*)_(s)>0 such that problem(P)has a least energy solution u_(a)(x)for each a∈(0,a^(*)_(s))and u_(a)blows up,as a↗a^(*)_(s),at some point x_(0)∈R^(N),which makes V(x_(0))be the minimum and m(x_(0))be the maximum.Moreover,the precise blowup rates for u_(a)are obtained under suitable conditions on V(x)and m(x).展开更多
Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this a...Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this article, we discuss approximate solutions to discontinuous Riemann-Hilbert boundary value problems, which have various applications in mechanics and physics. We first formulate the discontinuous Riemann-Hilbert problem for elliptic systems of first order complex equations in multiply connected domains and its modified well-posedness, then use the parameter extensional method to find approximate solutions to the modified boundary value problem for elliptic complex systems of first order equations, and then provide the error estimate of approximate solutions for the discontinuous boundary value problem.展开更多
This paper studies the Galerkin finite element approximations of a class of stochas- tic fractionM differential equations. The discretization in space is done by a standard continuous finite element method and almost ...This paper studies the Galerkin finite element approximations of a class of stochas- tic fractionM differential equations. The discretization in space is done by a standard continuous finite element method and almost optimal order error estimates are obtained. The discretization in time is achieved via the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. We give strong convergence error estimates for both semidiscrete and fully discrete schemes. The proof is based on the error estimates for the corresponding deterministic problem. Finally, the numerical example is carried out to verify the theoretical results.展开更多
基金Supported by NSFC (Grant Nos.11931012,11871387,11871395 and 12171379)。
文摘We are interested in the existence and asymptotic behavior for the least energy solutions of the following fractional eigenvalue problem (P)(-△)^(s)u+V(x)u=μu+am(x)|u|^(4s/N)u,∫_(R^(N))|u|^(2)dx=1,u∈H^(s)(R^(N)),where s∈(0,1),μ∈R,a>0,V(x)and m(x)are L^(∞)(R^(N))functions with N≥2.We prove that there is a threshold a^(*)_(s)>0 such that problem(P)has a least energy solution u_(a)(x)for each a∈(0,a^(*)_(s))and u_(a)blows up,as a↗a^(*)_(s),at some point x_(0)∈R^(N),which makes V(x_(0))be the minimum and m(x_(0))be the maximum.Moreover,the precise blowup rates for u_(a)are obtained under suitable conditions on V(x)and m(x).
文摘Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this article, we discuss approximate solutions to discontinuous Riemann-Hilbert boundary value problems, which have various applications in mechanics and physics. We first formulate the discontinuous Riemann-Hilbert problem for elliptic systems of first order complex equations in multiply connected domains and its modified well-posedness, then use the parameter extensional method to find approximate solutions to the modified boundary value problem for elliptic complex systems of first order equations, and then provide the error estimate of approximate solutions for the discontinuous boundary value problem.
文摘This paper studies the Galerkin finite element approximations of a class of stochas- tic fractionM differential equations. The discretization in space is done by a standard continuous finite element method and almost optimal order error estimates are obtained. The discretization in time is achieved via the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. We give strong convergence error estimates for both semidiscrete and fully discrete schemes. The proof is based on the error estimates for the corresponding deterministic problem. Finally, the numerical example is carried out to verify the theoretical results.
