A new smooth gap function for the box constrained variational inequality problem （VIP） is proposed based on an integral global optimality condition. The smooth gap function is simple and has some good differentiable...A new smooth gap function for the box constrained variational inequality problem （VIP） is proposed based on an integral global optimality condition. The smooth gap function is simple and has some good differentiable properties. The box constrained VIP can be reformulated as a differentiable optimization problem by the proposed smooth gap function. The conditions, under which any stationary point of the optimization problem is the solution to the box constrained VIP, are discussed. A simple frictional contact problem is analyzed to show the applications of the smooth gap function. Finally, the numerical experiments confirm the good theoretical properties of the method.展开更多
We study the existence and stability of the standing waves of two coupled SchrSdinger equations with potentials ｜x｜bi（bi ∈ R,i = 1, 2）. Under suitable conditions on the growth of the nonlinear terms, we first est...We study the existence and stability of the standing waves of two coupled SchrSdinger equations with potentials ｜x｜bi（bi ∈ R,i = 1, 2）. Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the SchrSdinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1 = b2 = 2.展开更多
The global solution for a coupled nonlinear Klein-Gordon system in two- dimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cros...The global solution for a coupled nonlinear Klein-Gordon system in two- dimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.展开更多
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
基金Project supported by the National Natural Science Foundation of China(Nos.10902077,11172209, and 10572031)
文摘A new smooth gap function for the box constrained variational inequality problem （VIP） is proposed based on an integral global optimality condition. The smooth gap function is simple and has some good differentiable properties. The box constrained VIP can be reformulated as a differentiable optimization problem by the proposed smooth gap function. The conditions, under which any stationary point of the optimization problem is the solution to the box constrained VIP, are discussed. A simple frictional contact problem is analyzed to show the applications of the smooth gap function. Finally, the numerical experiments confirm the good theoretical properties of the method.
基金supported by NSFC(11471331,11101418 and 11271360)
文摘We study the existence and stability of the standing waves of two coupled SchrSdinger equations with potentials ｜x｜bi（bi ∈ R,i = 1, 2）. Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the SchrSdinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1 = b2 = 2.
基金Project supported by the National Natural Science Foundation of China (No.10271084)the Natural Science Foundation for Young Scholars of Sichuan Province of China (No.07JQ0094)
文摘The global solution for a coupled nonlinear Klein-Gordon system in two- dimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.
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).
