The classical Hardy-Littlewood-Sobolev theorems for Riesz potentials (?Δ)?α/2 are extended to the generalised fractional integrals L –α/2 for 0 < α < n, where L=?div A? is a uniformly complex elliptic opera...The classical Hardy-Littlewood-Sobolev theorems for Riesz potentials (?Δ)?α/2 are extended to the generalised fractional integrals L –α/2 for 0 < α < n, where L=?div A? is a uniformly complex elliptic operator with bounded measurable coefficients in ?n.展开更多
Let A be a symmetric and positive definite(1,1)tensor on a bounded domain Ω in an ndimensional metric measure space(R^n,<,>,e^-φdv).In this paper,we investigate the Dirichlet eigenvalue problem of a system of ...Let A be a symmetric and positive definite(1,1)tensor on a bounded domain Ω in an ndimensional metric measure space(R^n,<,>,e^-φdv).In this paper,we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form{LA,φu+α▽(divu)-▽φdivu]=-su,inΩ,u∣aΩ=0,where LA,φ=div(A▽(·))-(A▽φ,▽(·)),α is a nonnegative constant and u is a vector-valued function.Some universal inequalities for eigenvalues of this problem are established.Moreover,as applications of these results,we give some estimates for the upper bound of sk+1 and the gap of sk+1-sk in terms of the first k eigenvalues.Our results contain some results for the Lam′e system and a system of equations of the drifting Laplacian.展开更多
We establish the existence and multiplicity of weak solutions for equations which involve a uniformly convex elliptic operator in divergence form(in particular, a p-Laplacian operator), while the nonlinearity has a(p-...We establish the existence and multiplicity of weak solutions for equations which involve a uniformly convex elliptic operator in divergence form(in particular, a p-Laplacian operator), while the nonlinearity has a(p- 1)-superlinear growth at infinity. Our result completes and extends the relevant results of recent papers. The argument in the proof of our main result relies on the Z2-symmetric version of mountain pass lemma.展开更多
The authors consider a stochastic heat equation in dimension d=1 driven by an additive space time white noise and having a mild nonlinearity.It is proved that the functional law of its solution is absolutely continuou...The authors consider a stochastic heat equation in dimension d=1 driven by an additive space time white noise and having a mild nonlinearity.It is proved that the functional law of its solution is absolutely continuous and possesses a smooth density with respect to the functional law of the corresponding linear SPDE.展开更多
In this paper we classify the positive solutions of the divergent equation with Neumann boundary on the upper half space{-div(t^(α)∇u)=t^(β)f(u),(y,t)∈R^(n+1)_(+),lim t→0^(+)t^(α)■u/■t=0 by the method of moving...In this paper we classify the positive solutions of the divergent equation with Neumann boundary on the upper half space{-div(t^(α)∇u)=t^(β)f(u),(y,t)∈R^(n+1)_(+),lim t→0^(+)t^(α)■u/■t=0 by the method of moving spheres and Kelvin transformations,where n≥1,α>0,β>−1,n−1/n+1β≤α<β+2,and f:(0,∞)→(0,∞)is non-negative continuous function satisfying some conditions.This equation arises from a weighed Sobolev inequality involving divergent operator div(t^(α)∇u)on the upper half space.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant No.1017111) Foundation of Advanced Research Center,Zhongshan University.
文摘The classical Hardy-Littlewood-Sobolev theorems for Riesz potentials (?Δ)?α/2 are extended to the generalised fractional integrals L –α/2 for 0 < α < n, where L=?div A? is a uniformly complex elliptic operator with bounded measurable coefficients in ?n.
基金supported by the National Natural Science Foundation of China(Grant Nos.1100113011571361 and 11831005)the Fundamental Research Funds for the Central Universities(Grant No.30917011335)。
文摘Let A be a symmetric and positive definite(1,1)tensor on a bounded domain Ω in an ndimensional metric measure space(R^n,<,>,e^-φdv).In this paper,we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form{LA,φu+α▽(divu)-▽φdivu]=-su,inΩ,u∣aΩ=0,where LA,φ=div(A▽(·))-(A▽φ,▽(·)),α is a nonnegative constant and u is a vector-valued function.Some universal inequalities for eigenvalues of this problem are established.Moreover,as applications of these results,we give some estimates for the upper bound of sk+1 and the gap of sk+1-sk in terms of the first k eigenvalues.Our results contain some results for the Lam′e system and a system of equations of the drifting Laplacian.
基金Supported by the NNSF of China(11101145)Supported by the NSF of Henan Province(102102210216)
文摘We establish the existence and multiplicity of weak solutions for equations which involve a uniformly convex elliptic operator in divergence form(in particular, a p-Laplacian operator), while the nonlinearity has a(p- 1)-superlinear growth at infinity. Our result completes and extends the relevant results of recent papers. The argument in the proof of our main result relies on the Z2-symmetric version of mountain pass lemma.
基金the grant MTM 2006-01351 from the Dirección General de Investigación,Ministerio de Educación y Ciencia,Spain.
文摘The authors consider a stochastic heat equation in dimension d=1 driven by an additive space time white noise and having a mild nonlinearity.It is proved that the functional law of its solution is absolutely continuous and possesses a smooth density with respect to the functional law of the corresponding linear SPDE.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12071269,11971385)Natural Science Basic Research Plan for Distinguished Young Scholars in Shaanxi Province of China(Grant No.2019JC-19)the Fundamental Research Funds for the Central Universities(Grant No.GK202101008)。
文摘In this paper we classify the positive solutions of the divergent equation with Neumann boundary on the upper half space{-div(t^(α)∇u)=t^(β)f(u),(y,t)∈R^(n+1)_(+),lim t→0^(+)t^(α)■u/■t=0 by the method of moving spheres and Kelvin transformations,where n≥1,α>0,β>−1,n−1/n+1β≤α<β+2,and f:(0,∞)→(0,∞)is non-negative continuous function satisfying some conditions.This equation arises from a weighed Sobolev inequality involving divergent operator div(t^(α)∇u)on the upper half space.
