In this paper,we analyze the existence,multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by singular Monge-Ampère equations{det D^(2)u_(1)=λh_(1)f_(1)(-u_(2))in...In this paper,we analyze the existence,multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by singular Monge-Ampère equations{det D^(2)u_(1)=λh_(1)f_(1)(-u_(2))inΩ,det D^(2)u2=λh_(2)f_(2)(-u_(1)),u_(1)=u_(2)=0,onəΩinΩfor a certain range ofλ>0,hi are weight functions,f_(i)are continuous functions with possible singularity at 0 and satisfy a combined N-superlinear growth at∞,where i∈{1,2},Ωis the unit ball in N.We establish the existence of a nontrivial radial convex solution for smallλ,multiplicity results of nontrivial radial convex solutions for certain ranges ofλ,and nonexistence results of nontrivial radial solutions for the caseλ≥1.The asymptotic behavior of nontrivial radial convex solutions is also considered.展开更多
In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable function which is bounded away from 0 in Ω,...In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable function which is bounded away from 0 in Ω, and the domain Ω is a smooth bounded open set in R^N（N≥2）. Especially, under the condition that g（x, s） = 1/｜s｜^α （α〉0） is singular at s = 0, we obtain that α 〈 p is necessary and sufficient for the existence of solutions in W0^1,p（Ω） to problem （0.1） when f is sufficiently regular.展开更多
An improved interpolating complex variable element-frees Galerkin(IICVEFG)method for the two-dimensional elastic problems is developed.This method is based on the improved interpolating complex variable moving least-s...An improved interpolating complex variable element-frees Galerkin(IICVEFG)method for the two-dimensional elastic problems is developed.This method is based on the improved interpolating complex variable moving least-squares(IICVMLS)method and the integral form of the elastic problems.In the IICVEFG method,the proposed shape function has the interpolating feature.Therefore,the essential boundary conditions can be exerted directly.Additionally,the unnecessary t erms in the discrete mat rices are removed,which resul ts in a set of concise formulas.This method is verified by analyzing three elastic examples under different constraints and loads.The numerical results show that the IICVEFG method is superior in precision and efficiency to other non-interpolating meshless methods.展开更多
Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between...Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between x and p;for each j=1,...,m.In this paper,using a method of blow-up analysis,we prove that the functional J(u)=1/2∫∑|ΔgU|^(2)dV_(g)+8π(1+β)1/volg(∑)∫∑udvg-8π(1+β)log∫_(∑)he^(U)dv_(g)is bounded from below on the Sobolev space w^(1,2)(g).展开更多
Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R^n×R^m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz ...Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R^n×R^m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R^n× R^m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R^n× R^m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R^n× R^m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R^n× R^m) to L~φ(R^n× R^m)and from H~φ_A(R^n×R^m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R^n× R^m and are new even for classical product Orlicz-Hardy spaces.展开更多
We study （pseudo-）differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y. The typical operators A are corner degenerate in ...We study （pseudo-）differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y. The typical operators A are corner degenerate in a specific way. They are described Cmodulo ＇lower order terms＇） by a principal symbolic hierarchy σ（A） = （σψ （A）, σ∧ CA）, σ∧ （A））, where σψ is the interior symbol and σ∧（A） （y, η）, （y, η） ∈ T＊Y／0, the Coperator-valued） edge symbol of ＇first generation＇, cf. [1]. The novelty here is the edge symbol σ∧ of ＇second generation＇, parametrised by （z, ζ） ∈ T＊Z ／ 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.展开更多
基金supported by Beijing Natural Science Foundation under Grant No.1212003the Promoting the Classified Development of Colleges and Universities-application and Cultivation of Scientific Research Awards of BISTU under Grant No.2021JLPY408。
文摘In this paper,we analyze the existence,multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by singular Monge-Ampère equations{det D^(2)u_(1)=λh_(1)f_(1)(-u_(2))inΩ,det D^(2)u2=λh_(2)f_(2)(-u_(1)),u_(1)=u_(2)=0,onəΩinΩfor a certain range ofλ>0,hi are weight functions,f_(i)are continuous functions with possible singularity at 0 and satisfy a combined N-superlinear growth at∞,where i∈{1,2},Ωis the unit ball in N.We establish the existence of a nontrivial radial convex solution for smallλ,multiplicity results of nontrivial radial convex solutions for certain ranges ofλ,and nonexistence results of nontrivial radial solutions for the caseλ≥1.The asymptotic behavior of nontrivial radial convex solutions is also considered.
基金supported by the Natural Science Foundation of Henan Province(15A110050)
文摘In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable function which is bounded away from 0 in Ω, and the domain Ω is a smooth bounded open set in R^N（N≥2）. Especially, under the condition that g（x, s） = 1/｜s｜^α （α〉0） is singular at s = 0, we obtain that α 〈 p is necessary and sufficient for the existence of solutions in W0^1,p（Ω） to problem （0.1） when f is sufficiently regular.
基金The authors sincerely acknowledge the financial support from the National Science Foundation of China(No.12002240)the National Science and Technology Major Project(No.2017-IV-0003-0040).
文摘An improved interpolating complex variable element-frees Galerkin(IICVEFG)method for the two-dimensional elastic problems is developed.This method is based on the improved interpolating complex variable moving least-squares(IICVMLS)method and the integral form of the elastic problems.In the IICVEFG method,the proposed shape function has the interpolating feature.Therefore,the essential boundary conditions can be exerted directly.Additionally,the unnecessary t erms in the discrete mat rices are removed,which resul ts in a set of concise formulas.This method is verified by analyzing three elastic examples under different constraints and loads.The numerical results show that the IICVEFG method is superior in precision and efficiency to other non-interpolating meshless methods.
基金the National Science Foundation of China(GrantNo.11401575).
文摘Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between x and p;for each j=1,...,m.In this paper,using a method of blow-up analysis,we prove that the functional J(u)=1/2∫∑|ΔgU|^(2)dV_(g)+8π(1+β)1/volg(∑)∫∑udvg-8π(1+β)log∫_(∑)he^(U)dv_(g)is bounded from below on the Sobolev space w^(1,2)(g).
基金supported by National Natural Science Foundation of China (Grant Nos. 11671414, 11271091, 11471040, 11461065, 11661075, 11571039 and 11671185)
文摘Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R^n×R^m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R^n× R^m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R^n× R^m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R^n× R^m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R^n× R^m) to L~φ(R^n× R^m)and from H~φ_A(R^n×R^m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R^n× R^m and are new even for classical product Orlicz-Hardy spaces.
文摘We study （pseudo-）differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y. The typical operators A are corner degenerate in a specific way. They are described Cmodulo ＇lower order terms＇） by a principal symbolic hierarchy σ（A） = （σψ （A）, σ∧ CA）, σ∧ （A））, where σψ is the interior symbol and σ∧（A） （y, η）, （y, η） ∈ T＊Y／0, the Coperator-valued） edge symbol of ＇first generation＇, cf. [1]. The novelty here is the edge symbol σ∧ of ＇second generation＇, parametrised by （z, ζ） ∈ T＊Z ／ 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.
