The critical group C（G） of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r（G） be the minimum number of generators （i.e., the rank）...The critical group C（G） of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r（G） be the minimum number of generators （i.e., the rank） of the group C（G） and β（G） be the number of independent cycles of G. In this paper, some forbidden induced subgraphs axe given for r（G） = n - 3 and all graphs with r（G） = j3（G） = n - 3 are characterized展开更多
In this article several existence theorems on multiple solutions for the twopoint boundary value problem with resonance at, both infinity and zero are proved.
In this article, we study a semilinear p-Laplacian Dirichlet problem arising in population dynamics. We obtain the Morse critical groups at zero. The results show that the energy functional of the problem is trivial. ...In this article, we study a semilinear p-Laplacian Dirichlet problem arising in population dynamics. We obtain the Morse critical groups at zero. The results show that the energy functional of the problem is trivial. As a consequence, the existence and bifurcation of the nontrivial solutions to the problem are established.展开更多
In this article, under some conditions on the behaviors of the perturbed function f（x, s） or its primitive F（x,s） =∫so f（x,t）dt near infinity and near zero, a class of asymptotically linear elliptic equations i...In this article, under some conditions on the behaviors of the perturbed function f（x, s） or its primitive F（x,s） =∫so f（x,t）dt near infinity and near zero, a class of asymptotically linear elliptic equations involving natural growth term is studied. By computing the critical group, the existence of three nontrivial solutions is proved.展开更多
Using variational methods and Morse theory, we obtain some existence results of multiple solutions for certain semilinear problems associated with general Dirichlet forms.
The critical aggregation concentration（CAC） of four with three kinds of conventional surfactants, namely,two cationic surfactants [hexadecyltrimethyl ammonium bromide（CTAB） and tetradecyltrimethyl ammonium bromide...The critical aggregation concentration（CAC） of four with three kinds of conventional surfactants, namely,two cationic surfactants [hexadecyltrimethyl ammonium bromide（CTAB） and tetradecyltrimethyl ammonium bromide（TTAB）], one anionic surfactant [sodium dodecyl sulfate（SDS）], and a nonionic surfactant [Triton X-100（TX-100）], were determined by variation of ^1H chemical shifts with surfactant concentrations. Results show that the CAC values of protons at different positions of the same molecule are different, and those of the terminal methyl protons are the lowest, respectively, which suggests that the terminal groups of the alkyl chains aggregates first during micellization. Measurement of the transverse relaxation time（T2） of different protons in SDS also show that the terminal methyl protons start to decrease with the increase in concentration first, which supports the above mentioned tendency.展开更多
We consider a Dirichlet nonlinear equation driven by the(p,2)-Laplacian and with a reaction having the competing effects of a parametric asymmetric superlinear term and a resonant perturbation.We show that for all sma...We consider a Dirichlet nonlinear equation driven by the(p,2)-Laplacian and with a reaction having the competing effects of a parametric asymmetric superlinear term and a resonant perturbation.We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions all with sign information.展开更多
The aim of this paper is the study of a double phase problems involving superlinear nonlinearities with a growth that need not satisfy the Ambrosetti-Rabinowitz condition.Using variational tools together with suitable...The aim of this paper is the study of a double phase problems involving superlinear nonlinearities with a growth that need not satisfy the Ambrosetti-Rabinowitz condition.Using variational tools together with suitable truncation and minimax techniques with Morse theory,the authors prove the existence of one and three nontrivial weak solutions,respectively.展开更多
In this paper,Fucik spectrum,ordinary differential equation theory of Banach spaces and Morse theory are used to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity,and so...In this paper,Fucik spectrum,ordinary differential equation theory of Banach spaces and Morse theory are used to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity,and some new results on the existence of nontrivial solutions,multiple solutions and sign-changing solutions are obtained.In one case seven nontrivial solutions are got.The techniques have independent interest.展开更多
基金Supported by FRG, Hong Kong Baptist-University the first author is supported by National Natural Science Foundation of China （Grant No. 10671061） The authors would like to thank the anonymous referee for a number of helpful suggestions.
文摘The critical group C（G） of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r（G） be the minimum number of generators （i.e., the rank） of the group C（G） and β（G） be the number of independent cycles of G. In this paper, some forbidden induced subgraphs axe given for r（G） = n - 3 and all graphs with r（G） = j3（G） = n - 3 are characterized
基金Supported by the Natural Science Foundation of China(10471098)the Beijing Natural Science Foundation(1052004)the Foundation of Beijing's Educational Committee(KM200510028001)the Key-Project of NSFB-FBEC
文摘In this article several existence theorems on multiple solutions for the twopoint boundary value problem with resonance at, both infinity and zero are proved.
文摘In this article, we study a semilinear p-Laplacian Dirichlet problem arising in population dynamics. We obtain the Morse critical groups at zero. The results show that the energy functional of the problem is trivial. As a consequence, the existence and bifurcation of the nontrivial solutions to the problem are established.
基金supported by the National Science Foundation of China (1077107410801055)Doctoral Program of NEM of China (200805611026)
文摘In this article, under some conditions on the behaviors of the perturbed function f（x, s） or its primitive F（x,s） =∫so f（x,t）dt near infinity and near zero, a class of asymptotically linear elliptic equations involving natural growth term is studied. By computing the critical group, the existence of three nontrivial solutions is proved.
基金supported by National Natural Science Foundation of China - NSAF (10976026)National Natural Science Foundation of China (11271305)
文摘Using variational methods and Morse theory, we obtain some existence results of multiple solutions for certain semilinear problems associated with general Dirichlet forms.
基金supported by the National Natural Science Foundation of China(Nos.21375145,21221064)
文摘The critical aggregation concentration（CAC） of four with three kinds of conventional surfactants, namely,two cationic surfactants [hexadecyltrimethyl ammonium bromide（CTAB） and tetradecyltrimethyl ammonium bromide（TTAB）], one anionic surfactant [sodium dodecyl sulfate（SDS）], and a nonionic surfactant [Triton X-100（TX-100）], were determined by variation of ^1H chemical shifts with surfactant concentrations. Results show that the CAC values of protons at different positions of the same molecule are different, and those of the terminal methyl protons are the lowest, respectively, which suggests that the terminal groups of the alkyl chains aggregates first during micellization. Measurement of the transverse relaxation time（T2） of different protons in SDS also show that the terminal methyl protons start to decrease with the increase in concentration first, which supports the above mentioned tendency.
基金NNSF of China(Grant No.12071413)NSF of Guangxi(Grant No.2023GXNSFAA026085)the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No.823731 CONMECH。
文摘We consider a Dirichlet nonlinear equation driven by the(p,2)-Laplacian and with a reaction having the competing effects of a parametric asymmetric superlinear term and a resonant perturbation.We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions all with sign information.
基金supported by the National Natural Science Foundation of China (No. 11201095)the Fundamental Research Funds for the Central Universities (No. 3072022TS2402)+1 种基金the Postdoctoral research startup foundation of Heilongjiang (No. LBH-Q14044)the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502)
文摘The aim of this paper is the study of a double phase problems involving superlinear nonlinearities with a growth that need not satisfy the Ambrosetti-Rabinowitz condition.Using variational tools together with suitable truncation and minimax techniques with Morse theory,the authors prove the existence of one and three nontrivial weak solutions,respectively.
基金This work was supported by the Australian Research Council and the National Natural Science Foundation of China (Grant No. 2178200) the Foundation of State Education Commission for Returned Overseas Scholars.
文摘In this paper,Fucik spectrum,ordinary differential equation theory of Banach spaces and Morse theory are used to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity,and some new results on the existence of nontrivial solutions,multiple solutions and sign-changing solutions are obtained.In one case seven nontrivial solutions are got.The techniques have independent interest.
