Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical ...Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:{-div(|△u|p-2△u)=|x|s|u|p*(s)-2u+λ|x|t|u|p-2u, x∈B1, u|σB1 =0, where t, s〉-p, 2≤p〈N, p*(s)= (N+s)pN-p andλ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N 〉p(p-1)t+p(p2-p+1) andλ∈(0,λ1,t), whereλ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ (ps+p) min{1, p+t/p+s}+p2p-(p-1) min{1, p+tp+s} andλ〉0 is small.展开更多
Based on the precise integration method (PIM), a coupling technique of the high order multiplication perturbation method (HOMPM) and the reduction method is proposed to solve variable coefficient singularly pertur...Based on the precise integration method (PIM), a coupling technique of the high order multiplication perturbation method (HOMPM) and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems (TPBVPs) with one boundary layer. First, the inhomogeneous ordinary differential equations (ODEs) are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.展开更多
By means of the method of solid angle coefficients and the permutation formula on the building domain of complex biballs,direct solutions of some singular integral equations with variable coefficients are discussed an...By means of the method of solid angle coefficients and the permutation formula on the building domain of complex biballs,direct solutions of some singular integral equations with variable coefficients are discussed and the explicit formulas for these solutions are obtained.展开更多
The interaction between multiple curved rigid line and circular inclusion in antiplane loading condition is considered in this paper. By utilizing the point force elementary solutions and taking density function of tr...The interaction between multiple curved rigid line and circular inclusion in antiplane loading condition is considered in this paper. By utilizing the point force elementary solutions and taking density function of traction difference along curved rigid lines, a group of weakly singular integral equations with logarithmic kernels can be obtained. After the numerical solution of the integral equations, the discrete values of density functions of traction difference are obtainable. So the stress singularity coefficient at rigid line tips can be calculated, and two numerical examples are given.展开更多
In this paper we consider the existence of infinitely many positive solutions for second order nonlinear eigenvalue problem with singular coefficient function. By the use of Krasnosel'skii fixed point theorem of c...In this paper we consider the existence of infinitely many positive solutions for second order nonlinear eigenvalue problem with singular coefficient function. By the use of Krasnosel'skii fixed point theorem of cone expansion-compression type we give several sufficient conditions.展开更多
基金supported by the National Natural Science Foundation of China(11326139,11326145)Tian Yuan Foundation(KJLD12067)Hubei Provincial Department of Education(Q20122504)
文摘Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:{-div(|△u|p-2△u)=|x|s|u|p*(s)-2u+λ|x|t|u|p-2u, x∈B1, u|σB1 =0, where t, s〉-p, 2≤p〈N, p*(s)= (N+s)pN-p andλ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N 〉p(p-1)t+p(p2-p+1) andλ∈(0,λ1,t), whereλ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ (ps+p) min{1, p+t/p+s}+p2p-(p-1) min{1, p+tp+s} andλ〉0 is small.
基金Project supported by the National Natural Science Foundation of China(Key Program)(Nos.11132004 and 51078145)
文摘Based on the precise integration method (PIM), a coupling technique of the high order multiplication perturbation method (HOMPM) and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems (TPBVPs) with one boundary layer. First, the inhomogeneous ordinary differential equations (ODEs) are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.
基金Supported by the NNSF of china(11171298)SuppoSed by the Natural Science Foundation of Zhejiang Province(Y6110425,Y604563)
文摘By means of the method of solid angle coefficients and the permutation formula on the building domain of complex biballs,direct solutions of some singular integral equations with variable coefficients are discussed and the explicit formulas for these solutions are obtained.
文摘The interaction between multiple curved rigid line and circular inclusion in antiplane loading condition is considered in this paper. By utilizing the point force elementary solutions and taking density function of traction difference along curved rigid lines, a group of weakly singular integral equations with logarithmic kernels can be obtained. After the numerical solution of the integral equations, the discrete values of density functions of traction difference are obtainable. So the stress singularity coefficient at rigid line tips can be calculated, and two numerical examples are given.
文摘In this paper we consider the existence of infinitely many positive solutions for second order nonlinear eigenvalue problem with singular coefficient function. By the use of Krasnosel'skii fixed point theorem of cone expansion-compression type we give several sufficient conditions.