In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without th...In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang’s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.展开更多
The existence and uniqueness of singular solutions decaying like r^-m（see （1.4）） of the equation △u＋k∑i=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe sepa...The existence and uniqueness of singular solutions decaying like r^-m（see （1.4）） of the equation △u＋k∑i=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe separation structure of singular solutions decaying like r^-（n-2） of eq. （0.1） are discussed. moreover, we obtain the explicit critical exponent ps （l） （see （1.9））.展开更多
Some authors employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ?-n+ , of the solutions of scalar boundary value proble...Some authors employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ?-n+ , of the solutions of scalar boundary value problemsIn this paper, we extend these results to vector boundary value problems, under analogous stability conditions on the solution u = u(t) of the reduced equation 0 = h(t, u) Two types of asymptotic behavior are studied, depending on whether the reduced solution u(f) has or does not have a con tinuous first derivative in (a, b) leading to the phenomena of boundary and angular layers.展开更多
The shock solution for the semilinear singularly perturbed two-point boundary value problem was studied. Under suitable conditions and using the theory of differential inequalities, the existence and asymptotic behavi...The shock solution for the semilinear singularly perturbed two-point boundary value problem was studied. Under suitable conditions and using the theory of differential inequalities, the existence and asymptotic behavior of the solution for the original boundary value problems are discussed. The uniformly effective asymptotic expansion and estimation of solution u(x, ε) were obtained.展开更多
In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a ...In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a simple case of semilinear equations under the frame of the space of conormal distributions. We prove the result that the solution has the same singularities on the hypersurface in which the coefficients have the conormal singularities.展开更多
This paper considers a class of boundary value problems for the semilinear singularly perturbed fractional differential equation. Under the suitable conditions, first, the outer solution of the original problem is obt...This paper considers a class of boundary value problems for the semilinear singularly perturbed fractional differential equation. Under the suitable conditions, first, the outer solution of the original problem is obtained; secondly, using the stretched variable and the composing expansion method the boundary layer is constructed; finally, using the theory of differential inequalities the asymptotic behaviour of solution for the problem is studied and the uniformly valid asymptotic estimation is discussed.展开更多
In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condi...In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.展开更多
基金supported in part by NSF(Youth) of Shandong Province and NNSF of China
文摘In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang’s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.
基金Supported by the Natural Science Foundation of China(10901126)
文摘The existence and uniqueness of singular solutions decaying like r^-m（see （1.4）） of the equation △u＋k∑i=1ci｜x｜liupi=0,x∈R^N are obtained, wheren≥3, ci 〉0, li〉-2, i=1,2,..,k, pi〉 1, i=l,2,-..,kandthe separation structure of singular solutions decaying like r^-（n-2） of eq. （0.1） are discussed. moreover, we obtain the explicit critical exponent ps （l） （see （1.9））.
文摘Some authors employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ?-n+ , of the solutions of scalar boundary value problemsIn this paper, we extend these results to vector boundary value problems, under analogous stability conditions on the solution u = u(t) of the reduced equation 0 = h(t, u) Two types of asymptotic behavior are studied, depending on whether the reduced solution u(f) has or does not have a con tinuous first derivative in (a, b) leading to the phenomena of boundary and angular layers.
文摘The shock solution for the semilinear singularly perturbed two-point boundary value problem was studied. Under suitable conditions and using the theory of differential inequalities, the existence and asymptotic behavior of the solution for the original boundary value problems are discussed. The uniformly effective asymptotic expansion and estimation of solution u(x, ε) were obtained.
文摘In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a simple case of semilinear equations under the frame of the space of conormal distributions. We prove the result that the solution has the same singularities on the hypersurface in which the coefficients have the conormal singularities.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 40676016 and 40876010)the Knowledge Innovation Project of Chinese Academy of Sciences (Grant No. KZCX2-YW-Q03-08)LASG State Key Laboratory Special Fund and R&D Special Fund for Public Welfare Industry (meteorology) (Grant No. GYHY200806010)
文摘This paper considers a class of boundary value problems for the semilinear singularly perturbed fractional differential equation. Under the suitable conditions, first, the outer solution of the original problem is obtained; secondly, using the stretched variable and the composing expansion method the boundary layer is constructed; finally, using the theory of differential inequalities the asymptotic behaviour of solution for the problem is studied and the uniformly valid asymptotic estimation is discussed.
文摘In the poper, the method of separating singularity is applied to study the uniformly difference scheme of a singular perturbation problem for a semilinear ordinary differential equation with mixed boundary value condition. The uniform convergence on small parameter ε of order one for an IVin type difference scheme constructed is proved. At the end of the paper, a numerical example is given. The computing results coincide with the theoretical analysis.
