In this paper, a new viewpoint of the division by zero z/0 = 0 in matrices is introduced and the results will show that the division by zero is our elementary and fundamental mathematics. New and practical meanings fo...In this paper, a new viewpoint of the division by zero z/0 = 0 in matrices is introduced and the results will show that the division by zero is our elementary and fundamental mathematics. New and practical meanings for many mathematical and physical formulas for the denominator zero cases may be given. Furthermore, a new space idea for the point at infinity for the Eucleadian plane is also introduced.展开更多
In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions： △2 u = K（x）u p , u 〉 0 in Ω , △u = u = 0 on Ω , where Ω is ...In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions： △2 u = K（x）u p , u 〉 0 in Ω , △u = u = 0 on Ω , where Ω is a smooth domain in R n , n 5, and p ＋ 1 = 2 n n 4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.展开更多
We consider the problem of conformal metrics equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball Bn, n ≥ 4. By variational methods, we prove the ex...We consider the problem of conformal metrics equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball Bn, n ≥ 4. By variational methods, we prove the existence of two peak solutions that concentrate around a strict local maximum points of the mean curvature under certain conditions.展开更多
Abstract Let K be a given positive function on a bounded domain Ω of R^(n),n≥3.The authors consider a nonlinear variational problem of the form:-Δu=K|u|^(4/n-2)u in Ω with mixed Dirichlet-Neumann boundary conditio...Abstract Let K be a given positive function on a bounded domain Ω of R^(n),n≥3.The authors consider a nonlinear variational problem of the form:-Δu=K|u|^(4/n-2)u in Ω with mixed Dirichlet-Neumann boundary conditions.It is a non-compact variational problem,in the sense that the associated energy functional J fails to satisfy the Palais-Smale condition.This generates concentration and blow-up phenomena.By studying the behaviors of non-precompact flow lines of a decreasing pseudogradient of J,they characterize the points where blow-up phenomena occur,the so-called critical points at infinity.Such a characterization combined with tools of Morse theory,algebraic topology and dynamical system,allow them to prove critical perturbation results under geometrical hypothesis on the boundary part in which the Neumann condition is prescribed.展开更多
In this paper we show the distribution of critical points at infinity of n- dimensional polynomial differential systems, and give the conditions, under which the system is degenerate at infinity. Also, we discuss the...In this paper we show the distribution of critical points at infinity of n- dimensional polynomial differential systems, and give the conditions, under which the system is degenerate at infinity. Also, we discuss the quadratic systems with degenerate infinity, and obtain some similar properties to 2-dimensional quadratic systems.展开更多
By discussing the relation between the method of Poincare's transformation and the method of Bendixsion's transformation which are used to analyse the behavior of singular points at infinity, a formula for ind...By discussing the relation between the method of Poincare's transformation and the method of Bendixsion's transformation which are used to analyse the behavior of singular points at infinity, a formula for index of a singular point of polynomial differential system in the plane is deduced. It is convenient to use the formula in this paper to calculate the index of a singular point with higher order.展开更多
文摘In this paper, a new viewpoint of the division by zero z/0 = 0 in matrices is introduced and the results will show that the division by zero is our elementary and fundamental mathematics. New and practical meanings for many mathematical and physical formulas for the denominator zero cases may be given. Furthermore, a new space idea for the point at infinity for the Eucleadian plane is also introduced.
文摘In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions： △2 u = K（x）u p , u 〉 0 in Ω , △u = u = 0 on Ω , where Ω is a smooth domain in R n , n 5, and p ＋ 1 = 2 n n 4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.
文摘We consider the problem of conformal metrics equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball Bn, n ≥ 4. By variational methods, we prove the existence of two peak solutions that concentrate around a strict local maximum points of the mean curvature under certain conditions.
文摘Abstract Let K be a given positive function on a bounded domain Ω of R^(n),n≥3.The authors consider a nonlinear variational problem of the form:-Δu=K|u|^(4/n-2)u in Ω with mixed Dirichlet-Neumann boundary conditions.It is a non-compact variational problem,in the sense that the associated energy functional J fails to satisfy the Palais-Smale condition.This generates concentration and blow-up phenomena.By studying the behaviors of non-precompact flow lines of a decreasing pseudogradient of J,they characterize the points where blow-up phenomena occur,the so-called critical points at infinity.Such a characterization combined with tools of Morse theory,algebraic topology and dynamical system,allow them to prove critical perturbation results under geometrical hypothesis on the boundary part in which the Neumann condition is prescribed.
文摘In this paper we show the distribution of critical points at infinity of n- dimensional polynomial differential systems, and give the conditions, under which the system is degenerate at infinity. Also, we discuss the quadratic systems with degenerate infinity, and obtain some similar properties to 2-dimensional quadratic systems.
文摘By discussing the relation between the method of Poincare's transformation and the method of Bendixsion's transformation which are used to analyse the behavior of singular points at infinity, a formula for index of a singular point of polynomial differential system in the plane is deduced. It is convenient to use the formula in this paper to calculate the index of a singular point with higher order.
