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
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.
