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
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.
