In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory ...In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.展开更多
Positive entire solutions of the equation where 1 〈 p ≤ N, q 〉 0, are classified via their Morse indices. It is seen that there is a critical power q = qc such that this equation has no positive radial entire solut...Positive entire solutions of the equation where 1 〈 p ≤ N, q 〉 0, are classified via their Morse indices. It is seen that there is a critical power q = qc such that this equation has no positive radial entire solution that has finite Morse index when q 〉 qc but it admits a family of stable positive radial entire solutions when 0 〈 q ≤ qc- Proof of the stability of positive radial entire solutions of the equation when 1 〈 p 〈 2 and 0 〈 q ≤ qc relies on Caffarelli-Kohn Nirenberg＇s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 〈 p ≤ N and q 〉 qc. The case of 1 〈 p 〈 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.展开更多
The purpose of this paper is to use a very recent three critical points theorem due to Bonanno and Marano to establish the existence of at least three solutions for the quasilinear second order differential equation o...The purpose of this paper is to use a very recent three critical points theorem due to Bonanno and Marano to establish the existence of at least three solutions for the quasilinear second order differential equation on a compact interval[a,b] R{-u＇＇=（λf（x,u）＋g（u））h（u＇）,in（a,b）,u（a）=u（b）=0under ppropriate hypotheses.We exhibit the existence of at least three（weak）solutions and,and the results are illustrated by examples.展开更多
文摘In this paper, we study the existence result for degenerate elliptic equations with singular potential and critical cone sobolev exponents on singular manifolds. With the help of the variational method and the theory of genus, we obtain several results under different conditions.
基金supported by NSFC(Grant Nos.11171092 and 11571093)supported by NSFC(Grant No.11371117)
文摘Positive entire solutions of the equation where 1 〈 p ≤ N, q 〉 0, are classified via their Morse indices. It is seen that there is a critical power q = qc such that this equation has no positive radial entire solution that has finite Morse index when q 〉 qc but it admits a family of stable positive radial entire solutions when 0 〈 q ≤ qc- Proof of the stability of positive radial entire solutions of the equation when 1 〈 p 〈 2 and 0 〈 q ≤ qc relies on Caffarelli-Kohn Nirenberg＇s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 〈 p ≤ N and q 〉 qc. The case of 1 〈 p 〈 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.
基金supported in part by grant from IPM(No.89350020)
文摘The purpose of this paper is to use a very recent three critical points theorem due to Bonanno and Marano to establish the existence of at least three solutions for the quasilinear second order differential equation on a compact interval[a,b] R{-u＇＇=（λf（x,u）＋g（u））h（u＇）,in（a,b）,u（a）=u（b）=0under ppropriate hypotheses.We exhibit the existence of at least three（weak）solutions and,and the results are illustrated by examples.
